Wikiversity enwikiversity https://en.wikiversity.org/wiki/Wikiversity:Main_Page MediaWiki 1.47.0-wmf.5 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 Wikiversity:Colloquium 4 28 2812849 2812786 2026-06-05T14:55:49Z Codename Noreste 2969951 /* Create an autopatrolled user group? */ edit opening comment ([[mw:c:Special:MyLanguage/User:JWBTH/CD|CD]]) 2812849 wikitext text/x-wiki {{Wikiversity:Colloquium/Header}} <!-- MESSAGES GO BELOW --> == [[MediaWiki:Protectedpagetext#Protected edit request on 11 December 2025]] == I posted an edit request there 5 months ago, so I’ll be taking it to this page. [[Special:Contributions/&#126;2026-28640-56|&#126;2026-28640-56]] ([[User talk:&#126;2026-28640-56|talk]]) 23:33, 12 May 2026 (UTC) :What exactly is the problem? I don't understand what needs to change and why. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 23:35, 12 May 2026 (UTC) : Pinging @[[User:Atcovi|Atcovi]], @[[User:Jtneill|Jtneill]] and @[[User:Juandev|Juandev]] for further input. Someone is requesting a modification to [[MediaWiki:Protectedpagetext]] to use {{tlx|Protected page text}}, but we might need to discuss whether to use the template. In the meantime, I'll start a sandbox version of the protected page text template. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 23:19, 14 May 2026 (UTC) ::Sounds good -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 04:13, 15 May 2026 (UTC) :::+1 Jtneill. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 12:59, 19 May 2026 (UTC) == Proposal to rehost Wikinews here == As many of you know, and mentioned here at the Colloquium, our sister project Wikinews recently closed, with all 31 active editions made read-only. [[User:BigKrow]] has asked about the prospect of writing news stories here and I suggested that since we already have [[School:Journalism]] and some resources related to the [[:Category:Journalism|broader topic of journalism]]. I would like to propose that we have continued and indefinite space for {{w|citizen journalism}} by essentially repurposing Wikinews into a sub-project here. The only special infrastructure that Wikinews required was [[:mw:Extension:DynamicPageList]], which was deactivated and caused issues due to a lack of maintenance. I will add this proposal to the site banner, but I recognize that that may be a conflict of interest, so if anyone requests that I remove it, I will. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:30, 14 May 2026 (UTC) :I would like to see this conversation go for at least 30 days to establish a consensus. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:35, 14 May 2026 (UTC) ===Votes=== *{{support}} as proposer (with BK's inspiration). I think that an ongoing experiment in citizen journalism is a fit and appropriate use of this site. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:35, 14 May 2026 (UTC) *{{support}}, hope to seeing ideas about this, and thank you @[[User:Koavf|Koavf]] [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 11:08, 14 May 2026 (UTC) *{{support}} Other than perhaps inflating the total number of pages reported, I see the idea of "practicing journalism" a worthy and relevant activity within the domain of Wikiversity. [[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 21:41, 14 May 2026 (UTC) *{{support}} Conditional on development of (a) community guidelines that ensure alignment with Wikiversity's purpose, and (b) clear, nested page-naming structures for projects. More detail below. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:48, 15 May 2026 (UTC) *{{contra}} This proposal doesn't seem interested in expanding educational materials in journalism, but rather in providing space and protection for Wikinews contributors. But this is contrary to the goals of Wikiversity, and I'm not sure it's a good idea, even with regard to WMF. If WMF decides to close a project and another community lets it run on its domain, that's a bit of an undermining of WMF's and the community's decisions. Given that Wikiversity has had several conflicts with other communities and WMF in its history, I'm against it.--[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:59, 15 May 2026 (UTC) *{{contra}} This seems like a proposal to continue the mission of WikiNews, but not a proposal specifically to improve Wikiversity. I concur with Juandev's comments. --[[User:Mu301|mikeu]] <sup>[[User talk:Mu301|talk]]</sup> 20:29, 30 May 2026 (UTC) * {{oppose}} per above. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 19:05, 1 June 2026 (UTC) *{{oppose}} Wikiversity isn’t Wikinews and it also isn’t a dumping ground for anything not covered by other projects. It was already suggested, rather bafflingly, that Wikinews parasitize Wikipedia as a host. If it were allowed to freeload off of Wikiversity it would simply promote a view I and likely many others have— that Wikiversity (as it currently exists) has no standards and mostly just exists to host subpar content that wouldn’t be tolerated on any other Wikimedia site. Wikinews needs a new, non-Wikimedia host, and Wikiversity needs to get its act together by enforcing a minimum scope and standard for what it allows. --[[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 01:16, 4 June 2026 (UTC) ===Comments and questions=== :Definitely worthy of discussion, so I have no problem with the proposal in the sitenotice. :Initial questions: :* Does this proposal include importing English Wikinews content e.g., to [[Wikinews]] subpages? :* What are "active editions"? :* How can Wikiversity navigate the concerns that lead to the closure of Wikinews? :* Are any changes to the scope of Wikinews proposed? :* How does [[Wikinews]] fit with the [[Wikiversity:Mission]]? What aligns well? Where might there be tension? :** e.g., I'm not sure that a page like [[User:BigKrow/Manchester City moves two points behind Arsenal]] in and of itself will serve as an educational resource. :-- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 05:52, 14 May 2026 (UTC) :* Does this proposal include importing English Wikinews content e.g., to [[Wikinews]] subpages? ::*No, not at this time. :* What are "active editions"? ::*There were 30 other active editions of Wikinews in addition to English (e.g. [[:n:es:]]) at the time of universal closure (2026-05-04). :* How can Wikiversity navigate the concerns that lead to the closure of Wikinews? ::*One of the biggest issues was the problems with DPL, which is now irrelevant. Another was the lack of activity, which can be ameliorated by having it be part of an existing project instead of its own domain (e.g. some editions of Wikipedia host their own Wikinews already and those projects were not impacted by the closure). :* Are any changes to the scope of Wikinews proposed? ::*Not at this juncture. I would also propose as far as implemention goes that we would request a new namespace and that the material be more-or-less sequestered into its own ongoing project, like Wikijournal is or like the Cookbook and Wikijunior are at our sister [[:b:]]. :* How does [[Wikinews]] fit with the [[Wikiversity:Mission]]? What aligns well? Where might there be tension? :** e.g., I'm not sure that a page like [[Story/Manchester City moves two points behind Arsenal]] in and of itself will serve as an educational resource. ::*The process of citizen journalists practicing their craft in real-time and collaborating with others to do so is itself an education activity. We would essentially be hosting a real-time experiment in citizen journalism, online communities, and collaborative learning in addition to the prospect of spreading educational information from someone actually reading the news. I would propose that we could also make a more deliberate attempt to engage with learning <em>about</em> what does and doesn't work with collaborative news writing by experimentation (e.g. audio news, syndicating to other sites, incorporating freely-licensed news from other sources, writing hyper-local news, writing briefs versus longer-term reportage) and also seeing if the problems noted in the Task Force report that recommended closure can be overcome. Note that we have already done some local investigation about and learning about wiki-based journalism on Wikinews here at [[Journalism studies and Wikinews]]. We could continue that learning and refine the process, including incorporating journalism students from universities. As for tensions, Wikinews is the only sister project that must be done with a quick turn-around: if you take a long time to [[:s:|transcribe a book]], that's just how long it takes, but if you take a long time to write news, it ceases to be news entirely. Wikiversity has been a very slow-growing project that has definitely had some successes but has generally come together over a long period with most learning resources being individual passion projects (or sometimes, frankly, crankery) which would not work with collaborative news that requires more than just a single editor writing whatever he feels like. ::Please let me know any other questions/concerns and any other editors feel free to give your own perspective. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 06:13, 14 May 2026 (UTC) :::Thanks, Justin — it is food for thought. :::In attempting to understand how we've arrived here, I've summarised some of the background on this page: [[Wikinews]]. :::Perhaps it could be helpful to flesh out more of the vision / ideas / possibilities / challenges on that page? -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:49, 14 May 2026 (UTC) :::*Having given it some thought, in principle, I support hosting [[citizen journalism]] on Wikiversity where it is clearly connected to a learning project and/or constitutes original research, both of which align strongly with [[Wikiversity:Mission|Wikiversity’s educational mission]]. :::*My chief concern is the potential for news content that is not clearly linked to the purpose of Wikiversity. To avoid this, some community-agreed guidelines would be prudent. These need not be overly restrictive; they should support boldness and experimentation while helping ensure alignment with Wikiversity's purpose. :::*Given the reported low and declining activity on Wikinews, it seems unlikely that English Wikiversity would be overwhelmed by an influx of news-related editing. My impression is that English Wikinews was the most active edition, but even so, many contributors are likely to disperse to other projects or cease editing altogether. A modest migration of interested editors to Wikiversity seems manageable. :::*At this stage, I do not think a dedicated namespace is necessary. Subpages under [[Wikinews]] or nested pages under relevant learning or research projects, or user-space draft pages should be suitable. I agree that [[Wikijournal]] offers a useful model, as do several existing course structures on Wikiversity. :::*I support [[User:Koavf]]’s suggestions about framing Wikinews activity explicitly around learning. This would create a distinctive space for experimenting with collaborative news production in ways that are pedagogically meaningful. I agree that the [[journalism studies and Wikinews]] project developed by David and Leigh Blackall through the University of Wollongong is an excellent example of the intersection between Wikiversity and Wikinews. The [[Wikinews]] page could evolve into a hub for such projects. :::*I've tidied the [[:Category:Wikinews|Wikinews category]] and merged some content into the [[Wikinews]] page. As part of a reinvigoration effort, please review these and related resources such as [[:Category:Journalism]] and [[School:Journalism]]. :::*A further argument in favour of this initiative is that Wikipedia explicitly excludes both news reporting and original research. So, there is value in maintaining spaces within the Wikimedia ecosystem where these forms of knowledge production can be openly developed and curated. Such work can, in turn, generate valuable evidence and source material that may later inform Wikipedia articles. :::*The closure of WMF-hosted Wikinews does not imply that open wiki-based news curation lacks value. Indeed, the closure documentation appears supportive of experimentation with alternative news models across Wikimedia projects, including through Wikipedia and Wikidata. In that context, Wikiversity seems a natural home for a Wikinews experiment, provided it is clearly grounded in learning and/or research. :::-- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:39, 15 May 2026 (UTC) My understanding towards Wikinews' failure is that everything takes too long to be approved for the publish status, which means that any breaking news would have already become days-old stale news. Wikinews has a brand recognition (for right or wrong reasons) than Wikiversity and I wonder how effective Wikiversity can attract the "Wikinews refugees" to edit here. And just a quick note on the governance. Since each Wikiversity language operates independently, each language has to vote & adopt this proposal independently. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;"><sup>Talk page</sup></span></b>]] 13:47, 15 May 2026 (UTC) :Your assessment about Wikinews is partially correct. I referenced it earlier, but to be explicit, there is a [[:m:Proposal for Closing Wikinews|report by a task force on sister projects]] that outlines their concerns. There are a few, one of which was the nature of the staleness of news. Thanks also for clarifying that this proposal is only relevant to en.wv and is not binding or even proposed for other editions of Wikiversity. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 18:54, 15 May 2026 (UTC) *Note: I am not a regular here, and just visit Wikiversity for the WikiJournal project. Challenges of Wikinews included that it required timely reporting and fact-checking processes which differed greatly from the well-established ones in Wikipedia. Here in Wikiversity, there is the WikiJournal project, and that can take some some forms of journalism, just not breaking news reporting. I am in favor of salvaging parts of Wikinews if helpful. Could it, would it be feasible to adapt Wikijournal to accept some forms of news journalism, but just not the timed news reporting? For example, WikiJournal already is doing conference proceedings, and could likely do related event reports even months after the event ended. It could probably accept long-form investigative reporting, which is a sort of news that is not breaking news. I am not sure what the possibilities are, but I would prefer to build up systems that already work rather than import systems which had problems elsewhere. Thanks. [[User:Bluerasberry|<span style="background:#cedff2;color:#11e">''' Blue Rasberry '''</span>]][[User talk:Bluerasberry|<span style="cursor:help"><span style="background:#cedff2;color:#11e">(talk)</span></span>]] 19:17, 22 May 2026 (UTC) *:I agree that there are certain kinds of journalism that are perfectly valid and not time-bound like breaking news reporting, so that won't suffer from the issues noted before. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 21:15, 22 May 2026 (UTC) == [[Wikiversity:Deletion policy]] proposed as policy == {{archive top|Consensus to promote to an official policy. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 22:30, 1 June 2026 (UTC)}} [[Wikiversity:Deletions]] has been operating as a [[Wikiversity:Guidelines|guideline]]. It has been revised and moved to [[Wikiversity:Deletion policy]], consistent with naming conventions used across sister projects such as Wikipedia, Wikibooks, and Wikiquote. The speedy deletion criteria have also been updated for consistency with [[MediaWiki:Deletereason-dropdown]]. This proposal is for the page to be formally adopted as [[Wikiversity:Policies|Wikiversity policy]]. Community feedback is invited, including suggestions for further improvements that may strengthen the proposed policy. === Voting === *{{support}} Seems reasonable. If there's somehow something missed here, we can just amend it later. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:33, 18 May 2026 (UTC) *{{support}} I don't see any issues with the policy. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 16:07, 18 May 2026 (UTC) === Comments === {{archive bottom}} == May 2026 Wikimedia Café meetups regarding the Wikimedia Foundation Annual Plan == <div class="border-box" style="background-color: var(--background-color-warning-subtle, #f8eaba); max-width: 875px; padding: 5px; border: 1px solid black; margin: 5px; color: var(--clr-dark)"> <div class="box" style="float:left; padding-top: 15px; padding-right: 15px;">[[File:Wikimedia Café logo in plain SVG format.svg|75px|alt=The logo for the Wikimedia Café]]</div> Hello! There will be two '''[https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9 Wikimedia Café]''' discussion opportunities during the last weekend of May. Both sessions will focus on the [https://meta.wikimedia.org/wiki/Wikimedia_Foundation_Annual_Plan/2026-2027 the 2026-2027 Wikimedia Foundation Annual Plan]. Participants may attend either or both sessions. #'''Saturday, 30 May 2026 at 15:00 UTC''' ([https://zonestamp.toolforge.org/1780153200 timestamp converter]), at a time friendly to the Americas, Africa, and Europe #'''Sunday, 31 May 2026 at 05:00 UTC''' ([https://zonestamp.toolforge.org/1780203600 timestamp converter]), at a time friendly to Asia and the Pacific Café participants are highly encouraged to read in advance [https://en.wikipedia.org/wiki/User:Sohom_Datta/annual_plan_guide at least this summary of the plan]. Optionally, Café participants are encouraged to read portions of the plan that interest them and [https://meta.wikimedia.org/wiki/Talk:Wikimedia_Foundation_Annual_Plan/2026-2027 ask questions or provide feedback on the Annual Plan talk page]. Please see the Café page for more information, including [https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9#May_2026_meetings_with_a_focus_on_Wikimedia_Foundation_Annual_Plan/2026-2027 tables of timestamp conversions for both sessions], [https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9#Agenda._This_will_be_an_approximately_1_hour_Caf%C3%A9_session,_and_is_extendible_for_an_additional_30_minutes_if_needed. the agenda], and [https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9#How_to_attend_the_session how to register]! <br /> [[File:Buntstifte Eberhard Faber crop 64h.jpg|860px|alt=cropped image of colored pencils]]</div> <span style="white-space:nowrap;">[[User:Pine|<span style="color:#01796f; text-shadow:#00BFFF 0 0 1.0em">↠Pine</span>]] [[User talk:Pine|<span style="color:DeepSkyBlue">(<b style="color:#FFDF00;text-shadow:#FFDF00 0 0 1.0em">✉</b>)</span>]]</span> 19:46, 21 May 2026 (UTC) == Vote now in the 2026 U4C election == <section begin="announcement-content" /> Eligible voters are asked to participate in the 2026 [[m:Special:MyLanguage/Universal_Code_of_Conduct/Coordinating_Committee|Universal Code of Conduct Coordinating Committee]] election. More information–including an eligibility check, voting process information, candidate information, and a link to the vote–are available on Meta at the [[m:Special:MyLanguage/Universal_Code_of_Conduct/Coordinating_Committee/Election/2026|2026 Election information page]]. The vote closes on 2 June 2026 at [https://zonestamp.toolforge.org/1780358400 00:00 UTC]. Please vote if your account is eligible. Results will be available by 14 June 2026. -- In cooperation with the U4C,<section end="announcement-content" /> [[m:User:Keegan (WMF)|Keegan (WMF)]] ([[m:User talk:Keegan (WMF)|talk]]) 17:15, 27 May 2026 (UTC) <!-- Message sent by User:Keegan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Global_message_delivery&oldid=30513860 --> == Create an autopatrolled user group? == {{tracked|T428269}} I would like to propose creating the user group <code>autopatrolled</code> (autopatrolled user), in which for non-curators and non-custodians, their page creations and file uploads would be automatically marked as patrolled by the MediaWiki software. Custodians may grant the user group, at their discretion, to users who create good quality pages that do not need frequent patrolling. On a side note, the term {{tq|autopatroller}} would be used, but because we don't have non-curator/custodian patrollers (as we rely on curators and custodians to patrol), I suggest on using the term {{tq|autopatrolled user}}. Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:31, 29 May 2026 (UTC) :'''Support''' re: the name, I don't really understand the reasoning, so I am '''neutral''' on that. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 15:45, 29 May 2026 (UTC) :: Regarding the name, this is because as we don't have the patroller user group, we rely on curators and custodians to patrol new pages and file uploads. Does that make sense? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:39, 29 May 2026 (UTC) :::Not really, but I don't think it's the most important thing. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 16:42, 29 May 2026 (UTC) :::: We'll decide on the name later. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 01:48, 30 May 2026 (UTC) :::::Oh, please don't let me stand in the way. I'm just not very smart, so don't hold up a matter on my account. I didn't want to derail the proposal, which is a fine and sensible one. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 04:16, 30 May 2026 (UTC) : '''Support''' - sounds like a good idea :* Suggest adding a draft section about this group to [[Wikiversity:Patrolling]]. There is a statement in the Introduction of the page that I'm not sure if its correct and at least could be improved: "Wikiversity also uses an autopatrol right, meaning trusted users' contributions are automatically marked as checked so patrollers can focus on reviewing newer or anonymous editors." :* Regarding autopatroller vs autropatrolled user, what terms are used on similar WMF wiki projects? : -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:28, 30 May 2026 (UTC) ::# I would create a starting page about the user groups, with experienced editors expanding the page. A summarized part of that page would also be added to [[Wikiversity:Patrolling]]. ::# For a similar example, English Wikipedia uses the term {{tq|Autopatrolled}}, just that term only. :: [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:22, 30 May 2026 (UTC) == How much of Wikiversity’s content is LLM slop? == Because it seems like a non-trivial amount, along with AI slop images as well. Is there some kind of AI cleanup project established yet? [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 01:20, 4 June 2026 (UTC) :We have discussed AI but I don't know of any explicit initiative to find and delete AI-generated noise. Individual modules have been deleted for having been made by AI. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 08:50, 4 June 2026 (UTC) == Draft inactivity policy == I created [[Wikiversity:Inactivity policy]] as a start. Any experienced Wikiversity user may feel free to expand it. This is also one-to-two step(s) towards opting out of the [[m:Admin activity review|AAR process]]. However, I made a bold change to reduce the response timeframe from one month to two weeks. In addition, should we reduce the inactivity timeframe to one year? For the latter, most projects use that timeframe and I suggested this for consistency. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:57, 4 June 2026 (UTC) :I support those suggestions. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 17:55, 4 June 2026 (UTC) abvo17xjlt6kz5ut7t0r1ylp0cvr8ey 2812901 2812849 2026-06-05T17:55:07Z Codename Noreste 2969951 /* Proposed user group and/or possible policy changes */ new topic ([[mw:c:Special:MyLanguage/User:JWBTH/CD|CD]]) 2812901 wikitext text/x-wiki {{Wikiversity:Colloquium/Header}} <!-- MESSAGES GO BELOW --> == [[MediaWiki:Protectedpagetext#Protected edit request on 11 December 2025]] == I posted an edit request there 5 months ago, so I’ll be taking it to this page. [[Special:Contributions/&#126;2026-28640-56|&#126;2026-28640-56]] ([[User talk:&#126;2026-28640-56|talk]]) 23:33, 12 May 2026 (UTC) :What exactly is the problem? I don't understand what needs to change and why. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 23:35, 12 May 2026 (UTC) : Pinging @[[User:Atcovi|Atcovi]], @[[User:Jtneill|Jtneill]] and @[[User:Juandev|Juandev]] for further input. Someone is requesting a modification to [[MediaWiki:Protectedpagetext]] to use {{tlx|Protected page text}}, but we might need to discuss whether to use the template. In the meantime, I'll start a sandbox version of the protected page text template. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 23:19, 14 May 2026 (UTC) ::Sounds good -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 04:13, 15 May 2026 (UTC) :::+1 Jtneill. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 12:59, 19 May 2026 (UTC) == Proposal to rehost Wikinews here == As many of you know, and mentioned here at the Colloquium, our sister project Wikinews recently closed, with all 31 active editions made read-only. [[User:BigKrow]] has asked about the prospect of writing news stories here and I suggested that since we already have [[School:Journalism]] and some resources related to the [[:Category:Journalism|broader topic of journalism]]. I would like to propose that we have continued and indefinite space for {{w|citizen journalism}} by essentially repurposing Wikinews into a sub-project here. The only special infrastructure that Wikinews required was [[:mw:Extension:DynamicPageList]], which was deactivated and caused issues due to a lack of maintenance. I will add this proposal to the site banner, but I recognize that that may be a conflict of interest, so if anyone requests that I remove it, I will. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:30, 14 May 2026 (UTC) :I would like to see this conversation go for at least 30 days to establish a consensus. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:35, 14 May 2026 (UTC) ===Votes=== *{{support}} as proposer (with BK's inspiration). I think that an ongoing experiment in citizen journalism is a fit and appropriate use of this site. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:35, 14 May 2026 (UTC) *{{support}}, hope to seeing ideas about this, and thank you @[[User:Koavf|Koavf]] [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 11:08, 14 May 2026 (UTC) *{{support}} Other than perhaps inflating the total number of pages reported, I see the idea of "practicing journalism" a worthy and relevant activity within the domain of Wikiversity. [[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 21:41, 14 May 2026 (UTC) *{{support}} Conditional on development of (a) community guidelines that ensure alignment with Wikiversity's purpose, and (b) clear, nested page-naming structures for projects. More detail below. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:48, 15 May 2026 (UTC) *{{contra}} This proposal doesn't seem interested in expanding educational materials in journalism, but rather in providing space and protection for Wikinews contributors. But this is contrary to the goals of Wikiversity, and I'm not sure it's a good idea, even with regard to WMF. If WMF decides to close a project and another community lets it run on its domain, that's a bit of an undermining of WMF's and the community's decisions. Given that Wikiversity has had several conflicts with other communities and WMF in its history, I'm against it.--[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:59, 15 May 2026 (UTC) *{{contra}} This seems like a proposal to continue the mission of WikiNews, but not a proposal specifically to improve Wikiversity. I concur with Juandev's comments. --[[User:Mu301|mikeu]] <sup>[[User talk:Mu301|talk]]</sup> 20:29, 30 May 2026 (UTC) * {{oppose}} per above. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 19:05, 1 June 2026 (UTC) *{{oppose}} Wikiversity isn’t Wikinews and it also isn’t a dumping ground for anything not covered by other projects. It was already suggested, rather bafflingly, that Wikinews parasitize Wikipedia as a host. If it were allowed to freeload off of Wikiversity it would simply promote a view I and likely many others have— that Wikiversity (as it currently exists) has no standards and mostly just exists to host subpar content that wouldn’t be tolerated on any other Wikimedia site. Wikinews needs a new, non-Wikimedia host, and Wikiversity needs to get its act together by enforcing a minimum scope and standard for what it allows. --[[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 01:16, 4 June 2026 (UTC) ===Comments and questions=== :Definitely worthy of discussion, so I have no problem with the proposal in the sitenotice. :Initial questions: :* Does this proposal include importing English Wikinews content e.g., to [[Wikinews]] subpages? :* What are "active editions"? :* How can Wikiversity navigate the concerns that lead to the closure of Wikinews? :* Are any changes to the scope of Wikinews proposed? :* How does [[Wikinews]] fit with the [[Wikiversity:Mission]]? What aligns well? Where might there be tension? :** e.g., I'm not sure that a page like [[User:BigKrow/Manchester City moves two points behind Arsenal]] in and of itself will serve as an educational resource. :-- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 05:52, 14 May 2026 (UTC) :* Does this proposal include importing English Wikinews content e.g., to [[Wikinews]] subpages? ::*No, not at this time. :* What are "active editions"? ::*There were 30 other active editions of Wikinews in addition to English (e.g. [[:n:es:]]) at the time of universal closure (2026-05-04). :* How can Wikiversity navigate the concerns that lead to the closure of Wikinews? ::*One of the biggest issues was the problems with DPL, which is now irrelevant. Another was the lack of activity, which can be ameliorated by having it be part of an existing project instead of its own domain (e.g. some editions of Wikipedia host their own Wikinews already and those projects were not impacted by the closure). :* Are any changes to the scope of Wikinews proposed? ::*Not at this juncture. I would also propose as far as implemention goes that we would request a new namespace and that the material be more-or-less sequestered into its own ongoing project, like Wikijournal is or like the Cookbook and Wikijunior are at our sister [[:b:]]. :* How does [[Wikinews]] fit with the [[Wikiversity:Mission]]? What aligns well? Where might there be tension? :** e.g., I'm not sure that a page like [[Story/Manchester City moves two points behind Arsenal]] in and of itself will serve as an educational resource. ::*The process of citizen journalists practicing their craft in real-time and collaborating with others to do so is itself an education activity. We would essentially be hosting a real-time experiment in citizen journalism, online communities, and collaborative learning in addition to the prospect of spreading educational information from someone actually reading the news. I would propose that we could also make a more deliberate attempt to engage with learning <em>about</em> what does and doesn't work with collaborative news writing by experimentation (e.g. audio news, syndicating to other sites, incorporating freely-licensed news from other sources, writing hyper-local news, writing briefs versus longer-term reportage) and also seeing if the problems noted in the Task Force report that recommended closure can be overcome. Note that we have already done some local investigation about and learning about wiki-based journalism on Wikinews here at [[Journalism studies and Wikinews]]. We could continue that learning and refine the process, including incorporating journalism students from universities. As for tensions, Wikinews is the only sister project that must be done with a quick turn-around: if you take a long time to [[:s:|transcribe a book]], that's just how long it takes, but if you take a long time to write news, it ceases to be news entirely. Wikiversity has been a very slow-growing project that has definitely had some successes but has generally come together over a long period with most learning resources being individual passion projects (or sometimes, frankly, crankery) which would not work with collaborative news that requires more than just a single editor writing whatever he feels like. ::Please let me know any other questions/concerns and any other editors feel free to give your own perspective. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 06:13, 14 May 2026 (UTC) :::Thanks, Justin — it is food for thought. :::In attempting to understand how we've arrived here, I've summarised some of the background on this page: [[Wikinews]]. :::Perhaps it could be helpful to flesh out more of the vision / ideas / possibilities / challenges on that page? -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:49, 14 May 2026 (UTC) :::*Having given it some thought, in principle, I support hosting [[citizen journalism]] on Wikiversity where it is clearly connected to a learning project and/or constitutes original research, both of which align strongly with [[Wikiversity:Mission|Wikiversity’s educational mission]]. :::*My chief concern is the potential for news content that is not clearly linked to the purpose of Wikiversity. To avoid this, some community-agreed guidelines would be prudent. These need not be overly restrictive; they should support boldness and experimentation while helping ensure alignment with Wikiversity's purpose. :::*Given the reported low and declining activity on Wikinews, it seems unlikely that English Wikiversity would be overwhelmed by an influx of news-related editing. My impression is that English Wikinews was the most active edition, but even so, many contributors are likely to disperse to other projects or cease editing altogether. A modest migration of interested editors to Wikiversity seems manageable. :::*At this stage, I do not think a dedicated namespace is necessary. Subpages under [[Wikinews]] or nested pages under relevant learning or research projects, or user-space draft pages should be suitable. I agree that [[Wikijournal]] offers a useful model, as do several existing course structures on Wikiversity. :::*I support [[User:Koavf]]’s suggestions about framing Wikinews activity explicitly around learning. This would create a distinctive space for experimenting with collaborative news production in ways that are pedagogically meaningful. I agree that the [[journalism studies and Wikinews]] project developed by David and Leigh Blackall through the University of Wollongong is an excellent example of the intersection between Wikiversity and Wikinews. The [[Wikinews]] page could evolve into a hub for such projects. :::*I've tidied the [[:Category:Wikinews|Wikinews category]] and merged some content into the [[Wikinews]] page. As part of a reinvigoration effort, please review these and related resources such as [[:Category:Journalism]] and [[School:Journalism]]. :::*A further argument in favour of this initiative is that Wikipedia explicitly excludes both news reporting and original research. So, there is value in maintaining spaces within the Wikimedia ecosystem where these forms of knowledge production can be openly developed and curated. Such work can, in turn, generate valuable evidence and source material that may later inform Wikipedia articles. :::*The closure of WMF-hosted Wikinews does not imply that open wiki-based news curation lacks value. Indeed, the closure documentation appears supportive of experimentation with alternative news models across Wikimedia projects, including through Wikipedia and Wikidata. In that context, Wikiversity seems a natural home for a Wikinews experiment, provided it is clearly grounded in learning and/or research. :::-- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:39, 15 May 2026 (UTC) My understanding towards Wikinews' failure is that everything takes too long to be approved for the publish status, which means that any breaking news would have already become days-old stale news. Wikinews has a brand recognition (for right or wrong reasons) than Wikiversity and I wonder how effective Wikiversity can attract the "Wikinews refugees" to edit here. And just a quick note on the governance. Since each Wikiversity language operates independently, each language has to vote & adopt this proposal independently. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;"><sup>Talk page</sup></span></b>]] 13:47, 15 May 2026 (UTC) :Your assessment about Wikinews is partially correct. I referenced it earlier, but to be explicit, there is a [[:m:Proposal for Closing Wikinews|report by a task force on sister projects]] that outlines their concerns. There are a few, one of which was the nature of the staleness of news. Thanks also for clarifying that this proposal is only relevant to en.wv and is not binding or even proposed for other editions of Wikiversity. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 18:54, 15 May 2026 (UTC) *Note: I am not a regular here, and just visit Wikiversity for the WikiJournal project. Challenges of Wikinews included that it required timely reporting and fact-checking processes which differed greatly from the well-established ones in Wikipedia. Here in Wikiversity, there is the WikiJournal project, and that can take some some forms of journalism, just not breaking news reporting. I am in favor of salvaging parts of Wikinews if helpful. Could it, would it be feasible to adapt Wikijournal to accept some forms of news journalism, but just not the timed news reporting? For example, WikiJournal already is doing conference proceedings, and could likely do related event reports even months after the event ended. It could probably accept long-form investigative reporting, which is a sort of news that is not breaking news. I am not sure what the possibilities are, but I would prefer to build up systems that already work rather than import systems which had problems elsewhere. Thanks. [[User:Bluerasberry|<span style="background:#cedff2;color:#11e">''' Blue Rasberry '''</span>]][[User talk:Bluerasberry|<span style="cursor:help"><span style="background:#cedff2;color:#11e">(talk)</span></span>]] 19:17, 22 May 2026 (UTC) *:I agree that there are certain kinds of journalism that are perfectly valid and not time-bound like breaking news reporting, so that won't suffer from the issues noted before. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 21:15, 22 May 2026 (UTC) == [[Wikiversity:Deletion policy]] proposed as policy == {{archive top|Consensus to promote to an official policy. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 22:30, 1 June 2026 (UTC)}} [[Wikiversity:Deletions]] has been operating as a [[Wikiversity:Guidelines|guideline]]. It has been revised and moved to [[Wikiversity:Deletion policy]], consistent with naming conventions used across sister projects such as Wikipedia, Wikibooks, and Wikiquote. The speedy deletion criteria have also been updated for consistency with [[MediaWiki:Deletereason-dropdown]]. This proposal is for the page to be formally adopted as [[Wikiversity:Policies|Wikiversity policy]]. Community feedback is invited, including suggestions for further improvements that may strengthen the proposed policy. === Voting === *{{support}} Seems reasonable. If there's somehow something missed here, we can just amend it later. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:33, 18 May 2026 (UTC) *{{support}} I don't see any issues with the policy. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 16:07, 18 May 2026 (UTC) === Comments === {{archive bottom}} == May 2026 Wikimedia Café meetups regarding the Wikimedia Foundation Annual Plan == <div class="border-box" style="background-color: var(--background-color-warning-subtle, #f8eaba); max-width: 875px; padding: 5px; border: 1px solid black; margin: 5px; color: var(--clr-dark)"> <div class="box" style="float:left; padding-top: 15px; padding-right: 15px;">[[File:Wikimedia Café logo in plain SVG format.svg|75px|alt=The logo for the Wikimedia Café]]</div> Hello! There will be two '''[https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9 Wikimedia Café]''' discussion opportunities during the last weekend of May. Both sessions will focus on the [https://meta.wikimedia.org/wiki/Wikimedia_Foundation_Annual_Plan/2026-2027 the 2026-2027 Wikimedia Foundation Annual Plan]. Participants may attend either or both sessions. #'''Saturday, 30 May 2026 at 15:00 UTC''' ([https://zonestamp.toolforge.org/1780153200 timestamp converter]), at a time friendly to the Americas, Africa, and Europe #'''Sunday, 31 May 2026 at 05:00 UTC''' ([https://zonestamp.toolforge.org/1780203600 timestamp converter]), at a time friendly to Asia and the Pacific Café participants are highly encouraged to read in advance [https://en.wikipedia.org/wiki/User:Sohom_Datta/annual_plan_guide at least this summary of the plan]. Optionally, Café participants are encouraged to read portions of the plan that interest them and [https://meta.wikimedia.org/wiki/Talk:Wikimedia_Foundation_Annual_Plan/2026-2027 ask questions or provide feedback on the Annual Plan talk page]. Please see the Café page for more information, including [https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9#May_2026_meetings_with_a_focus_on_Wikimedia_Foundation_Annual_Plan/2026-2027 tables of timestamp conversions for both sessions], [https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9#Agenda._This_will_be_an_approximately_1_hour_Caf%C3%A9_session,_and_is_extendible_for_an_additional_30_minutes_if_needed. the agenda], and [https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9#How_to_attend_the_session how to register]! <br /> [[File:Buntstifte Eberhard Faber crop 64h.jpg|860px|alt=cropped image of colored pencils]]</div> <span style="white-space:nowrap;">[[User:Pine|<span style="color:#01796f; text-shadow:#00BFFF 0 0 1.0em">↠Pine</span>]] [[User talk:Pine|<span style="color:DeepSkyBlue">(<b style="color:#FFDF00;text-shadow:#FFDF00 0 0 1.0em">✉</b>)</span>]]</span> 19:46, 21 May 2026 (UTC) == Vote now in the 2026 U4C election == <section begin="announcement-content" /> Eligible voters are asked to participate in the 2026 [[m:Special:MyLanguage/Universal_Code_of_Conduct/Coordinating_Committee|Universal Code of Conduct Coordinating Committee]] election. More information–including an eligibility check, voting process information, candidate information, and a link to the vote–are available on Meta at the [[m:Special:MyLanguage/Universal_Code_of_Conduct/Coordinating_Committee/Election/2026|2026 Election information page]]. The vote closes on 2 June 2026 at [https://zonestamp.toolforge.org/1780358400 00:00 UTC]. Please vote if your account is eligible. Results will be available by 14 June 2026. -- In cooperation with the U4C,<section end="announcement-content" /> [[m:User:Keegan (WMF)|Keegan (WMF)]] ([[m:User talk:Keegan (WMF)|talk]]) 17:15, 27 May 2026 (UTC) <!-- Message sent by User:Keegan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Global_message_delivery&oldid=30513860 --> == Create an autopatrolled user group? == {{tracked|T428269}} I would like to propose creating the user group <code>autopatrolled</code> (autopatrolled user), in which for non-curators and non-custodians, their page creations and file uploads would be automatically marked as patrolled by the MediaWiki software. Custodians may grant the user group, at their discretion, to users who create good quality pages that do not need frequent patrolling. On a side note, the term {{tq|autopatroller}} would be used, but because we don't have non-curator/custodian patrollers (as we rely on curators and custodians to patrol), I suggest on using the term {{tq|autopatrolled user}}. Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:31, 29 May 2026 (UTC) :'''Support''' re: the name, I don't really understand the reasoning, so I am '''neutral''' on that. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 15:45, 29 May 2026 (UTC) :: Regarding the name, this is because as we don't have the patroller user group, we rely on curators and custodians to patrol new pages and file uploads. Does that make sense? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:39, 29 May 2026 (UTC) :::Not really, but I don't think it's the most important thing. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 16:42, 29 May 2026 (UTC) :::: We'll decide on the name later. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 01:48, 30 May 2026 (UTC) :::::Oh, please don't let me stand in the way. I'm just not very smart, so don't hold up a matter on my account. I didn't want to derail the proposal, which is a fine and sensible one. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 04:16, 30 May 2026 (UTC) : '''Support''' - sounds like a good idea :* Suggest adding a draft section about this group to [[Wikiversity:Patrolling]]. There is a statement in the Introduction of the page that I'm not sure if its correct and at least could be improved: "Wikiversity also uses an autopatrol right, meaning trusted users' contributions are automatically marked as checked so patrollers can focus on reviewing newer or anonymous editors." :* Regarding autopatroller vs autropatrolled user, what terms are used on similar WMF wiki projects? : -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:28, 30 May 2026 (UTC) ::# I would create a starting page about the user groups, with experienced editors expanding the page. A summarized part of that page would also be added to [[Wikiversity:Patrolling]]. ::# For a similar example, English Wikipedia uses the term {{tq|Autopatrolled}}, just that term only. :: [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:22, 30 May 2026 (UTC) == How much of Wikiversity’s content is LLM slop? == Because it seems like a non-trivial amount, along with AI slop images as well. Is there some kind of AI cleanup project established yet? [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 01:20, 4 June 2026 (UTC) :We have discussed AI but I don't know of any explicit initiative to find and delete AI-generated noise. Individual modules have been deleted for having been made by AI. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 08:50, 4 June 2026 (UTC) == Draft inactivity policy == I created [[Wikiversity:Inactivity policy]] as a start. Any experienced Wikiversity user may feel free to expand it. This is also one-to-two step(s) towards opting out of the [[m:Admin activity review|AAR process]]. However, I made a bold change to reduce the response timeframe from one month to two weeks. In addition, should we reduce the inactivity timeframe to one year? For the latter, most projects use that timeframe and I suggested this for consistency. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:57, 4 June 2026 (UTC) :I support those suggestions. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 17:55, 4 June 2026 (UTC) == Proposed user group and/or possible policy changes == I want to discuss about user group and possible policy changes. # First, interface administrator. I don't think we should allow interface administrators to remove their permission from their own account, since we have multiple active bureaucrats and we can ask them to remove the permission when done, or for them to add a temporary grant. This is according to the [[Wikiversity:IA|current IA policy]]. I also left [[Wikiversity talk:Interface administrators#My thoughts about this user group|my thoughts on the relevant talk page]]. # Second, curators. Given that curators have some sensitive custodian rights (such as <code>delete</code> [but not <code>undelete</code> or similar rights that allow viewing deleted content, unless the curatorship process is RFA-like] and <code>protect</code>), it would probably make more sense only for bureaucrats to grant and remove it, on par with them granting (but not removing) custodian permissions. # Third, about probationary custodians. [[Wikiversity:Probationary custodians]] is currently marked as historical, and the process might still exist on [[Wikiversity:Custodianship]]. Therefore, to maintain consistency with [[Wikiversity:Curatorship#How does one become a curator?]], I propose that we repeal the probationary custodianship process and change it more or less to align with the curatorship process, effectively making probationary custodians permanent ones. However, custodian mentors would still be retained. Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 17:55, 5 June 2026 (UTC) 2jq8sq0dy5ex0s6brc1wxy0s0c5gthc 2812902 2812901 2026-06-05T17:55:57Z Codename Noreste 2969951 /* Proposed user group and/or possible policy changes */ edit opening comment ([[mw:c:Special:MyLanguage/User:JWBTH/CD|CD]]) 2812902 wikitext text/x-wiki {{Wikiversity:Colloquium/Header}} <!-- MESSAGES GO BELOW --> == [[MediaWiki:Protectedpagetext#Protected edit request on 11 December 2025]] == I posted an edit request there 5 months ago, so I’ll be taking it to this page. [[Special:Contributions/&#126;2026-28640-56|&#126;2026-28640-56]] ([[User talk:&#126;2026-28640-56|talk]]) 23:33, 12 May 2026 (UTC) :What exactly is the problem? I don't understand what needs to change and why. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 23:35, 12 May 2026 (UTC) : Pinging @[[User:Atcovi|Atcovi]], @[[User:Jtneill|Jtneill]] and @[[User:Juandev|Juandev]] for further input. Someone is requesting a modification to [[MediaWiki:Protectedpagetext]] to use {{tlx|Protected page text}}, but we might need to discuss whether to use the template. In the meantime, I'll start a sandbox version of the protected page text template. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 23:19, 14 May 2026 (UTC) ::Sounds good -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 04:13, 15 May 2026 (UTC) :::+1 Jtneill. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 12:59, 19 May 2026 (UTC) == Proposal to rehost Wikinews here == As many of you know, and mentioned here at the Colloquium, our sister project Wikinews recently closed, with all 31 active editions made read-only. [[User:BigKrow]] has asked about the prospect of writing news stories here and I suggested that since we already have [[School:Journalism]] and some resources related to the [[:Category:Journalism|broader topic of journalism]]. I would like to propose that we have continued and indefinite space for {{w|citizen journalism}} by essentially repurposing Wikinews into a sub-project here. The only special infrastructure that Wikinews required was [[:mw:Extension:DynamicPageList]], which was deactivated and caused issues due to a lack of maintenance. I will add this proposal to the site banner, but I recognize that that may be a conflict of interest, so if anyone requests that I remove it, I will. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:30, 14 May 2026 (UTC) :I would like to see this conversation go for at least 30 days to establish a consensus. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:35, 14 May 2026 (UTC) ===Votes=== *{{support}} as proposer (with BK's inspiration). I think that an ongoing experiment in citizen journalism is a fit and appropriate use of this site. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:35, 14 May 2026 (UTC) *{{support}}, hope to seeing ideas about this, and thank you @[[User:Koavf|Koavf]] [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 11:08, 14 May 2026 (UTC) *{{support}} Other than perhaps inflating the total number of pages reported, I see the idea of "practicing journalism" a worthy and relevant activity within the domain of Wikiversity. [[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 21:41, 14 May 2026 (UTC) *{{support}} Conditional on development of (a) community guidelines that ensure alignment with Wikiversity's purpose, and (b) clear, nested page-naming structures for projects. More detail below. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:48, 15 May 2026 (UTC) *{{contra}} This proposal doesn't seem interested in expanding educational materials in journalism, but rather in providing space and protection for Wikinews contributors. But this is contrary to the goals of Wikiversity, and I'm not sure it's a good idea, even with regard to WMF. If WMF decides to close a project and another community lets it run on its domain, that's a bit of an undermining of WMF's and the community's decisions. Given that Wikiversity has had several conflicts with other communities and WMF in its history, I'm against it.--[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:59, 15 May 2026 (UTC) *{{contra}} This seems like a proposal to continue the mission of WikiNews, but not a proposal specifically to improve Wikiversity. I concur with Juandev's comments. --[[User:Mu301|mikeu]] <sup>[[User talk:Mu301|talk]]</sup> 20:29, 30 May 2026 (UTC) * {{oppose}} per above. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 19:05, 1 June 2026 (UTC) *{{oppose}} Wikiversity isn’t Wikinews and it also isn’t a dumping ground for anything not covered by other projects. It was already suggested, rather bafflingly, that Wikinews parasitize Wikipedia as a host. If it were allowed to freeload off of Wikiversity it would simply promote a view I and likely many others have— that Wikiversity (as it currently exists) has no standards and mostly just exists to host subpar content that wouldn’t be tolerated on any other Wikimedia site. Wikinews needs a new, non-Wikimedia host, and Wikiversity needs to get its act together by enforcing a minimum scope and standard for what it allows. --[[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 01:16, 4 June 2026 (UTC) ===Comments and questions=== :Definitely worthy of discussion, so I have no problem with the proposal in the sitenotice. :Initial questions: :* Does this proposal include importing English Wikinews content e.g., to [[Wikinews]] subpages? :* What are "active editions"? :* How can Wikiversity navigate the concerns that lead to the closure of Wikinews? :* Are any changes to the scope of Wikinews proposed? :* How does [[Wikinews]] fit with the [[Wikiversity:Mission]]? What aligns well? Where might there be tension? :** e.g., I'm not sure that a page like [[User:BigKrow/Manchester City moves two points behind Arsenal]] in and of itself will serve as an educational resource. :-- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 05:52, 14 May 2026 (UTC) :* Does this proposal include importing English Wikinews content e.g., to [[Wikinews]] subpages? ::*No, not at this time. :* What are "active editions"? ::*There were 30 other active editions of Wikinews in addition to English (e.g. [[:n:es:]]) at the time of universal closure (2026-05-04). :* How can Wikiversity navigate the concerns that lead to the closure of Wikinews? ::*One of the biggest issues was the problems with DPL, which is now irrelevant. Another was the lack of activity, which can be ameliorated by having it be part of an existing project instead of its own domain (e.g. some editions of Wikipedia host their own Wikinews already and those projects were not impacted by the closure). :* Are any changes to the scope of Wikinews proposed? ::*Not at this juncture. I would also propose as far as implemention goes that we would request a new namespace and that the material be more-or-less sequestered into its own ongoing project, like Wikijournal is or like the Cookbook and Wikijunior are at our sister [[:b:]]. :* How does [[Wikinews]] fit with the [[Wikiversity:Mission]]? What aligns well? Where might there be tension? :** e.g., I'm not sure that a page like [[Story/Manchester City moves two points behind Arsenal]] in and of itself will serve as an educational resource. ::*The process of citizen journalists practicing their craft in real-time and collaborating with others to do so is itself an education activity. We would essentially be hosting a real-time experiment in citizen journalism, online communities, and collaborative learning in addition to the prospect of spreading educational information from someone actually reading the news. I would propose that we could also make a more deliberate attempt to engage with learning <em>about</em> what does and doesn't work with collaborative news writing by experimentation (e.g. audio news, syndicating to other sites, incorporating freely-licensed news from other sources, writing hyper-local news, writing briefs versus longer-term reportage) and also seeing if the problems noted in the Task Force report that recommended closure can be overcome. Note that we have already done some local investigation about and learning about wiki-based journalism on Wikinews here at [[Journalism studies and Wikinews]]. We could continue that learning and refine the process, including incorporating journalism students from universities. As for tensions, Wikinews is the only sister project that must be done with a quick turn-around: if you take a long time to [[:s:|transcribe a book]], that's just how long it takes, but if you take a long time to write news, it ceases to be news entirely. Wikiversity has been a very slow-growing project that has definitely had some successes but has generally come together over a long period with most learning resources being individual passion projects (or sometimes, frankly, crankery) which would not work with collaborative news that requires more than just a single editor writing whatever he feels like. ::Please let me know any other questions/concerns and any other editors feel free to give your own perspective. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 06:13, 14 May 2026 (UTC) :::Thanks, Justin — it is food for thought. :::In attempting to understand how we've arrived here, I've summarised some of the background on this page: [[Wikinews]]. :::Perhaps it could be helpful to flesh out more of the vision / ideas / possibilities / challenges on that page? -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:49, 14 May 2026 (UTC) :::*Having given it some thought, in principle, I support hosting [[citizen journalism]] on Wikiversity where it is clearly connected to a learning project and/or constitutes original research, both of which align strongly with [[Wikiversity:Mission|Wikiversity’s educational mission]]. :::*My chief concern is the potential for news content that is not clearly linked to the purpose of Wikiversity. To avoid this, some community-agreed guidelines would be prudent. These need not be overly restrictive; they should support boldness and experimentation while helping ensure alignment with Wikiversity's purpose. :::*Given the reported low and declining activity on Wikinews, it seems unlikely that English Wikiversity would be overwhelmed by an influx of news-related editing. My impression is that English Wikinews was the most active edition, but even so, many contributors are likely to disperse to other projects or cease editing altogether. A modest migration of interested editors to Wikiversity seems manageable. :::*At this stage, I do not think a dedicated namespace is necessary. Subpages under [[Wikinews]] or nested pages under relevant learning or research projects, or user-space draft pages should be suitable. I agree that [[Wikijournal]] offers a useful model, as do several existing course structures on Wikiversity. :::*I support [[User:Koavf]]’s suggestions about framing Wikinews activity explicitly around learning. This would create a distinctive space for experimenting with collaborative news production in ways that are pedagogically meaningful. I agree that the [[journalism studies and Wikinews]] project developed by David and Leigh Blackall through the University of Wollongong is an excellent example of the intersection between Wikiversity and Wikinews. The [[Wikinews]] page could evolve into a hub for such projects. :::*I've tidied the [[:Category:Wikinews|Wikinews category]] and merged some content into the [[Wikinews]] page. As part of a reinvigoration effort, please review these and related resources such as [[:Category:Journalism]] and [[School:Journalism]]. :::*A further argument in favour of this initiative is that Wikipedia explicitly excludes both news reporting and original research. So, there is value in maintaining spaces within the Wikimedia ecosystem where these forms of knowledge production can be openly developed and curated. Such work can, in turn, generate valuable evidence and source material that may later inform Wikipedia articles. :::*The closure of WMF-hosted Wikinews does not imply that open wiki-based news curation lacks value. Indeed, the closure documentation appears supportive of experimentation with alternative news models across Wikimedia projects, including through Wikipedia and Wikidata. In that context, Wikiversity seems a natural home for a Wikinews experiment, provided it is clearly grounded in learning and/or research. :::-- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:39, 15 May 2026 (UTC) My understanding towards Wikinews' failure is that everything takes too long to be approved for the publish status, which means that any breaking news would have already become days-old stale news. Wikinews has a brand recognition (for right or wrong reasons) than Wikiversity and I wonder how effective Wikiversity can attract the "Wikinews refugees" to edit here. And just a quick note on the governance. Since each Wikiversity language operates independently, each language has to vote & adopt this proposal independently. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;"><sup>Talk page</sup></span></b>]] 13:47, 15 May 2026 (UTC) :Your assessment about Wikinews is partially correct. I referenced it earlier, but to be explicit, there is a [[:m:Proposal for Closing Wikinews|report by a task force on sister projects]] that outlines their concerns. There are a few, one of which was the nature of the staleness of news. Thanks also for clarifying that this proposal is only relevant to en.wv and is not binding or even proposed for other editions of Wikiversity. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 18:54, 15 May 2026 (UTC) *Note: I am not a regular here, and just visit Wikiversity for the WikiJournal project. Challenges of Wikinews included that it required timely reporting and fact-checking processes which differed greatly from the well-established ones in Wikipedia. Here in Wikiversity, there is the WikiJournal project, and that can take some some forms of journalism, just not breaking news reporting. I am in favor of salvaging parts of Wikinews if helpful. Could it, would it be feasible to adapt Wikijournal to accept some forms of news journalism, but just not the timed news reporting? For example, WikiJournal already is doing conference proceedings, and could likely do related event reports even months after the event ended. It could probably accept long-form investigative reporting, which is a sort of news that is not breaking news. I am not sure what the possibilities are, but I would prefer to build up systems that already work rather than import systems which had problems elsewhere. Thanks. [[User:Bluerasberry|<span style="background:#cedff2;color:#11e">''' Blue Rasberry '''</span>]][[User talk:Bluerasberry|<span style="cursor:help"><span style="background:#cedff2;color:#11e">(talk)</span></span>]] 19:17, 22 May 2026 (UTC) *:I agree that there are certain kinds of journalism that are perfectly valid and not time-bound like breaking news reporting, so that won't suffer from the issues noted before. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 21:15, 22 May 2026 (UTC) == [[Wikiversity:Deletion policy]] proposed as policy == {{archive top|Consensus to promote to an official policy. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 22:30, 1 June 2026 (UTC)}} [[Wikiversity:Deletions]] has been operating as a [[Wikiversity:Guidelines|guideline]]. It has been revised and moved to [[Wikiversity:Deletion policy]], consistent with naming conventions used across sister projects such as Wikipedia, Wikibooks, and Wikiquote. The speedy deletion criteria have also been updated for consistency with [[MediaWiki:Deletereason-dropdown]]. This proposal is for the page to be formally adopted as [[Wikiversity:Policies|Wikiversity policy]]. Community feedback is invited, including suggestions for further improvements that may strengthen the proposed policy. === Voting === *{{support}} Seems reasonable. If there's somehow something missed here, we can just amend it later. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:33, 18 May 2026 (UTC) *{{support}} I don't see any issues with the policy. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 16:07, 18 May 2026 (UTC) === Comments === {{archive bottom}} == May 2026 Wikimedia Café meetups regarding the Wikimedia Foundation Annual Plan == <div class="border-box" style="background-color: var(--background-color-warning-subtle, #f8eaba); max-width: 875px; padding: 5px; border: 1px solid black; margin: 5px; color: var(--clr-dark)"> <div class="box" style="float:left; padding-top: 15px; padding-right: 15px;">[[File:Wikimedia Café logo in plain SVG format.svg|75px|alt=The logo for the Wikimedia Café]]</div> Hello! There will be two '''[https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9 Wikimedia Café]''' discussion opportunities during the last weekend of May. Both sessions will focus on the [https://meta.wikimedia.org/wiki/Wikimedia_Foundation_Annual_Plan/2026-2027 the 2026-2027 Wikimedia Foundation Annual Plan]. Participants may attend either or both sessions. #'''Saturday, 30 May 2026 at 15:00 UTC''' ([https://zonestamp.toolforge.org/1780153200 timestamp converter]), at a time friendly to the Americas, Africa, and Europe #'''Sunday, 31 May 2026 at 05:00 UTC''' ([https://zonestamp.toolforge.org/1780203600 timestamp converter]), at a time friendly to Asia and the Pacific Café participants are highly encouraged to read in advance [https://en.wikipedia.org/wiki/User:Sohom_Datta/annual_plan_guide at least this summary of the plan]. Optionally, Café participants are encouraged to read portions of the plan that interest them and [https://meta.wikimedia.org/wiki/Talk:Wikimedia_Foundation_Annual_Plan/2026-2027 ask questions or provide feedback on the Annual Plan talk page]. Please see the Café page for more information, including [https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9#May_2026_meetings_with_a_focus_on_Wikimedia_Foundation_Annual_Plan/2026-2027 tables of timestamp conversions for both sessions], [https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9#Agenda._This_will_be_an_approximately_1_hour_Caf%C3%A9_session,_and_is_extendible_for_an_additional_30_minutes_if_needed. the agenda], and [https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9#How_to_attend_the_session how to register]! <br /> [[File:Buntstifte Eberhard Faber crop 64h.jpg|860px|alt=cropped image of colored pencils]]</div> <span style="white-space:nowrap;">[[User:Pine|<span style="color:#01796f; text-shadow:#00BFFF 0 0 1.0em">↠Pine</span>]] [[User talk:Pine|<span style="color:DeepSkyBlue">(<b style="color:#FFDF00;text-shadow:#FFDF00 0 0 1.0em">✉</b>)</span>]]</span> 19:46, 21 May 2026 (UTC) == Vote now in the 2026 U4C election == <section begin="announcement-content" /> Eligible voters are asked to participate in the 2026 [[m:Special:MyLanguage/Universal_Code_of_Conduct/Coordinating_Committee|Universal Code of Conduct Coordinating Committee]] election. More information–including an eligibility check, voting process information, candidate information, and a link to the vote–are available on Meta at the [[m:Special:MyLanguage/Universal_Code_of_Conduct/Coordinating_Committee/Election/2026|2026 Election information page]]. The vote closes on 2 June 2026 at [https://zonestamp.toolforge.org/1780358400 00:00 UTC]. Please vote if your account is eligible. Results will be available by 14 June 2026. -- In cooperation with the U4C,<section end="announcement-content" /> [[m:User:Keegan (WMF)|Keegan (WMF)]] ([[m:User talk:Keegan (WMF)|talk]]) 17:15, 27 May 2026 (UTC) <!-- Message sent by User:Keegan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Global_message_delivery&oldid=30513860 --> == Create an autopatrolled user group? == {{tracked|T428269}} I would like to propose creating the user group <code>autopatrolled</code> (autopatrolled user), in which for non-curators and non-custodians, their page creations and file uploads would be automatically marked as patrolled by the MediaWiki software. Custodians may grant the user group, at their discretion, to users who create good quality pages that do not need frequent patrolling. On a side note, the term {{tq|autopatroller}} would be used, but because we don't have non-curator/custodian patrollers (as we rely on curators and custodians to patrol), I suggest on using the term {{tq|autopatrolled user}}. Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:31, 29 May 2026 (UTC) :'''Support''' re: the name, I don't really understand the reasoning, so I am '''neutral''' on that. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 15:45, 29 May 2026 (UTC) :: Regarding the name, this is because as we don't have the patroller user group, we rely on curators and custodians to patrol new pages and file uploads. Does that make sense? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:39, 29 May 2026 (UTC) :::Not really, but I don't think it's the most important thing. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 16:42, 29 May 2026 (UTC) :::: We'll decide on the name later. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 01:48, 30 May 2026 (UTC) :::::Oh, please don't let me stand in the way. I'm just not very smart, so don't hold up a matter on my account. I didn't want to derail the proposal, which is a fine and sensible one. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 04:16, 30 May 2026 (UTC) : '''Support''' - sounds like a good idea :* Suggest adding a draft section about this group to [[Wikiversity:Patrolling]]. There is a statement in the Introduction of the page that I'm not sure if its correct and at least could be improved: "Wikiversity also uses an autopatrol right, meaning trusted users' contributions are automatically marked as checked so patrollers can focus on reviewing newer or anonymous editors." :* Regarding autopatroller vs autropatrolled user, what terms are used on similar WMF wiki projects? : -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:28, 30 May 2026 (UTC) ::# I would create a starting page about the user groups, with experienced editors expanding the page. A summarized part of that page would also be added to [[Wikiversity:Patrolling]]. ::# For a similar example, English Wikipedia uses the term {{tq|Autopatrolled}}, just that term only. :: [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:22, 30 May 2026 (UTC) == How much of Wikiversity’s content is LLM slop? == Because it seems like a non-trivial amount, along with AI slop images as well. Is there some kind of AI cleanup project established yet? [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 01:20, 4 June 2026 (UTC) :We have discussed AI but I don't know of any explicit initiative to find and delete AI-generated noise. Individual modules have been deleted for having been made by AI. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 08:50, 4 June 2026 (UTC) == Draft inactivity policy == I created [[Wikiversity:Inactivity policy]] as a start. Any experienced Wikiversity user may feel free to expand it. This is also one-to-two step(s) towards opting out of the [[m:Admin activity review|AAR process]]. However, I made a bold change to reduce the response timeframe from one month to two weeks. In addition, should we reduce the inactivity timeframe to one year? For the latter, most projects use that timeframe and I suggested this for consistency. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:57, 4 June 2026 (UTC) :I support those suggestions. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 17:55, 4 June 2026 (UTC) == Proposed user group and/or possible policy changes == I want to discuss about user group and possible policy changes. # First, interface administrators. I don't think we should allow interface administrators to remove their permission from their own account, since we have multiple active bureaucrats and we can ask them to remove the permission when done, or for them to add a temporary grant. This is according to the [[Wikiversity:IA|current IA policy]]. I also left [[Wikiversity talk:Interface administrators#My thoughts about this user group|my thoughts on the relevant talk page]]. # Second, curators. Given that curators have some sensitive custodian rights (such as <code>delete</code> [but not <code>undelete</code> or similar rights that allow viewing deleted content, unless the curatorship process is RFA-like] and <code>protect</code>), it would probably make more sense only for bureaucrats to grant and remove it, on par with them granting (but not removing) custodian permissions. # Third, about probationary custodians. [[Wikiversity:Probationary custodians]] is currently marked as historical, and the process might still exist on [[Wikiversity:Custodianship]]. Therefore, to maintain consistency with [[Wikiversity:Curatorship#How does one become a curator?]], I propose that we repeal the probationary custodianship process and change it more or less to align with the curatorship process, effectively making probationary custodians permanent ones. However, custodian mentors would still be retained. Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 17:55, 5 June 2026 (UTC) fi07mm4ostnm7e4d3k1ny10w3koj18i 2812993 2812902 2026-06-05T22:43:41Z OhanaUnited 18921 /* Comments and questions */ Reply 2812993 wikitext text/x-wiki {{Wikiversity:Colloquium/Header}} <!-- MESSAGES GO BELOW --> == [[MediaWiki:Protectedpagetext#Protected edit request on 11 December 2025]] == I posted an edit request there 5 months ago, so I’ll be taking it to this page. [[Special:Contributions/&#126;2026-28640-56|&#126;2026-28640-56]] ([[User talk:&#126;2026-28640-56|talk]]) 23:33, 12 May 2026 (UTC) :What exactly is the problem? I don't understand what needs to change and why. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 23:35, 12 May 2026 (UTC) : Pinging @[[User:Atcovi|Atcovi]], @[[User:Jtneill|Jtneill]] and @[[User:Juandev|Juandev]] for further input. Someone is requesting a modification to [[MediaWiki:Protectedpagetext]] to use {{tlx|Protected page text}}, but we might need to discuss whether to use the template. In the meantime, I'll start a sandbox version of the protected page text template. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 23:19, 14 May 2026 (UTC) ::Sounds good -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 04:13, 15 May 2026 (UTC) :::+1 Jtneill. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 12:59, 19 May 2026 (UTC) == Proposal to rehost Wikinews here == As many of you know, and mentioned here at the Colloquium, our sister project Wikinews recently closed, with all 31 active editions made read-only. [[User:BigKrow]] has asked about the prospect of writing news stories here and I suggested that since we already have [[School:Journalism]] and some resources related to the [[:Category:Journalism|broader topic of journalism]]. I would like to propose that we have continued and indefinite space for {{w|citizen journalism}} by essentially repurposing Wikinews into a sub-project here. The only special infrastructure that Wikinews required was [[:mw:Extension:DynamicPageList]], which was deactivated and caused issues due to a lack of maintenance. I will add this proposal to the site banner, but I recognize that that may be a conflict of interest, so if anyone requests that I remove it, I will. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:30, 14 May 2026 (UTC) :I would like to see this conversation go for at least 30 days to establish a consensus. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:35, 14 May 2026 (UTC) ===Votes=== *{{support}} as proposer (with BK's inspiration). I think that an ongoing experiment in citizen journalism is a fit and appropriate use of this site. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:35, 14 May 2026 (UTC) *{{support}}, hope to seeing ideas about this, and thank you @[[User:Koavf|Koavf]] [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 11:08, 14 May 2026 (UTC) *{{support}} Other than perhaps inflating the total number of pages reported, I see the idea of "practicing journalism" a worthy and relevant activity within the domain of Wikiversity. [[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 21:41, 14 May 2026 (UTC) *{{support}} Conditional on development of (a) community guidelines that ensure alignment with Wikiversity's purpose, and (b) clear, nested page-naming structures for projects. More detail below. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:48, 15 May 2026 (UTC) *{{contra}} This proposal doesn't seem interested in expanding educational materials in journalism, but rather in providing space and protection for Wikinews contributors. But this is contrary to the goals of Wikiversity, and I'm not sure it's a good idea, even with regard to WMF. If WMF decides to close a project and another community lets it run on its domain, that's a bit of an undermining of WMF's and the community's decisions. Given that Wikiversity has had several conflicts with other communities and WMF in its history, I'm against it.--[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:59, 15 May 2026 (UTC) *{{contra}} This seems like a proposal to continue the mission of WikiNews, but not a proposal specifically to improve Wikiversity. I concur with Juandev's comments. --[[User:Mu301|mikeu]] <sup>[[User talk:Mu301|talk]]</sup> 20:29, 30 May 2026 (UTC) * {{oppose}} per above. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 19:05, 1 June 2026 (UTC) *{{oppose}} Wikiversity isn’t Wikinews and it also isn’t a dumping ground for anything not covered by other projects. It was already suggested, rather bafflingly, that Wikinews parasitize Wikipedia as a host. If it were allowed to freeload off of Wikiversity it would simply promote a view I and likely many others have— that Wikiversity (as it currently exists) has no standards and mostly just exists to host subpar content that wouldn’t be tolerated on any other Wikimedia site. Wikinews needs a new, non-Wikimedia host, and Wikiversity needs to get its act together by enforcing a minimum scope and standard for what it allows. --[[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 01:16, 4 June 2026 (UTC) ===Comments and questions=== :Definitely worthy of discussion, so I have no problem with the proposal in the sitenotice. :Initial questions: :* Does this proposal include importing English Wikinews content e.g., to [[Wikinews]] subpages? :* What are "active editions"? :* How can Wikiversity navigate the concerns that lead to the closure of Wikinews? :* Are any changes to the scope of Wikinews proposed? :* How does [[Wikinews]] fit with the [[Wikiversity:Mission]]? What aligns well? Where might there be tension? :** e.g., I'm not sure that a page like [[User:BigKrow/Manchester City moves two points behind Arsenal]] in and of itself will serve as an educational resource. :-- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 05:52, 14 May 2026 (UTC) :* Does this proposal include importing English Wikinews content e.g., to [[Wikinews]] subpages? ::*No, not at this time. :* What are "active editions"? ::*There were 30 other active editions of Wikinews in addition to English (e.g. [[:n:es:]]) at the time of universal closure (2026-05-04). :* How can Wikiversity navigate the concerns that lead to the closure of Wikinews? ::*One of the biggest issues was the problems with DPL, which is now irrelevant. Another was the lack of activity, which can be ameliorated by having it be part of an existing project instead of its own domain (e.g. some editions of Wikipedia host their own Wikinews already and those projects were not impacted by the closure). :* Are any changes to the scope of Wikinews proposed? ::*Not at this juncture. I would also propose as far as implemention goes that we would request a new namespace and that the material be more-or-less sequestered into its own ongoing project, like Wikijournal is or like the Cookbook and Wikijunior are at our sister [[:b:]]. :* How does [[Wikinews]] fit with the [[Wikiversity:Mission]]? What aligns well? Where might there be tension? :** e.g., I'm not sure that a page like [[Story/Manchester City moves two points behind Arsenal]] in and of itself will serve as an educational resource. ::*The process of citizen journalists practicing their craft in real-time and collaborating with others to do so is itself an education activity. We would essentially be hosting a real-time experiment in citizen journalism, online communities, and collaborative learning in addition to the prospect of spreading educational information from someone actually reading the news. I would propose that we could also make a more deliberate attempt to engage with learning <em>about</em> what does and doesn't work with collaborative news writing by experimentation (e.g. audio news, syndicating to other sites, incorporating freely-licensed news from other sources, writing hyper-local news, writing briefs versus longer-term reportage) and also seeing if the problems noted in the Task Force report that recommended closure can be overcome. Note that we have already done some local investigation about and learning about wiki-based journalism on Wikinews here at [[Journalism studies and Wikinews]]. We could continue that learning and refine the process, including incorporating journalism students from universities. As for tensions, Wikinews is the only sister project that must be done with a quick turn-around: if you take a long time to [[:s:|transcribe a book]], that's just how long it takes, but if you take a long time to write news, it ceases to be news entirely. Wikiversity has been a very slow-growing project that has definitely had some successes but has generally come together over a long period with most learning resources being individual passion projects (or sometimes, frankly, crankery) which would not work with collaborative news that requires more than just a single editor writing whatever he feels like. ::Please let me know any other questions/concerns and any other editors feel free to give your own perspective. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 06:13, 14 May 2026 (UTC) :::Thanks, Justin — it is food for thought. :::In attempting to understand how we've arrived here, I've summarised some of the background on this page: [[Wikinews]]. :::Perhaps it could be helpful to flesh out more of the vision / ideas / possibilities / challenges on that page? -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:49, 14 May 2026 (UTC) :::*Having given it some thought, in principle, I support hosting [[citizen journalism]] on Wikiversity where it is clearly connected to a learning project and/or constitutes original research, both of which align strongly with [[Wikiversity:Mission|Wikiversity’s educational mission]]. :::*My chief concern is the potential for news content that is not clearly linked to the purpose of Wikiversity. To avoid this, some community-agreed guidelines would be prudent. These need not be overly restrictive; they should support boldness and experimentation while helping ensure alignment with Wikiversity's purpose. :::*Given the reported low and declining activity on Wikinews, it seems unlikely that English Wikiversity would be overwhelmed by an influx of news-related editing. My impression is that English Wikinews was the most active edition, but even so, many contributors are likely to disperse to other projects or cease editing altogether. A modest migration of interested editors to Wikiversity seems manageable. :::*At this stage, I do not think a dedicated namespace is necessary. Subpages under [[Wikinews]] or nested pages under relevant learning or research projects, or user-space draft pages should be suitable. I agree that [[Wikijournal]] offers a useful model, as do several existing course structures on Wikiversity. :::*I support [[User:Koavf]]’s suggestions about framing Wikinews activity explicitly around learning. This would create a distinctive space for experimenting with collaborative news production in ways that are pedagogically meaningful. I agree that the [[journalism studies and Wikinews]] project developed by David and Leigh Blackall through the University of Wollongong is an excellent example of the intersection between Wikiversity and Wikinews. The [[Wikinews]] page could evolve into a hub for such projects. :::*I've tidied the [[:Category:Wikinews|Wikinews category]] and merged some content into the [[Wikinews]] page. As part of a reinvigoration effort, please review these and related resources such as [[:Category:Journalism]] and [[School:Journalism]]. :::*A further argument in favour of this initiative is that Wikipedia explicitly excludes both news reporting and original research. So, there is value in maintaining spaces within the Wikimedia ecosystem where these forms of knowledge production can be openly developed and curated. Such work can, in turn, generate valuable evidence and source material that may later inform Wikipedia articles. :::*The closure of WMF-hosted Wikinews does not imply that open wiki-based news curation lacks value. Indeed, the closure documentation appears supportive of experimentation with alternative news models across Wikimedia projects, including through Wikipedia and Wikidata. In that context, Wikiversity seems a natural home for a Wikinews experiment, provided it is clearly grounded in learning and/or research. :::-- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:39, 15 May 2026 (UTC) My understanding towards Wikinews' failure is that everything takes too long to be approved for the publish status, which means that any breaking news would have already become days-old stale news. Wikinews has a brand recognition (for right or wrong reasons) than Wikiversity and I wonder how effective Wikiversity can attract the "Wikinews refugees" to edit here. And just a quick note on the governance. Since each Wikiversity language operates independently, each language has to vote & adopt this proposal independently. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;"><sup>Talk page</sup></span></b>]] 13:47, 15 May 2026 (UTC) :Your assessment about Wikinews is partially correct. I referenced it earlier, but to be explicit, there is a [[:m:Proposal for Closing Wikinews|report by a task force on sister projects]] that outlines their concerns. There are a few, one of which was the nature of the staleness of news. Thanks also for clarifying that this proposal is only relevant to en.wv and is not binding or even proposed for other editions of Wikiversity. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 18:54, 15 May 2026 (UTC) *Note: I am not a regular here, and just visit Wikiversity for the WikiJournal project. Challenges of Wikinews included that it required timely reporting and fact-checking processes which differed greatly from the well-established ones in Wikipedia. Here in Wikiversity, there is the WikiJournal project, and that can take some some forms of journalism, just not breaking news reporting. I am in favor of salvaging parts of Wikinews if helpful. Could it, would it be feasible to adapt Wikijournal to accept some forms of news journalism, but just not the timed news reporting? For example, WikiJournal already is doing conference proceedings, and could likely do related event reports even months after the event ended. It could probably accept long-form investigative reporting, which is a sort of news that is not breaking news. I am not sure what the possibilities are, but I would prefer to build up systems that already work rather than import systems which had problems elsewhere. Thanks. [[User:Bluerasberry|<span style="background:#cedff2;color:#11e">''' Blue Rasberry '''</span>]][[User talk:Bluerasberry|<span style="cursor:help"><span style="background:#cedff2;color:#11e">(talk)</span></span>]] 19:17, 22 May 2026 (UTC) *:I agree that there are certain kinds of journalism that are perfectly valid and not time-bound like breaking news reporting, so that won't suffer from the issues noted before. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 21:15, 22 May 2026 (UTC) *::@[[User:Bluerasberry|Bluerasberry]] WikiJournal is not interested in taking on news journalism. WikiJournal is publishing conference proceedings at the request of some Wikimedian educators, and conference proceedings is what a "regular" journal publishes. News journalism is quite different from this, and if WikiJournal starts to deviate towards publishing news journalism, it will create barrier towards future initiatives like being indexed in Medline or Web of Science, and may risk being delisted from Scopus. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;"><sup>Talk page</sup></span></b>]] 22:43, 5 June 2026 (UTC) == [[Wikiversity:Deletion policy]] proposed as policy == {{archive top|Consensus to promote to an official policy. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 22:30, 1 June 2026 (UTC)}} [[Wikiversity:Deletions]] has been operating as a [[Wikiversity:Guidelines|guideline]]. It has been revised and moved to [[Wikiversity:Deletion policy]], consistent with naming conventions used across sister projects such as Wikipedia, Wikibooks, and Wikiquote. The speedy deletion criteria have also been updated for consistency with [[MediaWiki:Deletereason-dropdown]]. This proposal is for the page to be formally adopted as [[Wikiversity:Policies|Wikiversity policy]]. Community feedback is invited, including suggestions for further improvements that may strengthen the proposed policy. === Voting === *{{support}} Seems reasonable. If there's somehow something missed here, we can just amend it later. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:33, 18 May 2026 (UTC) *{{support}} I don't see any issues with the policy. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 16:07, 18 May 2026 (UTC) === Comments === {{archive bottom}} == May 2026 Wikimedia Café meetups regarding the Wikimedia Foundation Annual Plan == <div class="border-box" style="background-color: var(--background-color-warning-subtle, #f8eaba); max-width: 875px; padding: 5px; border: 1px solid black; margin: 5px; color: var(--clr-dark)"> <div class="box" style="float:left; padding-top: 15px; padding-right: 15px;">[[File:Wikimedia Café logo in plain SVG format.svg|75px|alt=The logo for the Wikimedia Café]]</div> Hello! There will be two '''[https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9 Wikimedia Café]''' discussion opportunities during the last weekend of May. Both sessions will focus on the [https://meta.wikimedia.org/wiki/Wikimedia_Foundation_Annual_Plan/2026-2027 the 2026-2027 Wikimedia Foundation Annual Plan]. Participants may attend either or both sessions. #'''Saturday, 30 May 2026 at 15:00 UTC''' ([https://zonestamp.toolforge.org/1780153200 timestamp converter]), at a time friendly to the Americas, Africa, and Europe #'''Sunday, 31 May 2026 at 05:00 UTC''' ([https://zonestamp.toolforge.org/1780203600 timestamp converter]), at a time friendly to Asia and the Pacific Café participants are highly encouraged to read in advance [https://en.wikipedia.org/wiki/User:Sohom_Datta/annual_plan_guide at least this summary of the plan]. Optionally, Café participants are encouraged to read portions of the plan that interest them and [https://meta.wikimedia.org/wiki/Talk:Wikimedia_Foundation_Annual_Plan/2026-2027 ask questions or provide feedback on the Annual Plan talk page]. Please see the Café page for more information, including [https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9#May_2026_meetings_with_a_focus_on_Wikimedia_Foundation_Annual_Plan/2026-2027 tables of timestamp conversions for both sessions], [https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9#Agenda._This_will_be_an_approximately_1_hour_Caf%C3%A9_session,_and_is_extendible_for_an_additional_30_minutes_if_needed. the agenda], and [https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9#How_to_attend_the_session how to register]! <br /> [[File:Buntstifte Eberhard Faber crop 64h.jpg|860px|alt=cropped image of colored pencils]]</div> <span style="white-space:nowrap;">[[User:Pine|<span style="color:#01796f; text-shadow:#00BFFF 0 0 1.0em">↠Pine</span>]] [[User talk:Pine|<span style="color:DeepSkyBlue">(<b style="color:#FFDF00;text-shadow:#FFDF00 0 0 1.0em">✉</b>)</span>]]</span> 19:46, 21 May 2026 (UTC) == Vote now in the 2026 U4C election == <section begin="announcement-content" /> Eligible voters are asked to participate in the 2026 [[m:Special:MyLanguage/Universal_Code_of_Conduct/Coordinating_Committee|Universal Code of Conduct Coordinating Committee]] election. More information–including an eligibility check, voting process information, candidate information, and a link to the vote–are available on Meta at the [[m:Special:MyLanguage/Universal_Code_of_Conduct/Coordinating_Committee/Election/2026|2026 Election information page]]. The vote closes on 2 June 2026 at [https://zonestamp.toolforge.org/1780358400 00:00 UTC]. Please vote if your account is eligible. Results will be available by 14 June 2026. -- In cooperation with the U4C,<section end="announcement-content" /> [[m:User:Keegan (WMF)|Keegan (WMF)]] ([[m:User talk:Keegan (WMF)|talk]]) 17:15, 27 May 2026 (UTC) <!-- Message sent by User:Keegan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Global_message_delivery&oldid=30513860 --> == Create an autopatrolled user group? == {{tracked|T428269}} I would like to propose creating the user group <code>autopatrolled</code> (autopatrolled user), in which for non-curators and non-custodians, their page creations and file uploads would be automatically marked as patrolled by the MediaWiki software. Custodians may grant the user group, at their discretion, to users who create good quality pages that do not need frequent patrolling. On a side note, the term {{tq|autopatroller}} would be used, but because we don't have non-curator/custodian patrollers (as we rely on curators and custodians to patrol), I suggest on using the term {{tq|autopatrolled user}}. Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:31, 29 May 2026 (UTC) :'''Support''' re: the name, I don't really understand the reasoning, so I am '''neutral''' on that. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 15:45, 29 May 2026 (UTC) :: Regarding the name, this is because as we don't have the patroller user group, we rely on curators and custodians to patrol new pages and file uploads. Does that make sense? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:39, 29 May 2026 (UTC) :::Not really, but I don't think it's the most important thing. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 16:42, 29 May 2026 (UTC) :::: We'll decide on the name later. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 01:48, 30 May 2026 (UTC) :::::Oh, please don't let me stand in the way. I'm just not very smart, so don't hold up a matter on my account. I didn't want to derail the proposal, which is a fine and sensible one. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 04:16, 30 May 2026 (UTC) : '''Support''' - sounds like a good idea :* Suggest adding a draft section about this group to [[Wikiversity:Patrolling]]. There is a statement in the Introduction of the page that I'm not sure if its correct and at least could be improved: "Wikiversity also uses an autopatrol right, meaning trusted users' contributions are automatically marked as checked so patrollers can focus on reviewing newer or anonymous editors." :* Regarding autopatroller vs autropatrolled user, what terms are used on similar WMF wiki projects? : -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:28, 30 May 2026 (UTC) ::# I would create a starting page about the user groups, with experienced editors expanding the page. A summarized part of that page would also be added to [[Wikiversity:Patrolling]]. ::# For a similar example, English Wikipedia uses the term {{tq|Autopatrolled}}, just that term only. :: [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:22, 30 May 2026 (UTC) == How much of Wikiversity’s content is LLM slop? == Because it seems like a non-trivial amount, along with AI slop images as well. Is there some kind of AI cleanup project established yet? [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 01:20, 4 June 2026 (UTC) :We have discussed AI but I don't know of any explicit initiative to find and delete AI-generated noise. Individual modules have been deleted for having been made by AI. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 08:50, 4 June 2026 (UTC) == Draft inactivity policy == I created [[Wikiversity:Inactivity policy]] as a start. Any experienced Wikiversity user may feel free to expand it. This is also one-to-two step(s) towards opting out of the [[m:Admin activity review|AAR process]]. However, I made a bold change to reduce the response timeframe from one month to two weeks. In addition, should we reduce the inactivity timeframe to one year? For the latter, most projects use that timeframe and I suggested this for consistency. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:57, 4 June 2026 (UTC) :I support those suggestions. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 17:55, 4 June 2026 (UTC) == Proposed user group and/or possible policy changes == I want to discuss about user group and possible policy changes. # First, interface administrators. I don't think we should allow interface administrators to remove their permission from their own account, since we have multiple active bureaucrats and we can ask them to remove the permission when done, or for them to add a temporary grant. This is according to the [[Wikiversity:IA|current IA policy]]. I also left [[Wikiversity talk:Interface administrators#My thoughts about this user group|my thoughts on the relevant talk page]]. # Second, curators. Given that curators have some sensitive custodian rights (such as <code>delete</code> [but not <code>undelete</code> or similar rights that allow viewing deleted content, unless the curatorship process is RFA-like] and <code>protect</code>), it would probably make more sense only for bureaucrats to grant and remove it, on par with them granting (but not removing) custodian permissions. # Third, about probationary custodians. [[Wikiversity:Probationary custodians]] is currently marked as historical, and the process might still exist on [[Wikiversity:Custodianship]]. Therefore, to maintain consistency with [[Wikiversity:Curatorship#How does one become a curator?]], I propose that we repeal the probationary custodianship process and change it more or less to align with the curatorship process, effectively making probationary custodians permanent ones. However, custodian mentors would still be retained. Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 17:55, 5 June 2026 (UTC) 6ktccp2tsdbaclo982xsxk98bop47ev 2813020 2812993 2026-06-06T01:20:42Z Chet cunningham 3089504 /* Goverment */ new section 2813020 wikitext text/x-wiki {{Wikiversity:Colloquium/Header}} <!-- MESSAGES GO BELOW --> == [[MediaWiki:Protectedpagetext#Protected edit request on 11 December 2025]] == I posted an edit request there 5 months ago, so I’ll be taking it to this page. [[Special:Contributions/&#126;2026-28640-56|&#126;2026-28640-56]] ([[User talk:&#126;2026-28640-56|talk]]) 23:33, 12 May 2026 (UTC) :What exactly is the problem? I don't understand what needs to change and why. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 23:35, 12 May 2026 (UTC) : Pinging @[[User:Atcovi|Atcovi]], @[[User:Jtneill|Jtneill]] and @[[User:Juandev|Juandev]] for further input. Someone is requesting a modification to [[MediaWiki:Protectedpagetext]] to use {{tlx|Protected page text}}, but we might need to discuss whether to use the template. In the meantime, I'll start a sandbox version of the protected page text template. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 23:19, 14 May 2026 (UTC) ::Sounds good -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 04:13, 15 May 2026 (UTC) :::+1 Jtneill. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 12:59, 19 May 2026 (UTC) == Proposal to rehost Wikinews here == As many of you know, and mentioned here at the Colloquium, our sister project Wikinews recently closed, with all 31 active editions made read-only. [[User:BigKrow]] has asked about the prospect of writing news stories here and I suggested that since we already have [[School:Journalism]] and some resources related to the [[:Category:Journalism|broader topic of journalism]]. I would like to propose that we have continued and indefinite space for {{w|citizen journalism}} by essentially repurposing Wikinews into a sub-project here. The only special infrastructure that Wikinews required was [[:mw:Extension:DynamicPageList]], which was deactivated and caused issues due to a lack of maintenance. I will add this proposal to the site banner, but I recognize that that may be a conflict of interest, so if anyone requests that I remove it, I will. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:30, 14 May 2026 (UTC) :I would like to see this conversation go for at least 30 days to establish a consensus. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:35, 14 May 2026 (UTC) ===Votes=== *{{support}} as proposer (with BK's inspiration). I think that an ongoing experiment in citizen journalism is a fit and appropriate use of this site. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:35, 14 May 2026 (UTC) *{{support}}, hope to seeing ideas about this, and thank you @[[User:Koavf|Koavf]] [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 11:08, 14 May 2026 (UTC) *{{support}} Other than perhaps inflating the total number of pages reported, I see the idea of "practicing journalism" a worthy and relevant activity within the domain of Wikiversity. [[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 21:41, 14 May 2026 (UTC) *{{support}} Conditional on development of (a) community guidelines that ensure alignment with Wikiversity's purpose, and (b) clear, nested page-naming structures for projects. More detail below. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:48, 15 May 2026 (UTC) *{{contra}} This proposal doesn't seem interested in expanding educational materials in journalism, but rather in providing space and protection for Wikinews contributors. But this is contrary to the goals of Wikiversity, and I'm not sure it's a good idea, even with regard to WMF. If WMF decides to close a project and another community lets it run on its domain, that's a bit of an undermining of WMF's and the community's decisions. Given that Wikiversity has had several conflicts with other communities and WMF in its history, I'm against it.--[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:59, 15 May 2026 (UTC) *{{contra}} This seems like a proposal to continue the mission of WikiNews, but not a proposal specifically to improve Wikiversity. I concur with Juandev's comments. --[[User:Mu301|mikeu]] <sup>[[User talk:Mu301|talk]]</sup> 20:29, 30 May 2026 (UTC) * {{oppose}} per above. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 19:05, 1 June 2026 (UTC) *{{oppose}} Wikiversity isn’t Wikinews and it also isn’t a dumping ground for anything not covered by other projects. It was already suggested, rather bafflingly, that Wikinews parasitize Wikipedia as a host. If it were allowed to freeload off of Wikiversity it would simply promote a view I and likely many others have— that Wikiversity (as it currently exists) has no standards and mostly just exists to host subpar content that wouldn’t be tolerated on any other Wikimedia site. Wikinews needs a new, non-Wikimedia host, and Wikiversity needs to get its act together by enforcing a minimum scope and standard for what it allows. --[[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 01:16, 4 June 2026 (UTC) ===Comments and questions=== :Definitely worthy of discussion, so I have no problem with the proposal in the sitenotice. :Initial questions: :* Does this proposal include importing English Wikinews content e.g., to [[Wikinews]] subpages? :* What are "active editions"? :* How can Wikiversity navigate the concerns that lead to the closure of Wikinews? :* Are any changes to the scope of Wikinews proposed? :* How does [[Wikinews]] fit with the [[Wikiversity:Mission]]? What aligns well? Where might there be tension? :** e.g., I'm not sure that a page like [[User:BigKrow/Manchester City moves two points behind Arsenal]] in and of itself will serve as an educational resource. :-- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 05:52, 14 May 2026 (UTC) :* Does this proposal include importing English Wikinews content e.g., to [[Wikinews]] subpages? ::*No, not at this time. :* What are "active editions"? ::*There were 30 other active editions of Wikinews in addition to English (e.g. [[:n:es:]]) at the time of universal closure (2026-05-04). :* How can Wikiversity navigate the concerns that lead to the closure of Wikinews? ::*One of the biggest issues was the problems with DPL, which is now irrelevant. Another was the lack of activity, which can be ameliorated by having it be part of an existing project instead of its own domain (e.g. some editions of Wikipedia host their own Wikinews already and those projects were not impacted by the closure). :* Are any changes to the scope of Wikinews proposed? ::*Not at this juncture. I would also propose as far as implemention goes that we would request a new namespace and that the material be more-or-less sequestered into its own ongoing project, like Wikijournal is or like the Cookbook and Wikijunior are at our sister [[:b:]]. :* How does [[Wikinews]] fit with the [[Wikiversity:Mission]]? What aligns well? Where might there be tension? :** e.g., I'm not sure that a page like [[Story/Manchester City moves two points behind Arsenal]] in and of itself will serve as an educational resource. ::*The process of citizen journalists practicing their craft in real-time and collaborating with others to do so is itself an education activity. We would essentially be hosting a real-time experiment in citizen journalism, online communities, and collaborative learning in addition to the prospect of spreading educational information from someone actually reading the news. I would propose that we could also make a more deliberate attempt to engage with learning <em>about</em> what does and doesn't work with collaborative news writing by experimentation (e.g. audio news, syndicating to other sites, incorporating freely-licensed news from other sources, writing hyper-local news, writing briefs versus longer-term reportage) and also seeing if the problems noted in the Task Force report that recommended closure can be overcome. Note that we have already done some local investigation about and learning about wiki-based journalism on Wikinews here at [[Journalism studies and Wikinews]]. We could continue that learning and refine the process, including incorporating journalism students from universities. As for tensions, Wikinews is the only sister project that must be done with a quick turn-around: if you take a long time to [[:s:|transcribe a book]], that's just how long it takes, but if you take a long time to write news, it ceases to be news entirely. Wikiversity has been a very slow-growing project that has definitely had some successes but has generally come together over a long period with most learning resources being individual passion projects (or sometimes, frankly, crankery) which would not work with collaborative news that requires more than just a single editor writing whatever he feels like. ::Please let me know any other questions/concerns and any other editors feel free to give your own perspective. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 06:13, 14 May 2026 (UTC) :::Thanks, Justin — it is food for thought. :::In attempting to understand how we've arrived here, I've summarised some of the background on this page: [[Wikinews]]. :::Perhaps it could be helpful to flesh out more of the vision / ideas / possibilities / challenges on that page? -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:49, 14 May 2026 (UTC) :::*Having given it some thought, in principle, I support hosting [[citizen journalism]] on Wikiversity where it is clearly connected to a learning project and/or constitutes original research, both of which align strongly with [[Wikiversity:Mission|Wikiversity’s educational mission]]. :::*My chief concern is the potential for news content that is not clearly linked to the purpose of Wikiversity. To avoid this, some community-agreed guidelines would be prudent. These need not be overly restrictive; they should support boldness and experimentation while helping ensure alignment with Wikiversity's purpose. :::*Given the reported low and declining activity on Wikinews, it seems unlikely that English Wikiversity would be overwhelmed by an influx of news-related editing. My impression is that English Wikinews was the most active edition, but even so, many contributors are likely to disperse to other projects or cease editing altogether. A modest migration of interested editors to Wikiversity seems manageable. :::*At this stage, I do not think a dedicated namespace is necessary. Subpages under [[Wikinews]] or nested pages under relevant learning or research projects, or user-space draft pages should be suitable. I agree that [[Wikijournal]] offers a useful model, as do several existing course structures on Wikiversity. :::*I support [[User:Koavf]]’s suggestions about framing Wikinews activity explicitly around learning. This would create a distinctive space for experimenting with collaborative news production in ways that are pedagogically meaningful. I agree that the [[journalism studies and Wikinews]] project developed by David and Leigh Blackall through the University of Wollongong is an excellent example of the intersection between Wikiversity and Wikinews. The [[Wikinews]] page could evolve into a hub for such projects. :::*I've tidied the [[:Category:Wikinews|Wikinews category]] and merged some content into the [[Wikinews]] page. As part of a reinvigoration effort, please review these and related resources such as [[:Category:Journalism]] and [[School:Journalism]]. :::*A further argument in favour of this initiative is that Wikipedia explicitly excludes both news reporting and original research. So, there is value in maintaining spaces within the Wikimedia ecosystem where these forms of knowledge production can be openly developed and curated. Such work can, in turn, generate valuable evidence and source material that may later inform Wikipedia articles. :::*The closure of WMF-hosted Wikinews does not imply that open wiki-based news curation lacks value. Indeed, the closure documentation appears supportive of experimentation with alternative news models across Wikimedia projects, including through Wikipedia and Wikidata. In that context, Wikiversity seems a natural home for a Wikinews experiment, provided it is clearly grounded in learning and/or research. :::-- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:39, 15 May 2026 (UTC) My understanding towards Wikinews' failure is that everything takes too long to be approved for the publish status, which means that any breaking news would have already become days-old stale news. Wikinews has a brand recognition (for right or wrong reasons) than Wikiversity and I wonder how effective Wikiversity can attract the "Wikinews refugees" to edit here. And just a quick note on the governance. Since each Wikiversity language operates independently, each language has to vote & adopt this proposal independently. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;"><sup>Talk page</sup></span></b>]] 13:47, 15 May 2026 (UTC) :Your assessment about Wikinews is partially correct. I referenced it earlier, but to be explicit, there is a [[:m:Proposal for Closing Wikinews|report by a task force on sister projects]] that outlines their concerns. There are a few, one of which was the nature of the staleness of news. Thanks also for clarifying that this proposal is only relevant to en.wv and is not binding or even proposed for other editions of Wikiversity. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 18:54, 15 May 2026 (UTC) *Note: I am not a regular here, and just visit Wikiversity for the WikiJournal project. Challenges of Wikinews included that it required timely reporting and fact-checking processes which differed greatly from the well-established ones in Wikipedia. Here in Wikiversity, there is the WikiJournal project, and that can take some some forms of journalism, just not breaking news reporting. I am in favor of salvaging parts of Wikinews if helpful. Could it, would it be feasible to adapt Wikijournal to accept some forms of news journalism, but just not the timed news reporting? For example, WikiJournal already is doing conference proceedings, and could likely do related event reports even months after the event ended. It could probably accept long-form investigative reporting, which is a sort of news that is not breaking news. I am not sure what the possibilities are, but I would prefer to build up systems that already work rather than import systems which had problems elsewhere. Thanks. [[User:Bluerasberry|<span style="background:#cedff2;color:#11e">''' Blue Rasberry '''</span>]][[User talk:Bluerasberry|<span style="cursor:help"><span style="background:#cedff2;color:#11e">(talk)</span></span>]] 19:17, 22 May 2026 (UTC) *:I agree that there are certain kinds of journalism that are perfectly valid and not time-bound like breaking news reporting, so that won't suffer from the issues noted before. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 21:15, 22 May 2026 (UTC) *::@[[User:Bluerasberry|Bluerasberry]] WikiJournal is not interested in taking on news journalism. WikiJournal is publishing conference proceedings at the request of some Wikimedian educators, and conference proceedings is what a "regular" journal publishes. News journalism is quite different from this, and if WikiJournal starts to deviate towards publishing news journalism, it will create barrier towards future initiatives like being indexed in Medline or Web of Science, and may risk being delisted from Scopus. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;"><sup>Talk page</sup></span></b>]] 22:43, 5 June 2026 (UTC) == [[Wikiversity:Deletion policy]] proposed as policy == {{archive top|Consensus to promote to an official policy. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 22:30, 1 June 2026 (UTC)}} [[Wikiversity:Deletions]] has been operating as a [[Wikiversity:Guidelines|guideline]]. It has been revised and moved to [[Wikiversity:Deletion policy]], consistent with naming conventions used across sister projects such as Wikipedia, Wikibooks, and Wikiquote. The speedy deletion criteria have also been updated for consistency with [[MediaWiki:Deletereason-dropdown]]. This proposal is for the page to be formally adopted as [[Wikiversity:Policies|Wikiversity policy]]. Community feedback is invited, including suggestions for further improvements that may strengthen the proposed policy. === Voting === *{{support}} Seems reasonable. If there's somehow something missed here, we can just amend it later. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:33, 18 May 2026 (UTC) *{{support}} I don't see any issues with the policy. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 16:07, 18 May 2026 (UTC) === Comments === {{archive bottom}} == May 2026 Wikimedia Café meetups regarding the Wikimedia Foundation Annual Plan == <div class="border-box" style="background-color: var(--background-color-warning-subtle, #f8eaba); max-width: 875px; padding: 5px; border: 1px solid black; margin: 5px; color: var(--clr-dark)"> <div class="box" style="float:left; padding-top: 15px; padding-right: 15px;">[[File:Wikimedia Café logo in plain SVG format.svg|75px|alt=The logo for the Wikimedia Café]]</div> Hello! There will be two '''[https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9 Wikimedia Café]''' discussion opportunities during the last weekend of May. Both sessions will focus on the [https://meta.wikimedia.org/wiki/Wikimedia_Foundation_Annual_Plan/2026-2027 the 2026-2027 Wikimedia Foundation Annual Plan]. Participants may attend either or both sessions. #'''Saturday, 30 May 2026 at 15:00 UTC''' ([https://zonestamp.toolforge.org/1780153200 timestamp converter]), at a time friendly to the Americas, Africa, and Europe #'''Sunday, 31 May 2026 at 05:00 UTC''' ([https://zonestamp.toolforge.org/1780203600 timestamp converter]), at a time friendly to Asia and the Pacific Café participants are highly encouraged to read in advance [https://en.wikipedia.org/wiki/User:Sohom_Datta/annual_plan_guide at least this summary of the plan]. Optionally, Café participants are encouraged to read portions of the plan that interest them and [https://meta.wikimedia.org/wiki/Talk:Wikimedia_Foundation_Annual_Plan/2026-2027 ask questions or provide feedback on the Annual Plan talk page]. Please see the Café page for more information, including [https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9#May_2026_meetings_with_a_focus_on_Wikimedia_Foundation_Annual_Plan/2026-2027 tables of timestamp conversions for both sessions], [https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9#Agenda._This_will_be_an_approximately_1_hour_Caf%C3%A9_session,_and_is_extendible_for_an_additional_30_minutes_if_needed. the agenda], and [https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9#How_to_attend_the_session how to register]! <br /> [[File:Buntstifte Eberhard Faber crop 64h.jpg|860px|alt=cropped image of colored pencils]]</div> <span style="white-space:nowrap;">[[User:Pine|<span style="color:#01796f; text-shadow:#00BFFF 0 0 1.0em">↠Pine</span>]] [[User talk:Pine|<span style="color:DeepSkyBlue">(<b style="color:#FFDF00;text-shadow:#FFDF00 0 0 1.0em">✉</b>)</span>]]</span> 19:46, 21 May 2026 (UTC) == Vote now in the 2026 U4C election == <section begin="announcement-content" /> Eligible voters are asked to participate in the 2026 [[m:Special:MyLanguage/Universal_Code_of_Conduct/Coordinating_Committee|Universal Code of Conduct Coordinating Committee]] election. More information–including an eligibility check, voting process information, candidate information, and a link to the vote–are available on Meta at the [[m:Special:MyLanguage/Universal_Code_of_Conduct/Coordinating_Committee/Election/2026|2026 Election information page]]. The vote closes on 2 June 2026 at [https://zonestamp.toolforge.org/1780358400 00:00 UTC]. Please vote if your account is eligible. Results will be available by 14 June 2026. -- In cooperation with the U4C,<section end="announcement-content" /> [[m:User:Keegan (WMF)|Keegan (WMF)]] ([[m:User talk:Keegan (WMF)|talk]]) 17:15, 27 May 2026 (UTC) <!-- Message sent by User:Keegan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Global_message_delivery&oldid=30513860 --> == Create an autopatrolled user group? == {{tracked|T428269}} I would like to propose creating the user group <code>autopatrolled</code> (autopatrolled user), in which for non-curators and non-custodians, their page creations and file uploads would be automatically marked as patrolled by the MediaWiki software. Custodians may grant the user group, at their discretion, to users who create good quality pages that do not need frequent patrolling. On a side note, the term {{tq|autopatroller}} would be used, but because we don't have non-curator/custodian patrollers (as we rely on curators and custodians to patrol), I suggest on using the term {{tq|autopatrolled user}}. Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:31, 29 May 2026 (UTC) :'''Support''' re: the name, I don't really understand the reasoning, so I am '''neutral''' on that. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 15:45, 29 May 2026 (UTC) :: Regarding the name, this is because as we don't have the patroller user group, we rely on curators and custodians to patrol new pages and file uploads. Does that make sense? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:39, 29 May 2026 (UTC) :::Not really, but I don't think it's the most important thing. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 16:42, 29 May 2026 (UTC) :::: We'll decide on the name later. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 01:48, 30 May 2026 (UTC) :::::Oh, please don't let me stand in the way. I'm just not very smart, so don't hold up a matter on my account. I didn't want to derail the proposal, which is a fine and sensible one. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 04:16, 30 May 2026 (UTC) : '''Support''' - sounds like a good idea :* Suggest adding a draft section about this group to [[Wikiversity:Patrolling]]. There is a statement in the Introduction of the page that I'm not sure if its correct and at least could be improved: "Wikiversity also uses an autopatrol right, meaning trusted users' contributions are automatically marked as checked so patrollers can focus on reviewing newer or anonymous editors." :* Regarding autopatroller vs autropatrolled user, what terms are used on similar WMF wiki projects? : -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:28, 30 May 2026 (UTC) ::# I would create a starting page about the user groups, with experienced editors expanding the page. A summarized part of that page would also be added to [[Wikiversity:Patrolling]]. ::# For a similar example, English Wikipedia uses the term {{tq|Autopatrolled}}, just that term only. :: [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:22, 30 May 2026 (UTC) == How much of Wikiversity’s content is LLM slop? == Because it seems like a non-trivial amount, along with AI slop images as well. Is there some kind of AI cleanup project established yet? [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 01:20, 4 June 2026 (UTC) :We have discussed AI but I don't know of any explicit initiative to find and delete AI-generated noise. Individual modules have been deleted for having been made by AI. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 08:50, 4 June 2026 (UTC) == Draft inactivity policy == I created [[Wikiversity:Inactivity policy]] as a start. Any experienced Wikiversity user may feel free to expand it. This is also one-to-two step(s) towards opting out of the [[m:Admin activity review|AAR process]]. However, I made a bold change to reduce the response timeframe from one month to two weeks. In addition, should we reduce the inactivity timeframe to one year? For the latter, most projects use that timeframe and I suggested this for consistency. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:57, 4 June 2026 (UTC) :I support those suggestions. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 17:55, 4 June 2026 (UTC) == Proposed user group and/or possible policy changes == I want to discuss about user group and possible policy changes. # First, interface administrators. I don't think we should allow interface administrators to remove their permission from their own account, since we have multiple active bureaucrats and we can ask them to remove the permission when done, or for them to add a temporary grant. This is according to the [[Wikiversity:IA|current IA policy]]. I also left [[Wikiversity talk:Interface administrators#My thoughts about this user group|my thoughts on the relevant talk page]]. # Second, curators. Given that curators have some sensitive custodian rights (such as <code>delete</code> [but not <code>undelete</code> or similar rights that allow viewing deleted content, unless the curatorship process is RFA-like] and <code>protect</code>), it would probably make more sense only for bureaucrats to grant and remove it, on par with them granting (but not removing) custodian permissions. # Third, about probationary custodians. [[Wikiversity:Probationary custodians]] is currently marked as historical, and the process might still exist on [[Wikiversity:Custodianship]]. Therefore, to maintain consistency with [[Wikiversity:Curatorship#How does one become a curator?]], I propose that we repeal the probationary custodianship process and change it more or less to align with the curatorship process, effectively making probationary custodians permanent ones. However, custodian mentors would still be retained. Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 17:55, 5 June 2026 (UTC) == Goverment == does anyone not like TRUMP!?!? 8fwkssr6i75r43q9ah39ldr6bpfggsj 2813021 2813020 2026-06-06T01:31:18Z Codename Noreste 2969951 Reverted edit by [[Special:Contributions/Chet cunningham|Chet cunningham]] ([[User_talk:Chet cunningham|talk]]) to last version by [[User:OhanaUnited|OhanaUnited]] using [[Wikiversity:Rollback|rollback]] 2812993 wikitext text/x-wiki {{Wikiversity:Colloquium/Header}} <!-- MESSAGES GO BELOW --> == [[MediaWiki:Protectedpagetext#Protected edit request on 11 December 2025]] == I posted an edit request there 5 months ago, so I’ll be taking it to this page. [[Special:Contributions/&#126;2026-28640-56|&#126;2026-28640-56]] ([[User talk:&#126;2026-28640-56|talk]]) 23:33, 12 May 2026 (UTC) :What exactly is the problem? I don't understand what needs to change and why. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 23:35, 12 May 2026 (UTC) : Pinging @[[User:Atcovi|Atcovi]], @[[User:Jtneill|Jtneill]] and @[[User:Juandev|Juandev]] for further input. Someone is requesting a modification to [[MediaWiki:Protectedpagetext]] to use {{tlx|Protected page text}}, but we might need to discuss whether to use the template. In the meantime, I'll start a sandbox version of the protected page text template. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 23:19, 14 May 2026 (UTC) ::Sounds good -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 04:13, 15 May 2026 (UTC) :::+1 Jtneill. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 12:59, 19 May 2026 (UTC) == Proposal to rehost Wikinews here == As many of you know, and mentioned here at the Colloquium, our sister project Wikinews recently closed, with all 31 active editions made read-only. [[User:BigKrow]] has asked about the prospect of writing news stories here and I suggested that since we already have [[School:Journalism]] and some resources related to the [[:Category:Journalism|broader topic of journalism]]. I would like to propose that we have continued and indefinite space for {{w|citizen journalism}} by essentially repurposing Wikinews into a sub-project here. The only special infrastructure that Wikinews required was [[:mw:Extension:DynamicPageList]], which was deactivated and caused issues due to a lack of maintenance. I will add this proposal to the site banner, but I recognize that that may be a conflict of interest, so if anyone requests that I remove it, I will. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:30, 14 May 2026 (UTC) :I would like to see this conversation go for at least 30 days to establish a consensus. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:35, 14 May 2026 (UTC) ===Votes=== *{{support}} as proposer (with BK's inspiration). I think that an ongoing experiment in citizen journalism is a fit and appropriate use of this site. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:35, 14 May 2026 (UTC) *{{support}}, hope to seeing ideas about this, and thank you @[[User:Koavf|Koavf]] [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 11:08, 14 May 2026 (UTC) *{{support}} Other than perhaps inflating the total number of pages reported, I see the idea of "practicing journalism" a worthy and relevant activity within the domain of Wikiversity. [[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 21:41, 14 May 2026 (UTC) *{{support}} Conditional on development of (a) community guidelines that ensure alignment with Wikiversity's purpose, and (b) clear, nested page-naming structures for projects. More detail below. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:48, 15 May 2026 (UTC) *{{contra}} This proposal doesn't seem interested in expanding educational materials in journalism, but rather in providing space and protection for Wikinews contributors. But this is contrary to the goals of Wikiversity, and I'm not sure it's a good idea, even with regard to WMF. If WMF decides to close a project and another community lets it run on its domain, that's a bit of an undermining of WMF's and the community's decisions. Given that Wikiversity has had several conflicts with other communities and WMF in its history, I'm against it.--[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:59, 15 May 2026 (UTC) *{{contra}} This seems like a proposal to continue the mission of WikiNews, but not a proposal specifically to improve Wikiversity. I concur with Juandev's comments. --[[User:Mu301|mikeu]] <sup>[[User talk:Mu301|talk]]</sup> 20:29, 30 May 2026 (UTC) * {{oppose}} per above. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 19:05, 1 June 2026 (UTC) *{{oppose}} Wikiversity isn’t Wikinews and it also isn’t a dumping ground for anything not covered by other projects. It was already suggested, rather bafflingly, that Wikinews parasitize Wikipedia as a host. If it were allowed to freeload off of Wikiversity it would simply promote a view I and likely many others have— that Wikiversity (as it currently exists) has no standards and mostly just exists to host subpar content that wouldn’t be tolerated on any other Wikimedia site. Wikinews needs a new, non-Wikimedia host, and Wikiversity needs to get its act together by enforcing a minimum scope and standard for what it allows. --[[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 01:16, 4 June 2026 (UTC) ===Comments and questions=== :Definitely worthy of discussion, so I have no problem with the proposal in the sitenotice. :Initial questions: :* Does this proposal include importing English Wikinews content e.g., to [[Wikinews]] subpages? :* What are "active editions"? :* How can Wikiversity navigate the concerns that lead to the closure of Wikinews? :* Are any changes to the scope of Wikinews proposed? :* How does [[Wikinews]] fit with the [[Wikiversity:Mission]]? What aligns well? Where might there be tension? :** e.g., I'm not sure that a page like [[User:BigKrow/Manchester City moves two points behind Arsenal]] in and of itself will serve as an educational resource. :-- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 05:52, 14 May 2026 (UTC) :* Does this proposal include importing English Wikinews content e.g., to [[Wikinews]] subpages? ::*No, not at this time. :* What are "active editions"? ::*There were 30 other active editions of Wikinews in addition to English (e.g. [[:n:es:]]) at the time of universal closure (2026-05-04). :* How can Wikiversity navigate the concerns that lead to the closure of Wikinews? ::*One of the biggest issues was the problems with DPL, which is now irrelevant. Another was the lack of activity, which can be ameliorated by having it be part of an existing project instead of its own domain (e.g. some editions of Wikipedia host their own Wikinews already and those projects were not impacted by the closure). :* Are any changes to the scope of Wikinews proposed? ::*Not at this juncture. I would also propose as far as implemention goes that we would request a new namespace and that the material be more-or-less sequestered into its own ongoing project, like Wikijournal is or like the Cookbook and Wikijunior are at our sister [[:b:]]. :* How does [[Wikinews]] fit with the [[Wikiversity:Mission]]? What aligns well? Where might there be tension? :** e.g., I'm not sure that a page like [[Story/Manchester City moves two points behind Arsenal]] in and of itself will serve as an educational resource. ::*The process of citizen journalists practicing their craft in real-time and collaborating with others to do so is itself an education activity. We would essentially be hosting a real-time experiment in citizen journalism, online communities, and collaborative learning in addition to the prospect of spreading educational information from someone actually reading the news. I would propose that we could also make a more deliberate attempt to engage with learning <em>about</em> what does and doesn't work with collaborative news writing by experimentation (e.g. audio news, syndicating to other sites, incorporating freely-licensed news from other sources, writing hyper-local news, writing briefs versus longer-term reportage) and also seeing if the problems noted in the Task Force report that recommended closure can be overcome. Note that we have already done some local investigation about and learning about wiki-based journalism on Wikinews here at [[Journalism studies and Wikinews]]. We could continue that learning and refine the process, including incorporating journalism students from universities. As for tensions, Wikinews is the only sister project that must be done with a quick turn-around: if you take a long time to [[:s:|transcribe a book]], that's just how long it takes, but if you take a long time to write news, it ceases to be news entirely. Wikiversity has been a very slow-growing project that has definitely had some successes but has generally come together over a long period with most learning resources being individual passion projects (or sometimes, frankly, crankery) which would not work with collaborative news that requires more than just a single editor writing whatever he feels like. ::Please let me know any other questions/concerns and any other editors feel free to give your own perspective. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 06:13, 14 May 2026 (UTC) :::Thanks, Justin — it is food for thought. :::In attempting to understand how we've arrived here, I've summarised some of the background on this page: [[Wikinews]]. :::Perhaps it could be helpful to flesh out more of the vision / ideas / possibilities / challenges on that page? -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:49, 14 May 2026 (UTC) :::*Having given it some thought, in principle, I support hosting [[citizen journalism]] on Wikiversity where it is clearly connected to a learning project and/or constitutes original research, both of which align strongly with [[Wikiversity:Mission|Wikiversity’s educational mission]]. :::*My chief concern is the potential for news content that is not clearly linked to the purpose of Wikiversity. To avoid this, some community-agreed guidelines would be prudent. These need not be overly restrictive; they should support boldness and experimentation while helping ensure alignment with Wikiversity's purpose. :::*Given the reported low and declining activity on Wikinews, it seems unlikely that English Wikiversity would be overwhelmed by an influx of news-related editing. My impression is that English Wikinews was the most active edition, but even so, many contributors are likely to disperse to other projects or cease editing altogether. A modest migration of interested editors to Wikiversity seems manageable. :::*At this stage, I do not think a dedicated namespace is necessary. Subpages under [[Wikinews]] or nested pages under relevant learning or research projects, or user-space draft pages should be suitable. I agree that [[Wikijournal]] offers a useful model, as do several existing course structures on Wikiversity. :::*I support [[User:Koavf]]’s suggestions about framing Wikinews activity explicitly around learning. This would create a distinctive space for experimenting with collaborative news production in ways that are pedagogically meaningful. I agree that the [[journalism studies and Wikinews]] project developed by David and Leigh Blackall through the University of Wollongong is an excellent example of the intersection between Wikiversity and Wikinews. The [[Wikinews]] page could evolve into a hub for such projects. :::*I've tidied the [[:Category:Wikinews|Wikinews category]] and merged some content into the [[Wikinews]] page. As part of a reinvigoration effort, please review these and related resources such as [[:Category:Journalism]] and [[School:Journalism]]. :::*A further argument in favour of this initiative is that Wikipedia explicitly excludes both news reporting and original research. So, there is value in maintaining spaces within the Wikimedia ecosystem where these forms of knowledge production can be openly developed and curated. Such work can, in turn, generate valuable evidence and source material that may later inform Wikipedia articles. :::*The closure of WMF-hosted Wikinews does not imply that open wiki-based news curation lacks value. Indeed, the closure documentation appears supportive of experimentation with alternative news models across Wikimedia projects, including through Wikipedia and Wikidata. In that context, Wikiversity seems a natural home for a Wikinews experiment, provided it is clearly grounded in learning and/or research. :::-- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:39, 15 May 2026 (UTC) My understanding towards Wikinews' failure is that everything takes too long to be approved for the publish status, which means that any breaking news would have already become days-old stale news. Wikinews has a brand recognition (for right or wrong reasons) than Wikiversity and I wonder how effective Wikiversity can attract the "Wikinews refugees" to edit here. And just a quick note on the governance. Since each Wikiversity language operates independently, each language has to vote & adopt this proposal independently. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;"><sup>Talk page</sup></span></b>]] 13:47, 15 May 2026 (UTC) :Your assessment about Wikinews is partially correct. I referenced it earlier, but to be explicit, there is a [[:m:Proposal for Closing Wikinews|report by a task force on sister projects]] that outlines their concerns. There are a few, one of which was the nature of the staleness of news. Thanks also for clarifying that this proposal is only relevant to en.wv and is not binding or even proposed for other editions of Wikiversity. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 18:54, 15 May 2026 (UTC) *Note: I am not a regular here, and just visit Wikiversity for the WikiJournal project. Challenges of Wikinews included that it required timely reporting and fact-checking processes which differed greatly from the well-established ones in Wikipedia. Here in Wikiversity, there is the WikiJournal project, and that can take some some forms of journalism, just not breaking news reporting. I am in favor of salvaging parts of Wikinews if helpful. Could it, would it be feasible to adapt Wikijournal to accept some forms of news journalism, but just not the timed news reporting? For example, WikiJournal already is doing conference proceedings, and could likely do related event reports even months after the event ended. It could probably accept long-form investigative reporting, which is a sort of news that is not breaking news. I am not sure what the possibilities are, but I would prefer to build up systems that already work rather than import systems which had problems elsewhere. Thanks. [[User:Bluerasberry|<span style="background:#cedff2;color:#11e">''' Blue Rasberry '''</span>]][[User talk:Bluerasberry|<span style="cursor:help"><span style="background:#cedff2;color:#11e">(talk)</span></span>]] 19:17, 22 May 2026 (UTC) *:I agree that there are certain kinds of journalism that are perfectly valid and not time-bound like breaking news reporting, so that won't suffer from the issues noted before. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 21:15, 22 May 2026 (UTC) *::@[[User:Bluerasberry|Bluerasberry]] WikiJournal is not interested in taking on news journalism. WikiJournal is publishing conference proceedings at the request of some Wikimedian educators, and conference proceedings is what a "regular" journal publishes. News journalism is quite different from this, and if WikiJournal starts to deviate towards publishing news journalism, it will create barrier towards future initiatives like being indexed in Medline or Web of Science, and may risk being delisted from Scopus. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;"><sup>Talk page</sup></span></b>]] 22:43, 5 June 2026 (UTC) == [[Wikiversity:Deletion policy]] proposed as policy == {{archive top|Consensus to promote to an official policy. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 22:30, 1 June 2026 (UTC)}} [[Wikiversity:Deletions]] has been operating as a [[Wikiversity:Guidelines|guideline]]. It has been revised and moved to [[Wikiversity:Deletion policy]], consistent with naming conventions used across sister projects such as Wikipedia, Wikibooks, and Wikiquote. The speedy deletion criteria have also been updated for consistency with [[MediaWiki:Deletereason-dropdown]]. This proposal is for the page to be formally adopted as [[Wikiversity:Policies|Wikiversity policy]]. Community feedback is invited, including suggestions for further improvements that may strengthen the proposed policy. === Voting === *{{support}} Seems reasonable. If there's somehow something missed here, we can just amend it later. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:33, 18 May 2026 (UTC) *{{support}} I don't see any issues with the policy. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 16:07, 18 May 2026 (UTC) === Comments === {{archive bottom}} == May 2026 Wikimedia Café meetups regarding the Wikimedia Foundation Annual Plan == <div class="border-box" style="background-color: var(--background-color-warning-subtle, #f8eaba); max-width: 875px; padding: 5px; border: 1px solid black; margin: 5px; color: var(--clr-dark)"> <div class="box" style="float:left; padding-top: 15px; padding-right: 15px;">[[File:Wikimedia Café logo in plain SVG format.svg|75px|alt=The logo for the Wikimedia Café]]</div> Hello! There will be two '''[https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9 Wikimedia Café]''' discussion opportunities during the last weekend of May. Both sessions will focus on the [https://meta.wikimedia.org/wiki/Wikimedia_Foundation_Annual_Plan/2026-2027 the 2026-2027 Wikimedia Foundation Annual Plan]. Participants may attend either or both sessions. #'''Saturday, 30 May 2026 at 15:00 UTC''' ([https://zonestamp.toolforge.org/1780153200 timestamp converter]), at a time friendly to the Americas, Africa, and Europe #'''Sunday, 31 May 2026 at 05:00 UTC''' ([https://zonestamp.toolforge.org/1780203600 timestamp converter]), at a time friendly to Asia and the Pacific Café participants are highly encouraged to read in advance [https://en.wikipedia.org/wiki/User:Sohom_Datta/annual_plan_guide at least this summary of the plan]. Optionally, Café participants are encouraged to read portions of the plan that interest them and [https://meta.wikimedia.org/wiki/Talk:Wikimedia_Foundation_Annual_Plan/2026-2027 ask questions or provide feedback on the Annual Plan talk page]. Please see the Café page for more information, including [https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9#May_2026_meetings_with_a_focus_on_Wikimedia_Foundation_Annual_Plan/2026-2027 tables of timestamp conversions for both sessions], [https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9#Agenda._This_will_be_an_approximately_1_hour_Caf%C3%A9_session,_and_is_extendible_for_an_additional_30_minutes_if_needed. the agenda], and [https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9#How_to_attend_the_session how to register]! <br /> [[File:Buntstifte Eberhard Faber crop 64h.jpg|860px|alt=cropped image of colored pencils]]</div> <span style="white-space:nowrap;">[[User:Pine|<span style="color:#01796f; text-shadow:#00BFFF 0 0 1.0em">↠Pine</span>]] [[User talk:Pine|<span style="color:DeepSkyBlue">(<b style="color:#FFDF00;text-shadow:#FFDF00 0 0 1.0em">✉</b>)</span>]]</span> 19:46, 21 May 2026 (UTC) == Vote now in the 2026 U4C election == <section begin="announcement-content" /> Eligible voters are asked to participate in the 2026 [[m:Special:MyLanguage/Universal_Code_of_Conduct/Coordinating_Committee|Universal Code of Conduct Coordinating Committee]] election. More information–including an eligibility check, voting process information, candidate information, and a link to the vote–are available on Meta at the [[m:Special:MyLanguage/Universal_Code_of_Conduct/Coordinating_Committee/Election/2026|2026 Election information page]]. The vote closes on 2 June 2026 at [https://zonestamp.toolforge.org/1780358400 00:00 UTC]. Please vote if your account is eligible. Results will be available by 14 June 2026. -- In cooperation with the U4C,<section end="announcement-content" /> [[m:User:Keegan (WMF)|Keegan (WMF)]] ([[m:User talk:Keegan (WMF)|talk]]) 17:15, 27 May 2026 (UTC) <!-- Message sent by User:Keegan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Global_message_delivery&oldid=30513860 --> == Create an autopatrolled user group? == {{tracked|T428269}} I would like to propose creating the user group <code>autopatrolled</code> (autopatrolled user), in which for non-curators and non-custodians, their page creations and file uploads would be automatically marked as patrolled by the MediaWiki software. Custodians may grant the user group, at their discretion, to users who create good quality pages that do not need frequent patrolling. On a side note, the term {{tq|autopatroller}} would be used, but because we don't have non-curator/custodian patrollers (as we rely on curators and custodians to patrol), I suggest on using the term {{tq|autopatrolled user}}. Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:31, 29 May 2026 (UTC) :'''Support''' re: the name, I don't really understand the reasoning, so I am '''neutral''' on that. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 15:45, 29 May 2026 (UTC) :: Regarding the name, this is because as we don't have the patroller user group, we rely on curators and custodians to patrol new pages and file uploads. Does that make sense? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:39, 29 May 2026 (UTC) :::Not really, but I don't think it's the most important thing. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 16:42, 29 May 2026 (UTC) :::: We'll decide on the name later. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 01:48, 30 May 2026 (UTC) :::::Oh, please don't let me stand in the way. I'm just not very smart, so don't hold up a matter on my account. I didn't want to derail the proposal, which is a fine and sensible one. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 04:16, 30 May 2026 (UTC) : '''Support''' - sounds like a good idea :* Suggest adding a draft section about this group to [[Wikiversity:Patrolling]]. There is a statement in the Introduction of the page that I'm not sure if its correct and at least could be improved: "Wikiversity also uses an autopatrol right, meaning trusted users' contributions are automatically marked as checked so patrollers can focus on reviewing newer or anonymous editors." :* Regarding autopatroller vs autropatrolled user, what terms are used on similar WMF wiki projects? : -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:28, 30 May 2026 (UTC) ::# I would create a starting page about the user groups, with experienced editors expanding the page. A summarized part of that page would also be added to [[Wikiversity:Patrolling]]. ::# For a similar example, English Wikipedia uses the term {{tq|Autopatrolled}}, just that term only. :: [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:22, 30 May 2026 (UTC) == How much of Wikiversity’s content is LLM slop? == Because it seems like a non-trivial amount, along with AI slop images as well. Is there some kind of AI cleanup project established yet? [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 01:20, 4 June 2026 (UTC) :We have discussed AI but I don't know of any explicit initiative to find and delete AI-generated noise. Individual modules have been deleted for having been made by AI. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 08:50, 4 June 2026 (UTC) == Draft inactivity policy == I created [[Wikiversity:Inactivity policy]] as a start. Any experienced Wikiversity user may feel free to expand it. This is also one-to-two step(s) towards opting out of the [[m:Admin activity review|AAR process]]. However, I made a bold change to reduce the response timeframe from one month to two weeks. In addition, should we reduce the inactivity timeframe to one year? For the latter, most projects use that timeframe and I suggested this for consistency. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:57, 4 June 2026 (UTC) :I support those suggestions. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 17:55, 4 June 2026 (UTC) == Proposed user group and/or possible policy changes == I want to discuss about user group and possible policy changes. # First, interface administrators. I don't think we should allow interface administrators to remove their permission from their own account, since we have multiple active bureaucrats and we can ask them to remove the permission when done, or for them to add a temporary grant. This is according to the [[Wikiversity:IA|current IA policy]]. I also left [[Wikiversity talk:Interface administrators#My thoughts about this user group|my thoughts on the relevant talk page]]. # Second, curators. Given that curators have some sensitive custodian rights (such as <code>delete</code> [but not <code>undelete</code> or similar rights that allow viewing deleted content, unless the curatorship process is RFA-like] and <code>protect</code>), it would probably make more sense only for bureaucrats to grant and remove it, on par with them granting (but not removing) custodian permissions. # Third, about probationary custodians. [[Wikiversity:Probationary custodians]] is currently marked as historical, and the process might still exist on [[Wikiversity:Custodianship]]. Therefore, to maintain consistency with [[Wikiversity:Curatorship#How does one become a curator?]], I propose that we repeal the probationary custodianship process and change it more or less to align with the curatorship process, effectively making probationary custodians permanent ones. However, custodian mentors would still be retained. Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 17:55, 5 June 2026 (UTC) 6ktccp2tsdbaclo982xsxk98bop47ev School:Music and Dance 100 1684 2812973 2799685 2026-06-05T22:07:34Z Kirby - Electrotechnics 3074947 changed to 'category:Music related projects' instead of category:Music 2812973 wikitext text/x-wiki {| cellspacing="0" cellpadding="0" style="margin:0em 0em 0em 0em; width:100%" | colspan="1" style="width:100%; vertical-align:top; border:2px solid #7A39C8;; background-color:#D4C8FF;{{text color default}};" | <div style="padding:0em 1em 1em 1em;"> <!--Note: Top, right, bottom, left --> {{Portal:Humanities/SchoolHeader|{{PAGENAME}}}} </div> |} Welcome to the Wikiversity '''School of Music and Dance'''. This school has two departments listed below : {{RightTOC}} == Departments == * [[Portal:Music|Music department]] * [[Portal:Dance|Dance department]] ==See also== * [[Music production]] ==Learning resources== ===External learning resources=== ====Videos==== * [https://www.youtube.com/watch?v=uv9T1fYA-oA Learn Renoise Beginner 1. What is Renoise] - Learning about digital audio workstation Renoise. ==External links== * [https://www.reddit.com/r/WeAreTheMusicMakers/ We Are Music Makers] sub-Reddit - a community of about 172,555 music producers [[Category:Wikiversity schools]] [[Category:Music related projects]] [[Category:Dance]] [[ar:كلية علوم الموسيقى]] [[fr:Faculté:Musique]] [[it:Facoltà:Musica]] [[pt:Portal:Linguagens e Artes]] [[ru:Факультет журналистики]] jhf0mrsi579jx6uwju9z558qw76hfhb Wikiversity:Notices for custodians 4 1786 2812829 2812817 2026-06-05T12:00:50Z Koavf 147 /* Deleting subpages */ Reply 2812829 wikitext text/x-wiki {{/Header}} == 2FA requirement for bureaucrats == Per [[Special:ListGroupRights#bureaucrat]] and per [[phab:T423120|T423120]], you'll notice that two-factor authentication is required to use bureaucrat permissions (and will soon be enforced). Our existing bureaucrats should take a moment to verify and utilize two-factor authentication. Thank you. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 22:31, 27 May 2026 (UTC) : Thanks for the reminder. Bureaucrats should have received emails. I switched it on recently. Relatively painless and hasn't disrupted workflow, so seems to be well implemented. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 02:13, 28 May 2026 (UTC) ::Yes, I turned this on. I would highly recommend that anyone with rights (custodians, curators, etc.) enable this. --[[User:Mu301|mikeu]] <sup>[[User talk:Mu301|talk]]</sup> 19:42, 30 May 2026 (UTC) == Redundant user rights == I recently changed the user rights for community approved custodians and bureaucrats per consensus. I just realized that I removed curator for Atcovi when adding 'crat thinking that curator was redundant. I then realized that I haven't been consistent about removing old bits. I don't have a strong opinion on this. Just asking. Should curator rights be removed when adding custodian or 'crat? I've never been a curator and don't currently have that bit set. Some accounts still have curator with other rights and others (like mine) don't. --[[User:Mu301|mikeu]] <sup>[[User talk:Mu301|talk]]</sup> 00:00, 31 May 2026 (UTC) :If someone steps down as bureaucrat but wants to remain a custodian/curator, then having those rights as well ensures that they won't be accidentally removed. This exact scenario just happened on another wiki where I am a bureaucrat. It can't hurt to have the redundant ones, if you ask me. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 00:02, 31 May 2026 (UTC) ::[[Special:ListUsers/bureaucrat|Currently]], all the 'crats have custodian; Koavf additionally has curator, which none of the other 'crat accounts have. PieWriter, MathXplore, and Koavf are the only custodians to also have curator. [https://en.wikiversity.org/wiki/Special:ListUsers?username=&group=sysop&wpsubmit=&wpFormIdentifier=mw-listusers-form&limit=50] I propose that we should either a) add curator to all 'crats and custodians or b) remove the redundant bit from all accounts. I don't have a preference, I'm just advocating for consistency and clarity. --[[User:Mu301|mikeu]] <sup>[[User talk:Mu301|talk]]</sup> 00:49, 31 May 2026 (UTC) ::: I lean more on removing the curator bit from all custodians and bureaucrats, as custodians themselves have most, if not all curator user rights, followed by some additional user rights. I planned to remove the curator bit from custodians and to leave a note here about my action(s) for review, until I saw this message. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 00:54, 31 May 2026 (UTC) :::: I'm inclined to follow the [[w:Principle of least privilege]] and remove redundant bits. A custodian or 'crat doesn't need curator. Granting these bits later should be no big deal. --[[User:Mu301|mikeu]] <sup>[[User talk:Mu301|talk]]</sup> 01:13, 31 May 2026 (UTC) : Agree with principles of simplicity and consistency. Plus that agreed practice should be documented. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 03:18, 31 May 2026 (UTC) {{ping|Atcovi|PieWriter|MathXplore|Koavf}} Pinging contributors who may have an interest in discussion. --[[User:Mu301|mikeu]] <sup>[[User talk:Mu301|talk]]</sup> 01:59, 31 May 2026 (UTC) :I'm okay with whatever. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 02:19, 31 May 2026 (UTC) : Removing the curator is OK. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 02:33, 31 May 2026 (UTC) :Seems fine to me [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 02:37, 31 May 2026 (UTC) : {{done}} for all three above. Atcovi already removed his own curator rights as it was redundant to custodian rights. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 14:59, 31 May 2026 (UTC) == Deleting subpages == What script do you suggest for deleting a large number of subpages? I found: * [[Wikipedia:User:Splarka/ajaxbatchdelete.js]] * [[Wikisource:User:George Orwell III/ajaxbatchdelete.js]] -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:31, 5 June 2026 (UTC) :I use MassDelete. You can find a copy at [[User:Koavf/common.js]] and enable it at [[Special:MassDelete]] once you have added it to your JS page(s). ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 12:00, 5 June 2026 (UTC) br0kqseoike39q1tr3s9xe6x697w9bo 2812888 2812829 2026-06-05T17:14:36Z Codename Noreste 2969951 /* Deleting subpages */ reply ([[mw:c:Special:MyLanguage/User:JWBTH/CD|CD]]) 2812888 wikitext text/x-wiki {{/Header}} == 2FA requirement for bureaucrats == Per [[Special:ListGroupRights#bureaucrat]] and per [[phab:T423120|T423120]], you'll notice that two-factor authentication is required to use bureaucrat permissions (and will soon be enforced). Our existing bureaucrats should take a moment to verify and utilize two-factor authentication. Thank you. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 22:31, 27 May 2026 (UTC) : Thanks for the reminder. Bureaucrats should have received emails. I switched it on recently. Relatively painless and hasn't disrupted workflow, so seems to be well implemented. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 02:13, 28 May 2026 (UTC) ::Yes, I turned this on. I would highly recommend that anyone with rights (custodians, curators, etc.) enable this. --[[User:Mu301|mikeu]] <sup>[[User talk:Mu301|talk]]</sup> 19:42, 30 May 2026 (UTC) == Redundant user rights == I recently changed the user rights for community approved custodians and bureaucrats per consensus. I just realized that I removed curator for Atcovi when adding 'crat thinking that curator was redundant. I then realized that I haven't been consistent about removing old bits. I don't have a strong opinion on this. Just asking. Should curator rights be removed when adding custodian or 'crat? I've never been a curator and don't currently have that bit set. Some accounts still have curator with other rights and others (like mine) don't. --[[User:Mu301|mikeu]] <sup>[[User talk:Mu301|talk]]</sup> 00:00, 31 May 2026 (UTC) :If someone steps down as bureaucrat but wants to remain a custodian/curator, then having those rights as well ensures that they won't be accidentally removed. This exact scenario just happened on another wiki where I am a bureaucrat. It can't hurt to have the redundant ones, if you ask me. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 00:02, 31 May 2026 (UTC) ::[[Special:ListUsers/bureaucrat|Currently]], all the 'crats have custodian; Koavf additionally has curator, which none of the other 'crat accounts have. PieWriter, MathXplore, and Koavf are the only custodians to also have curator. [https://en.wikiversity.org/wiki/Special:ListUsers?username=&group=sysop&wpsubmit=&wpFormIdentifier=mw-listusers-form&limit=50] I propose that we should either a) add curator to all 'crats and custodians or b) remove the redundant bit from all accounts. I don't have a preference, I'm just advocating for consistency and clarity. --[[User:Mu301|mikeu]] <sup>[[User talk:Mu301|talk]]</sup> 00:49, 31 May 2026 (UTC) ::: I lean more on removing the curator bit from all custodians and bureaucrats, as custodians themselves have most, if not all curator user rights, followed by some additional user rights. I planned to remove the curator bit from custodians and to leave a note here about my action(s) for review, until I saw this message. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 00:54, 31 May 2026 (UTC) :::: I'm inclined to follow the [[w:Principle of least privilege]] and remove redundant bits. A custodian or 'crat doesn't need curator. Granting these bits later should be no big deal. --[[User:Mu301|mikeu]] <sup>[[User talk:Mu301|talk]]</sup> 01:13, 31 May 2026 (UTC) : Agree with principles of simplicity and consistency. Plus that agreed practice should be documented. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 03:18, 31 May 2026 (UTC) {{ping|Atcovi|PieWriter|MathXplore|Koavf}} Pinging contributors who may have an interest in discussion. --[[User:Mu301|mikeu]] <sup>[[User talk:Mu301|talk]]</sup> 01:59, 31 May 2026 (UTC) :I'm okay with whatever. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 02:19, 31 May 2026 (UTC) : Removing the curator is OK. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 02:33, 31 May 2026 (UTC) :Seems fine to me [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 02:37, 31 May 2026 (UTC) : {{done}} for all three above. Atcovi already removed his own curator rights as it was redundant to custodian rights. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 14:59, 31 May 2026 (UTC) == Deleting subpages == What script do you suggest for deleting a large number of subpages? I found: * [[Wikipedia:User:Splarka/ajaxbatchdelete.js]] * [[Wikisource:User:George Orwell III/ajaxbatchdelete.js]] -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:31, 5 June 2026 (UTC) :I use MassDelete. You can find a copy at [[User:Koavf/common.js]] and enable it at [[Special:MassDelete]] once you have added it to your JS page(s). ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 12:00, 5 June 2026 (UTC) : I use MassDelete by DreamRimmer, see [[m:User:DreamRimmer/MassDelete.js]]. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 17:14, 5 June 2026 (UTC) a87jsd6p9vm8zju65xehdvwpoa55nif Wikiversity:Requests for Deletion 4 1791 2813023 2812792 2026-06-06T04:33:14Z PieWriter 3039865 /* Pages by Harold Foppele */ 2813023 wikitext text/x-wiki {{/header}} == [[IMHA Research Archives]] == I propose to '''move to userspace''', including the subpages. I struggle to understand how Wikiversity readers are supposed to benefit from the material here and in the subpages. In the log, there is e.g. '10 February 2019 Marshallsumter discuss contribs deleted page IMHA Research Archives (content was: "{<nowiki/>{Delete|Author request}} Thanks! -")', so the page was deleted before, but not the subpages. We could also delete all the material if we have strong enough suspicion too much of it is copyright violation. In any case, moving to user space improves the matter a little by moving the content away from Google search. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 13:38, 9 November 2025 (UTC) :Looking at some sub-pages, they can be deleted e.g., because they only consist of broken links or are largely empty. I deleted a couple but haven't been through all to check. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 00:27, 10 November 2025 (UTC) As an example, let me give the wikitext content of [[IMHA Research Archives/3. Scientific litterature search, storage and use]]: <pre> ==[[/Medicina Maritima - the Spanish scientific maritime health journal/]]== ==[[/PubMed/]]== ==[[/Google and Google Scholar/]]== ==[[/Zotero/]]== ==[https://www.dropbox.com/sh/d91z7bcyelfvk42/AAAkIvjtBnnFMbiU9ZLOdVL9a/Andrioti_database%20sources0310.pptx?dl=0 Maritime health web portal ressources ]== </pre> The wikilinks are red; the external link to dropbox says "You don't have access". This was made in 2016. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 09:04, 11 November 2025 (UTC) :I suggest delete -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 03:27, 12 November 2025 (UTC) :: I think we should avoid deletion as much as possible, instead moving to user space (bar copyvio, ethics violation, etc.). This is a good general principle. It greatly improves auditability and makes it so much easier for anyone to request undeletion since they know what content they are requesting for undeletion. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 09:52, 12 November 2025 (UTC) :::Do not recreate Wikiversity from the educational and research project to the personal blog. That will lead to the cancelation of it by WMF. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 21:44, 20 November 2025 (UTC) :::: The English Wikiversity has a long tradition of moving problematic content to user space, as per evidence collected at [[User:Dan_Polansky/About Wikiversity#Moving pages to userspace]]. If Wikimedia Foundation finds this problematic, they can start a discussion in Colloquium and state their concerns. They do not need to make explicit threats at first; they can start a discussion and explain why it is problematic. They can even do it from an anonymous IP and provide a well-articulated reasoning. And anyone else can start a discussion in Colloquium to change this tradition. I do not see why we should not want to change that tradition based on well-articulated, compelling reasoning. I see no reason why Juandev should be making threats instead of them, on a per RFD basis. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 05:58, 21 November 2025 (UTC) :::: If Juandev is ''sincere'' about deleting very-low-value items ''from user space'', he should perhaps demonstrate that by asking his pages like [[:cs:Uživatel:Juandev/Problémy/Kov/Repase dvířek elektroskříně]] to be deleted; otherwise, I register a ''glaring inconsistence''. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 07:43, 21 November 2025 (UTC) ::What was the original delate page about @[[User:Jtneill|Jtneill]]? I guess that would be crucial for the decission. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 21:48, 20 November 2025 (UTC) :::@[[User:Juandev|Juandev]] the couple of pages I checked and deleted were much like @[[User:Dan Polansky|Dan Polansky]] posted above i.e., headings with empty sections and/or broken links but no substantive content. But I think each sub-page needs checking. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 21:59, 20 November 2025 (UTC) ::::So I'm saying that the main page usually determines what the other pages are for. But if I don't know the page because it's been deleted, or why was deleted (deletion based on the founder's request is probably not the rule), it's hard to judge. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 22:16, 20 November 2025 (UTC) :::::I've pasted the original content of the root page: [[IMHA Research Archives#Original page]] (i.e., prior to the content being removed and deletion requested) to help understand the context for the sub-pages. In 2018, Saltrabook blanked the page, indicating that the content had been moved elsewhere, and requested page deletion. Marshallsumter then deleted the main page but not the sub-pages. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 01:58, 21 November 2025 (UTC) ::::::I see, so if those subpages are usefull I would keept them, if not I would delete them. I dont see a point of providing free hosting to sombody, by moving many pages to their user space. The question is if we want to host (i.e. to have in the main ns) lists of links elsewhere. I have no opinion on that. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 10:11, 22 November 2025 (UTC) : Let me clarify that while many of the subpages are like the example above, [[IMHA Research Archives/Scientific litterature search, storage and use/Zotero]] is different: :: "A continuous critical and evidence based learning is a core issue in clinical practice, research, teaching, publication and prevention activities. The Zotero Program is just one of many scientific literature management programs, that should be used for these purposes. Of course one can live without such a database but it helps a lot and can save a lot of time that could be used for more interesting issues. Not only that, but it helps to create better publications and knowledge. Without this program it can be very time consuming to publish a scientific article with the requested style for the references. Further in daily practice when you want to collect and cite a few references for a specific evidence in a clinical colloquium and discussion, this program is excellent. Therefore we strongly recommend that all maritime health persons learn how to use this excellent tool in their daily maritime health practice of all different types. There are good online courses for self-instruction on how to use Zotero. For example this one: Zotero fast online course But in order to increase IMHAR´s collective scientific strength in the use of EBM we would like to give training sessions in every possible opportunity, IMHA Symposia, seminars and other types of meetings. The database is useful for personal purposes but especially also for collaborative aims. At the IMHAR meeting in Paris Oct 7th 2016 we will give an introduction to the program by showing how it can be used in the daily practice and discuss strength and weaknesses compared to other similar databases." : Even longer is e.g. [[IMHA Research Archives/Scientific litterature search, storage and use/Medicina Maritima - the Spanish scientific maritime health journal]]. : However, that does not mean these should be salvaged. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 07:53, 21 November 2025 (UTC) :{{ping|Saltrabook}} I'm wondering if you can respond here to help us decide about whether to delete the IMHA Research Archives sub-pages or perhaps move them to your user space? -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:58, 17 May 2026 (UTC) : [[Special:Diff/2811248]] provides confirmation from Saltrabook to go ahead and delete these archives -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 12:56, 23 May 2026 (UTC) == Undeletion request == {{archive top|'''Not done''' - discussion has been opened for over a month with no proper explanation as to why the page in question should be undeleted + author of the page failed to address any relevant arguments. I'd suggest reading [[Wikiversity:What is Wikiversity?]] and [[Wikiversity:Verifiability]] to find ways on how you can contribute to this site productively. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 21:49, 4 June 2026 (UTC)}} It was deleted by an admin without discussion and with untrue rationale. If people take offense with the question that doesn't mean it's not a valid question and the page was good. Please undelete the Wikidebate page [https://web.archive.org/web/20250810030352/https://en.wikiversity.org/wiki/Is_it_likely_that_Earth_has_been_visited_by_aliens_millions_of_years_ago%3F Is it likely that Earth has been visited by aliens millions of years ago?] There are lots of sources on the subject, the wikidebate is sourced very well compared to other wikidebates and wikiversity pages, and the page is educational, useful and of good quality. [[User:Prototyperspective|Prototyperspective]] ([[User talk:Prototyperspective|discuss]] • [[Special:Contributions/Prototyperspective|contribs]]) 23:57, 10 April 2026 (UTC) :Page: [[Is it likely that Earth has been visited by aliens millions of years ago?]] :Ping: [[User:Atcovi|Atcovi]] -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 00:21, 11 April 2026 (UTC) :There is no need for a discussion for straight garbage-level, pseudoscientific content. :For '''Is it likely that Earth has been visited by aliens millions of years ago?''', the flaws for this page wouldn't even take someone more than a few minutes to assess: :* Essentially, the "pro" arguments unproven claims being derived from irrelevant, established facts (basically: "it is likely aliens have came because Earth has existed for so long [sources proving Earth's longevity]"). These are not serious, scientifically-backed arguments - these are non sequiturs. It's as if I said Wikipedia has existed longer than my existence on Earth ([https://d1wqtxts1xzle7.cloudfront.net/74351725/eyJoIjogImZiODhmYzNkODU1N2UxMWExYzUyODJiYzgzZTRmZDM4OTBjODY5YWMzMjA3NDNmOWEyZTA0ZTU3ZGYwZjAyYTkiLCAidSI6ICJodHRwczovL3B1cmUuaHZhLm5sL3dz-libre.pdf?1636354596=&response-content-disposition=inline%3B+filename%3DCritical_Point_of_View_A_Wikipedia_Reade.pdf&Expires=1775872055&Signature=GqbUZboYRvUYWi~aW40LT5eZSHrLuDL3o0-DxAH8vSvcJcGAuyByZWLF2oHTY6GlB72TqvZxpE-v9d4gvsA6myriYqO~QQQZgWxjT2JXjUWC-yiPcTF4l~lroJSi4dY0v9eKiBcU03l-aeUdrX8~UPfi0TfW0IhsmzH-VBR6X6FrzRpIqc6uM6n9YXfr5FRB3aCqqokU690af3n0Hguaub1Zgmh9qjYYqzBS0VOOHjKTTEQnDuadX3jl5CQeXYTaeCC3H0hMeVwHlratbrnuFEKC1aN0-5znCUoSzMEg21ECzGPTrSDM1W05dcK-u0ZTCeUGKAuC-2yRFL3sY46MIw__&Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA#page=157 reputable source proving ''this'' fact]), therefore it's likely that my birth took place solely for the sake of me experiencing Wikipedia (0 backing). It makes no sense and no person with at least a high school-level of intelligence would take this seriously. :* What is worse is that the user is being misleading with their "[the page is] sourced very well" claim. The sources ''themselves'' don't even back up the claims. It's just used as proof for an established concept, where the user then uses this established concept to jump to an unsupported, laughable conclusion that is pulled out of thin air. It's utterly ridiculous to even consider such a page for mainspace since it clearly violates our [[Wikiversity:Verifiability]] policy. This is, once again, pseudoscientific content that has caused our website to reduce in quality over the last few years. :* Going source by source, we can see that: :#[https://web.archive.org/web/20250918011642/https://timesofindia.indiatimes.com/blogs/thebigd/compress-earths-history-into-24-hrs-humans-came-at-1158-pm-yet-killed-70-of-wildlife/ ‘Compress Earth’s history into 24 hrs. Humans came at 11:58 pm, yet killed 70% of wildlife’] is literally just a blog post which doesn't even mention aliens or extraterrestrial life. It just talks about Earth's history in accordance with the 24-hour metric of time, and the author tries to use this article as a 'piece in the puzzle' of aliens "possibly" visiting Earth... which, once again, is unsupported and is not backed up anywhere in the article. :#[https://web.archive.org/web/20250808053249/https://news.cornell.edu/stories/2023/11/jurassic-worlds-might-be-easier-spot-modern-earth The Cornell article does not even remotely support the idea that "aliens visited Earth"]. It mentions a ''chance'' of "life there [a habitable exoplanet] might not be limited to microbes, but could include creatures as large and varied as the megalosauruses or microraptors that once roamed Earth.", but again, no justification to take this article as proof that "aliens may have visited us!". There's no mention of aliens visiting Earth anywhere in the article. Once again this is only proving the background premise, but not the unsupported, nonsensical "alien likelihood" argument that the author of this garbage page is trying to push so desperately. :#The Parker Solar Probe WP article does not even mention aliens either. It follows the same issues as the previous argument. :And the other page this user complained about [https://en.wikiversity.org/wiki/User_talk:Atcovi#Deletion_of_educational_page_because_of_personal_opinion on my talk page] holds almost similar, maybe even more fatal mistakes, than this one. It has nothing to do with "taking offense", this is just low-quality, garbage content. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 00:56, 11 April 2026 (UTC) ::Why do you think pro claims are required to be proven? It's possible to object to them and these are arguments, not contextualized to be statements of proven facts. And it's not a strange or unreasonable argument to make that since Earth has existed for long, it's more likely that aliens have come here in the past than in recent times or the near future. Instead of insulting others' intelligence, maybe engage with the actual reasoning rather than censoring it away. And there are lots of sources, such as [https://interestingengineering.com/science/alien-civilizations-may-have-visited-earth-millions-of-years-ago-study-says Alien Civilizations May Have Visited Earth Millions of Years Ago, Study Says] etc etc. The sources are used for the arguments themselves individually. [[User:Prototyperspective|Prototyperspective]] ([[User talk:Prototyperspective|discuss]] • [[Special:Contributions/Prototyperspective|contribs]]) 12:30, 11 April 2026 (UTC) :::Because, once again, this is not a site that caters to rampant debating for the sake of "we need to employ rationality and logic to solve the world's problems", we have policies that we need to fulfill. The claims made in the pro argument clearly do not meet [[Wikiversity:Verifiability]], since you cannot verify these arguments with the sources because they are not relevant. :::''"And it's not a strange or unreasonable argument to make that since Earth has existed for long, it's more likely that aliens have come here in the past than in recent times or the near future."'' The point being is that these arguments are not supported by the sources. Even the article you mention poses the idea as a hypothetical model. This is just you twisting the article to fit your unsupported narrative. I'll bring direct quotes for you to show why the linked article does not help you: :::* ''One problem the researchers do make sure to point out is that '''they are working with only one data point: our own behaviors and capabilities for space exploration'''. “We tried to come up with a model that would involve the fewest assumptions about sociology that we could,” Carroll-Nellenback told Business Insider. '''We have no real way of knowing the motivations of an alien civilization'''.'' --> proves that this is just speculation and no evidence-based arguments have been provided for the idea that aliens likely visited Earth. :::And I'm not sure if you read my entire response, but I ''did'' engage with your "actual reasoning" and exposed its weaknesses and lack of adherence to Wikiversity policies. If we allowed content that was just filled with non sequiturs we would have content that fails Wikiversity's educational objectives and reduces the overall quality of this website, hence why such a harsh stance needs to be taken. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 13:50, 11 April 2026 (UTC) ::::Thanks for proving that the Wikimedia ecosystem is unfit to deliberate on controversial topics. The question is entirely valid and the content is far better sourced than nearly all Wikidebates and has no genuine flaws. The only possible issue with it as far as I can see is that now that Wikidebates has been paused people can't add objections if they do have sth specific to say about the topic that's not already included on that page which already had plenty of Cons and objections. ::::The page was more educational than most of Wikiversity and it was well-sourced – wikidebates was for arguments so people were invited to make arguments based on sourced things or outlined logic and the page met [[WV:V]] and most pages on Wikiversity aren't sourced as good. Doesn't look like people can see beyond their biases and personal views here but that's more evident in the marginalization and deletion of wikidebates and the low activity in that project than these selective deletions. A constructive thing to do would be to add reasoned Cons and objections not yet on the page and people had plenty of time to do that. There are and will be other sites for free constructive rational adversarial deliberation (not a big loss in that sense). [[User:Prototyperspective|Prototyperspective]] ([[User talk:Prototyperspective|discuss]] • [[Special:Contributions/Prototyperspective|contribs]]) 16:31, 22 April 2026 (UTC) :::::Thank you for failing to address any of my arguments and going on an unrelated, nonsensical tangent that has nothing to do with the discussion. Once you start producing work that aligns with Wikiversity's content policies instead of typing up laughable, pseudoscientific garbage, then maybe your work can be accepted and not removed. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 16:59, 22 April 2026 (UTC) ::::::I suggest you stop ridiculing things and learn respectfully forming genuine points about the subject at hand. {{tq|the idea as a hypothetical model}} but please learn first about what arguments are and why they're not the same as a statement of objective proven fact. [[User:Prototyperspective|Prototyperspective]] ([[User talk:Prototyperspective|discuss]] • [[Special:Contributions/Prototyperspective|contribs]]) 17:18, 22 April 2026 (UTC) {{archive bottom}} ==Pages by Harold Foppele== {{archive top|All deleted, user page blanked [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 04:33, 6 June 2026 (UTC)}} [[User:Harold Foppele]] is locally blocked indefinitely and globally banned for sockpuppetry. There were also WMF and local community concerns expressed about copyright violation and AI (over)use. As a result, I think the Wikiversity pages created by this account warrant review with regard what should be deleted, what should be retained etc.: * [[Completing the square]] * [[Number of independent spatial modes in a spherical volume]] * [[Quantum]] ** [[Quantum/Andrew N. Jordan]] * [[Quantum A Matter Of Size]] * [[Quantum A Spooky Action at a Distance]] * [[Quantum: A Walk Through the Universe]] * [[Quantum Computing Algorithms in the NISQ Era]] * [[Quantum Formulas Collection]] * [[Quantum harmonic oscillator]] * [[Quantum Matter Elements and Particles]] * [[Quantum mechanics]] ** [[Quantum mechanics/Timeline]] * [[Quantum mechanics learning module]] * [[Quantum mechanics measurements]] * [[Quantum Noisy Qubits]] * [[Quantum optics beam splitter experiments]] * [[Quantum: The Secret of Cohesion: How Waves Hold Matter Together]] * [[Quantum Ultra fast lasers]] * [[Speed of sound experiments]] * [[User:Harold Foppele]] -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 08:12, 17 May 2026 (UTC) :'''Delete all''' Not worth keeping. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 08:27, 17 May 2026 (UTC) {{Archive bottom}} == [[Classical guitar pedagogy]] == According to the talk page, the author of this page intended to create this page for Wikipedia. At this moment in time (nearly 20 years later), the page is still riddled with red links and doesn't seem to fit Wikiversity's learning modules. Therefore, I propose that this page should be deleted. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 13:03, 19 May 2026 (UTC) :'''Weak delete''' This at least has <em>something</em> that someone could use, but agreed that it's not particularly useful and not likely to be developed. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 00:25, 20 May 2026 (UTC) : '''Move''' to [[w:User:Grégory Leclair/Classical guitar pedagogy]] -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 13:18, 23 May 2026 (UTC) == [[Film writing]] == {{archive top|Consensus to keep after vote changes. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:20, 31 May 2026 (UTC)}} Undeveloped since 2007. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 13:05, 19 May 2026 (UTC) :<del>'''Delete''' Nothing here. Great idea in principle, tho. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 00:25, 20 May 2026 (UTC)</del><ins>'''Keep''': It's now at least developed enough to be something. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:42, 31 May 2026 (UTC)</ins> : '''Keep''' as part of [[:Category:Film]] resources. I've tidied the page, so it looks less abandoned. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 02:57, 20 May 2026 (UTC) :@[[User:Atcovi|Atcovi]] @[[User:Koavf|Koavf]] The page seems to have been tidied up. Do you want to reevaluate your votes? [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 02:40, 31 May 2026 (UTC) {{archive bottom}} ==[[United States UFO files]]== {{archive top|Deleted, but the author of the resource did not respond here. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:16, 31 May 2026 (UTC)}} Seems to be WP-like; material copied from [[w:United States UFO files]] -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 01:46, 21 May 2026 (UTC) :'''Delete''', but why would a PROD template not suffice? My logic was that it is a newly created page (made just today), and isn't a big project/difficult page to deal with. Do we not deal with newly created pages that appear to not satisfy Wikiversity's objectives/mission with a PROD template? Wouldn't we best reserve RFDs for long-standing pages (like the two pages above this section being listed for deletion) or ''after'' the PROD template isn't enough to determine the fate of such pages (per [[Wikiversity:Deletion policy#Proposed deletion (prod)|here]]: "Anyone still considering that the resource should be deleted [after the placement of the PROD template] may discuss deletion.")? A PROD template may also be useful in this case to alert the author that the page is not compatible with Wikiversity's learning objectives and communicates a concise opportunity to refine the page with the 90-day limit. Maybe even in this case, a speedy would've been enough (possibly fitting [[Wikiversity:Deletion policy#Criteria for speedy deletion|#12]]: "No research objectives or discussion in history. Welcome users and resources when likely to be expanded shortly."). :Interested to hear your thoughts as I want to make sure this is clear, as I've been cleaning up a lot of 'dead' pages around Wikiversity and find myself confused on whether to use PROD or RFD. Thanks, —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 02:08, 21 May 2026 (UTC) : Yes, could be speedy deleted. Otherwise, I don't know about the merits about leaving it around for 90 days, hence me bringing it to here. There is some comment in [[Wikiversity:Deletion policy]] about the specific deletion templates not being so important. More important I think is to flag for discussion. However, we could also improve the proposed policy to make the process clearer. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 02:20, 21 May 2026 (UTC) : Ping {{u|User:Realcosmixyt}} for comment -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 12:54, 24 May 2026 (UTC) {{archive bottom}} == [[Emergency Operation Centre GIS]] == {{archive top|Consensus to delete. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:39, 30 May 2026 (UTC)}} Undeveloped for over a decade (only thing present is just an outline). —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 14:44, 22 May 2026 (UTC) :*'''Delete''' :―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 15:59, 22 May 2026 (UTC) :* '''Delete'''. Insufficiently developed. Was moved from [[b:Emergency Operation Centre GIS]]. : -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 13:13, 23 May 2026 (UTC) {{archive bottom}} ==[[Mippedia]] == {{archive top|Consensus to delete, and the author of the template did not respond to Jtneill's comment. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:37, 30 May 2026 (UTC)}} I propose the deletion of the page "[[Mippedia]]", due to the subject not being backed by reputable sources. Pages with the same subject has been deleted multiple times on the Indonesian Wikipedia. The original writer of the page did it solely to promote his wiki site. [[User:ANNAFscience|ANNAFscience]] ([[User talk:ANNAFscience|discuss]] • [[Special:Contributions/ANNAFscience|contribs]]) 10:39, 23 May 2026 (UTC) : {{ping|Sevent Me}} any comment? -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 13:10, 23 May 2026 (UTC) :'''Delete''' I don't know what the point of this is. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 16:26, 23 May 2026 (UTC) : '''Delete'''. Advertising. Points to a non-English, copyright restricted website. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 12:58, 24 May 2026 (UTC) {{archive bottom}} ==[[Wikiphilosophers]]== Moving from {{tl|prod}} by {{at|Atcovi}}: "similar "philosophy"-related content has been removed in the past [issue of pseudoscience] + very little moderation (mirroring the issues of [[Wikidebates]]) + lacks educational value." The project has also been nominated for deletion on its talk page: [[Talk:Wikiphilosophers]]. There are many subpages: {{Special:PrefixIndex/Wikiphilosophers/}} -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 13:45, 24 May 2026 (UTC) :'''Delete'''. Unfortunately, this project wasn't as successful as I had hoped. Kind regards, [[User:Perquirius|Perquirius]] ([[User talk:Perquirius|overleg]] • [[Special:Contributions/Perquirius|bijdragen]]) 14:29, 24 May 2026 (UTC) ::Don't forget to delete [[Template:Wikiphilosophers]], [[Template:Wikiphilosophers/doc]] and [[Template:Wikiphilosophers topics]] also. [[User:Perquirius|Perquirius]] ([[User talk:Perquirius|overleg]] • [[Special:Contributions/Perquirius|bijdragen]]) 14:30, 24 May 2026 (UTC) == [[Template:UserSkype]] == Service was discontinued over a year ago. I suggest deleting the Userbox and [[:Category:Users familiar with Skype]], as it can only confuse or mislead. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 21:17, 30 May 2026 (UTC) :'''Delete''' per reasoning. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 22:40, 30 May 2026 (UTC) : '''Delete''' -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 06:48, 31 May 2026 (UTC) iuqyk23s61oax5255rmuwstgz7snvo1 2813026 2813023 2026-06-06T05:06:10Z Dronebogus 3054149 /* Wikiphilosophers */ Reply 2813026 wikitext text/x-wiki {{/header}} == [[IMHA Research Archives]] == I propose to '''move to userspace''', including the subpages. I struggle to understand how Wikiversity readers are supposed to benefit from the material here and in the subpages. In the log, there is e.g. '10 February 2019 Marshallsumter discuss contribs deleted page IMHA Research Archives (content was: "{<nowiki/>{Delete|Author request}} Thanks! -")', so the page was deleted before, but not the subpages. We could also delete all the material if we have strong enough suspicion too much of it is copyright violation. In any case, moving to user space improves the matter a little by moving the content away from Google search. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 13:38, 9 November 2025 (UTC) :Looking at some sub-pages, they can be deleted e.g., because they only consist of broken links or are largely empty. I deleted a couple but haven't been through all to check. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 00:27, 10 November 2025 (UTC) As an example, let me give the wikitext content of [[IMHA Research Archives/3. Scientific litterature search, storage and use]]: <pre> ==[[/Medicina Maritima - the Spanish scientific maritime health journal/]]== ==[[/PubMed/]]== ==[[/Google and Google Scholar/]]== ==[[/Zotero/]]== ==[https://www.dropbox.com/sh/d91z7bcyelfvk42/AAAkIvjtBnnFMbiU9ZLOdVL9a/Andrioti_database%20sources0310.pptx?dl=0 Maritime health web portal ressources ]== </pre> The wikilinks are red; the external link to dropbox says "You don't have access". This was made in 2016. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 09:04, 11 November 2025 (UTC) :I suggest delete -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 03:27, 12 November 2025 (UTC) :: I think we should avoid deletion as much as possible, instead moving to user space (bar copyvio, ethics violation, etc.). This is a good general principle. It greatly improves auditability and makes it so much easier for anyone to request undeletion since they know what content they are requesting for undeletion. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 09:52, 12 November 2025 (UTC) :::Do not recreate Wikiversity from the educational and research project to the personal blog. That will lead to the cancelation of it by WMF. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 21:44, 20 November 2025 (UTC) :::: The English Wikiversity has a long tradition of moving problematic content to user space, as per evidence collected at [[User:Dan_Polansky/About Wikiversity#Moving pages to userspace]]. If Wikimedia Foundation finds this problematic, they can start a discussion in Colloquium and state their concerns. They do not need to make explicit threats at first; they can start a discussion and explain why it is problematic. They can even do it from an anonymous IP and provide a well-articulated reasoning. And anyone else can start a discussion in Colloquium to change this tradition. I do not see why we should not want to change that tradition based on well-articulated, compelling reasoning. I see no reason why Juandev should be making threats instead of them, on a per RFD basis. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 05:58, 21 November 2025 (UTC) :::: If Juandev is ''sincere'' about deleting very-low-value items ''from user space'', he should perhaps demonstrate that by asking his pages like [[:cs:Uživatel:Juandev/Problémy/Kov/Repase dvířek elektroskříně]] to be deleted; otherwise, I register a ''glaring inconsistence''. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 07:43, 21 November 2025 (UTC) ::What was the original delate page about @[[User:Jtneill|Jtneill]]? I guess that would be crucial for the decission. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 21:48, 20 November 2025 (UTC) :::@[[User:Juandev|Juandev]] the couple of pages I checked and deleted were much like @[[User:Dan Polansky|Dan Polansky]] posted above i.e., headings with empty sections and/or broken links but no substantive content. But I think each sub-page needs checking. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 21:59, 20 November 2025 (UTC) ::::So I'm saying that the main page usually determines what the other pages are for. But if I don't know the page because it's been deleted, or why was deleted (deletion based on the founder's request is probably not the rule), it's hard to judge. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 22:16, 20 November 2025 (UTC) :::::I've pasted the original content of the root page: [[IMHA Research Archives#Original page]] (i.e., prior to the content being removed and deletion requested) to help understand the context for the sub-pages. In 2018, Saltrabook blanked the page, indicating that the content had been moved elsewhere, and requested page deletion. Marshallsumter then deleted the main page but not the sub-pages. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 01:58, 21 November 2025 (UTC) ::::::I see, so if those subpages are usefull I would keept them, if not I would delete them. I dont see a point of providing free hosting to sombody, by moving many pages to their user space. The question is if we want to host (i.e. to have in the main ns) lists of links elsewhere. I have no opinion on that. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 10:11, 22 November 2025 (UTC) : Let me clarify that while many of the subpages are like the example above, [[IMHA Research Archives/Scientific litterature search, storage and use/Zotero]] is different: :: "A continuous critical and evidence based learning is a core issue in clinical practice, research, teaching, publication and prevention activities. The Zotero Program is just one of many scientific literature management programs, that should be used for these purposes. Of course one can live without such a database but it helps a lot and can save a lot of time that could be used for more interesting issues. Not only that, but it helps to create better publications and knowledge. Without this program it can be very time consuming to publish a scientific article with the requested style for the references. Further in daily practice when you want to collect and cite a few references for a specific evidence in a clinical colloquium and discussion, this program is excellent. Therefore we strongly recommend that all maritime health persons learn how to use this excellent tool in their daily maritime health practice of all different types. There are good online courses for self-instruction on how to use Zotero. For example this one: Zotero fast online course But in order to increase IMHAR´s collective scientific strength in the use of EBM we would like to give training sessions in every possible opportunity, IMHA Symposia, seminars and other types of meetings. The database is useful for personal purposes but especially also for collaborative aims. At the IMHAR meeting in Paris Oct 7th 2016 we will give an introduction to the program by showing how it can be used in the daily practice and discuss strength and weaknesses compared to other similar databases." : Even longer is e.g. [[IMHA Research Archives/Scientific litterature search, storage and use/Medicina Maritima - the Spanish scientific maritime health journal]]. : However, that does not mean these should be salvaged. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 07:53, 21 November 2025 (UTC) :{{ping|Saltrabook}} I'm wondering if you can respond here to help us decide about whether to delete the IMHA Research Archives sub-pages or perhaps move them to your user space? -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:58, 17 May 2026 (UTC) : [[Special:Diff/2811248]] provides confirmation from Saltrabook to go ahead and delete these archives -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 12:56, 23 May 2026 (UTC) == Undeletion request == {{archive top|'''Not done''' - discussion has been opened for over a month with no proper explanation as to why the page in question should be undeleted + author of the page failed to address any relevant arguments. I'd suggest reading [[Wikiversity:What is Wikiversity?]] and [[Wikiversity:Verifiability]] to find ways on how you can contribute to this site productively. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 21:49, 4 June 2026 (UTC)}} It was deleted by an admin without discussion and with untrue rationale. If people take offense with the question that doesn't mean it's not a valid question and the page was good. Please undelete the Wikidebate page [https://web.archive.org/web/20250810030352/https://en.wikiversity.org/wiki/Is_it_likely_that_Earth_has_been_visited_by_aliens_millions_of_years_ago%3F Is it likely that Earth has been visited by aliens millions of years ago?] There are lots of sources on the subject, the wikidebate is sourced very well compared to other wikidebates and wikiversity pages, and the page is educational, useful and of good quality. [[User:Prototyperspective|Prototyperspective]] ([[User talk:Prototyperspective|discuss]] • [[Special:Contributions/Prototyperspective|contribs]]) 23:57, 10 April 2026 (UTC) :Page: [[Is it likely that Earth has been visited by aliens millions of years ago?]] :Ping: [[User:Atcovi|Atcovi]] -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 00:21, 11 April 2026 (UTC) :There is no need for a discussion for straight garbage-level, pseudoscientific content. :For '''Is it likely that Earth has been visited by aliens millions of years ago?''', the flaws for this page wouldn't even take someone more than a few minutes to assess: :* Essentially, the "pro" arguments unproven claims being derived from irrelevant, established facts (basically: "it is likely aliens have came because Earth has existed for so long [sources proving Earth's longevity]"). These are not serious, scientifically-backed arguments - these are non sequiturs. It's as if I said Wikipedia has existed longer than my existence on Earth ([https://d1wqtxts1xzle7.cloudfront.net/74351725/eyJoIjogImZiODhmYzNkODU1N2UxMWExYzUyODJiYzgzZTRmZDM4OTBjODY5YWMzMjA3NDNmOWEyZTA0ZTU3ZGYwZjAyYTkiLCAidSI6ICJodHRwczovL3B1cmUuaHZhLm5sL3dz-libre.pdf?1636354596=&response-content-disposition=inline%3B+filename%3DCritical_Point_of_View_A_Wikipedia_Reade.pdf&Expires=1775872055&Signature=GqbUZboYRvUYWi~aW40LT5eZSHrLuDL3o0-DxAH8vSvcJcGAuyByZWLF2oHTY6GlB72TqvZxpE-v9d4gvsA6myriYqO~QQQZgWxjT2JXjUWC-yiPcTF4l~lroJSi4dY0v9eKiBcU03l-aeUdrX8~UPfi0TfW0IhsmzH-VBR6X6FrzRpIqc6uM6n9YXfr5FRB3aCqqokU690af3n0Hguaub1Zgmh9qjYYqzBS0VOOHjKTTEQnDuadX3jl5CQeXYTaeCC3H0hMeVwHlratbrnuFEKC1aN0-5znCUoSzMEg21ECzGPTrSDM1W05dcK-u0ZTCeUGKAuC-2yRFL3sY46MIw__&Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA#page=157 reputable source proving ''this'' fact]), therefore it's likely that my birth took place solely for the sake of me experiencing Wikipedia (0 backing). It makes no sense and no person with at least a high school-level of intelligence would take this seriously. :* What is worse is that the user is being misleading with their "[the page is] sourced very well" claim. The sources ''themselves'' don't even back up the claims. It's just used as proof for an established concept, where the user then uses this established concept to jump to an unsupported, laughable conclusion that is pulled out of thin air. It's utterly ridiculous to even consider such a page for mainspace since it clearly violates our [[Wikiversity:Verifiability]] policy. This is, once again, pseudoscientific content that has caused our website to reduce in quality over the last few years. :* Going source by source, we can see that: :#[https://web.archive.org/web/20250918011642/https://timesofindia.indiatimes.com/blogs/thebigd/compress-earths-history-into-24-hrs-humans-came-at-1158-pm-yet-killed-70-of-wildlife/ ‘Compress Earth’s history into 24 hrs. Humans came at 11:58 pm, yet killed 70% of wildlife’] is literally just a blog post which doesn't even mention aliens or extraterrestrial life. It just talks about Earth's history in accordance with the 24-hour metric of time, and the author tries to use this article as a 'piece in the puzzle' of aliens "possibly" visiting Earth... which, once again, is unsupported and is not backed up anywhere in the article. :#[https://web.archive.org/web/20250808053249/https://news.cornell.edu/stories/2023/11/jurassic-worlds-might-be-easier-spot-modern-earth The Cornell article does not even remotely support the idea that "aliens visited Earth"]. It mentions a ''chance'' of "life there [a habitable exoplanet] might not be limited to microbes, but could include creatures as large and varied as the megalosauruses or microraptors that once roamed Earth.", but again, no justification to take this article as proof that "aliens may have visited us!". There's no mention of aliens visiting Earth anywhere in the article. Once again this is only proving the background premise, but not the unsupported, nonsensical "alien likelihood" argument that the author of this garbage page is trying to push so desperately. :#The Parker Solar Probe WP article does not even mention aliens either. It follows the same issues as the previous argument. :And the other page this user complained about [https://en.wikiversity.org/wiki/User_talk:Atcovi#Deletion_of_educational_page_because_of_personal_opinion on my talk page] holds almost similar, maybe even more fatal mistakes, than this one. It has nothing to do with "taking offense", this is just low-quality, garbage content. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 00:56, 11 April 2026 (UTC) ::Why do you think pro claims are required to be proven? It's possible to object to them and these are arguments, not contextualized to be statements of proven facts. And it's not a strange or unreasonable argument to make that since Earth has existed for long, it's more likely that aliens have come here in the past than in recent times or the near future. Instead of insulting others' intelligence, maybe engage with the actual reasoning rather than censoring it away. And there are lots of sources, such as [https://interestingengineering.com/science/alien-civilizations-may-have-visited-earth-millions-of-years-ago-study-says Alien Civilizations May Have Visited Earth Millions of Years Ago, Study Says] etc etc. The sources are used for the arguments themselves individually. [[User:Prototyperspective|Prototyperspective]] ([[User talk:Prototyperspective|discuss]] • [[Special:Contributions/Prototyperspective|contribs]]) 12:30, 11 April 2026 (UTC) :::Because, once again, this is not a site that caters to rampant debating for the sake of "we need to employ rationality and logic to solve the world's problems", we have policies that we need to fulfill. The claims made in the pro argument clearly do not meet [[Wikiversity:Verifiability]], since you cannot verify these arguments with the sources because they are not relevant. :::''"And it's not a strange or unreasonable argument to make that since Earth has existed for long, it's more likely that aliens have come here in the past than in recent times or the near future."'' The point being is that these arguments are not supported by the sources. Even the article you mention poses the idea as a hypothetical model. This is just you twisting the article to fit your unsupported narrative. I'll bring direct quotes for you to show why the linked article does not help you: :::* ''One problem the researchers do make sure to point out is that '''they are working with only one data point: our own behaviors and capabilities for space exploration'''. “We tried to come up with a model that would involve the fewest assumptions about sociology that we could,” Carroll-Nellenback told Business Insider. '''We have no real way of knowing the motivations of an alien civilization'''.'' --> proves that this is just speculation and no evidence-based arguments have been provided for the idea that aliens likely visited Earth. :::And I'm not sure if you read my entire response, but I ''did'' engage with your "actual reasoning" and exposed its weaknesses and lack of adherence to Wikiversity policies. If we allowed content that was just filled with non sequiturs we would have content that fails Wikiversity's educational objectives and reduces the overall quality of this website, hence why such a harsh stance needs to be taken. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 13:50, 11 April 2026 (UTC) ::::Thanks for proving that the Wikimedia ecosystem is unfit to deliberate on controversial topics. The question is entirely valid and the content is far better sourced than nearly all Wikidebates and has no genuine flaws. The only possible issue with it as far as I can see is that now that Wikidebates has been paused people can't add objections if they do have sth specific to say about the topic that's not already included on that page which already had plenty of Cons and objections. ::::The page was more educational than most of Wikiversity and it was well-sourced – wikidebates was for arguments so people were invited to make arguments based on sourced things or outlined logic and the page met [[WV:V]] and most pages on Wikiversity aren't sourced as good. Doesn't look like people can see beyond their biases and personal views here but that's more evident in the marginalization and deletion of wikidebates and the low activity in that project than these selective deletions. A constructive thing to do would be to add reasoned Cons and objections not yet on the page and people had plenty of time to do that. There are and will be other sites for free constructive rational adversarial deliberation (not a big loss in that sense). [[User:Prototyperspective|Prototyperspective]] ([[User talk:Prototyperspective|discuss]] • [[Special:Contributions/Prototyperspective|contribs]]) 16:31, 22 April 2026 (UTC) :::::Thank you for failing to address any of my arguments and going on an unrelated, nonsensical tangent that has nothing to do with the discussion. Once you start producing work that aligns with Wikiversity's content policies instead of typing up laughable, pseudoscientific garbage, then maybe your work can be accepted and not removed. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 16:59, 22 April 2026 (UTC) ::::::I suggest you stop ridiculing things and learn respectfully forming genuine points about the subject at hand. {{tq|the idea as a hypothetical model}} but please learn first about what arguments are and why they're not the same as a statement of objective proven fact. [[User:Prototyperspective|Prototyperspective]] ([[User talk:Prototyperspective|discuss]] • [[Special:Contributions/Prototyperspective|contribs]]) 17:18, 22 April 2026 (UTC) {{archive bottom}} ==Pages by Harold Foppele== {{archive top|All deleted, user page blanked [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 04:33, 6 June 2026 (UTC)}} [[User:Harold Foppele]] is locally blocked indefinitely and globally banned for sockpuppetry. There were also WMF and local community concerns expressed about copyright violation and AI (over)use. As a result, I think the Wikiversity pages created by this account warrant review with regard what should be deleted, what should be retained etc.: * [[Completing the square]] * [[Number of independent spatial modes in a spherical volume]] * [[Quantum]] ** [[Quantum/Andrew N. Jordan]] * [[Quantum A Matter Of Size]] * [[Quantum A Spooky Action at a Distance]] * [[Quantum: A Walk Through the Universe]] * [[Quantum Computing Algorithms in the NISQ Era]] * [[Quantum Formulas Collection]] * [[Quantum harmonic oscillator]] * [[Quantum Matter Elements and Particles]] * [[Quantum mechanics]] ** [[Quantum mechanics/Timeline]] * [[Quantum mechanics learning module]] * [[Quantum mechanics measurements]] * [[Quantum Noisy Qubits]] * [[Quantum optics beam splitter experiments]] * [[Quantum: The Secret of Cohesion: How Waves Hold Matter Together]] * [[Quantum Ultra fast lasers]] * [[Speed of sound experiments]] * [[User:Harold Foppele]] -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 08:12, 17 May 2026 (UTC) :'''Delete all''' Not worth keeping. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 08:27, 17 May 2026 (UTC) {{Archive bottom}} == [[Classical guitar pedagogy]] == According to the talk page, the author of this page intended to create this page for Wikipedia. At this moment in time (nearly 20 years later), the page is still riddled with red links and doesn't seem to fit Wikiversity's learning modules. Therefore, I propose that this page should be deleted. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 13:03, 19 May 2026 (UTC) :'''Weak delete''' This at least has <em>something</em> that someone could use, but agreed that it's not particularly useful and not likely to be developed. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 00:25, 20 May 2026 (UTC) : '''Move''' to [[w:User:Grégory Leclair/Classical guitar pedagogy]] -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 13:18, 23 May 2026 (UTC) == [[Film writing]] == {{archive top|Consensus to keep after vote changes. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:20, 31 May 2026 (UTC)}} Undeveloped since 2007. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 13:05, 19 May 2026 (UTC) :<del>'''Delete''' Nothing here. Great idea in principle, tho. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 00:25, 20 May 2026 (UTC)</del><ins>'''Keep''': It's now at least developed enough to be something. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:42, 31 May 2026 (UTC)</ins> : '''Keep''' as part of [[:Category:Film]] resources. I've tidied the page, so it looks less abandoned. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 02:57, 20 May 2026 (UTC) :@[[User:Atcovi|Atcovi]] @[[User:Koavf|Koavf]] The page seems to have been tidied up. Do you want to reevaluate your votes? [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 02:40, 31 May 2026 (UTC) {{archive bottom}} ==[[United States UFO files]]== {{archive top|Deleted, but the author of the resource did not respond here. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:16, 31 May 2026 (UTC)}} Seems to be WP-like; material copied from [[w:United States UFO files]] -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 01:46, 21 May 2026 (UTC) :'''Delete''', but why would a PROD template not suffice? My logic was that it is a newly created page (made just today), and isn't a big project/difficult page to deal with. Do we not deal with newly created pages that appear to not satisfy Wikiversity's objectives/mission with a PROD template? Wouldn't we best reserve RFDs for long-standing pages (like the two pages above this section being listed for deletion) or ''after'' the PROD template isn't enough to determine the fate of such pages (per [[Wikiversity:Deletion policy#Proposed deletion (prod)|here]]: "Anyone still considering that the resource should be deleted [after the placement of the PROD template] may discuss deletion.")? A PROD template may also be useful in this case to alert the author that the page is not compatible with Wikiversity's learning objectives and communicates a concise opportunity to refine the page with the 90-day limit. Maybe even in this case, a speedy would've been enough (possibly fitting [[Wikiversity:Deletion policy#Criteria for speedy deletion|#12]]: "No research objectives or discussion in history. Welcome users and resources when likely to be expanded shortly."). :Interested to hear your thoughts as I want to make sure this is clear, as I've been cleaning up a lot of 'dead' pages around Wikiversity and find myself confused on whether to use PROD or RFD. Thanks, —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 02:08, 21 May 2026 (UTC) : Yes, could be speedy deleted. Otherwise, I don't know about the merits about leaving it around for 90 days, hence me bringing it to here. There is some comment in [[Wikiversity:Deletion policy]] about the specific deletion templates not being so important. More important I think is to flag for discussion. However, we could also improve the proposed policy to make the process clearer. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 02:20, 21 May 2026 (UTC) : Ping {{u|User:Realcosmixyt}} for comment -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 12:54, 24 May 2026 (UTC) {{archive bottom}} == [[Emergency Operation Centre GIS]] == {{archive top|Consensus to delete. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:39, 30 May 2026 (UTC)}} Undeveloped for over a decade (only thing present is just an outline). —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 14:44, 22 May 2026 (UTC) :*'''Delete''' :―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 15:59, 22 May 2026 (UTC) :* '''Delete'''. Insufficiently developed. Was moved from [[b:Emergency Operation Centre GIS]]. : -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 13:13, 23 May 2026 (UTC) {{archive bottom}} ==[[Mippedia]] == {{archive top|Consensus to delete, and the author of the template did not respond to Jtneill's comment. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:37, 30 May 2026 (UTC)}} I propose the deletion of the page "[[Mippedia]]", due to the subject not being backed by reputable sources. Pages with the same subject has been deleted multiple times on the Indonesian Wikipedia. The original writer of the page did it solely to promote his wiki site. [[User:ANNAFscience|ANNAFscience]] ([[User talk:ANNAFscience|discuss]] • [[Special:Contributions/ANNAFscience|contribs]]) 10:39, 23 May 2026 (UTC) : {{ping|Sevent Me}} any comment? -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 13:10, 23 May 2026 (UTC) :'''Delete''' I don't know what the point of this is. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 16:26, 23 May 2026 (UTC) : '''Delete'''. Advertising. Points to a non-English, copyright restricted website. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 12:58, 24 May 2026 (UTC) {{archive bottom}} ==[[Wikiphilosophers]]== Moving from {{tl|prod}} by {{at|Atcovi}}: "similar "philosophy"-related content has been removed in the past [issue of pseudoscience] + very little moderation (mirroring the issues of [[Wikidebates]]) + lacks educational value." The project has also been nominated for deletion on its talk page: [[Talk:Wikiphilosophers]]. There are many subpages: {{Special:PrefixIndex/Wikiphilosophers/}} -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 13:45, 24 May 2026 (UTC) :'''Delete'''. Unfortunately, this project wasn't as successful as I had hoped. Kind regards, [[User:Perquirius|Perquirius]] ([[User talk:Perquirius|overleg]] • [[Special:Contributions/Perquirius|bijdragen]]) 14:29, 24 May 2026 (UTC) ::Don't forget to delete [[Template:Wikiphilosophers]], [[Template:Wikiphilosophers/doc]] and [[Template:Wikiphilosophers topics]] also. [[User:Perquirius|Perquirius]] ([[User talk:Perquirius|overleg]] • [[Special:Contributions/Perquirius|bijdragen]]) 14:30, 24 May 2026 (UTC) :'''Delete''' Wikiversity should not be used as a host for failed sister projects. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 05:06, 6 June 2026 (UTC) == [[Template:UserSkype]] == Service was discontinued over a year ago. I suggest deleting the Userbox and [[:Category:Users familiar with Skype]], as it can only confuse or mislead. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 21:17, 30 May 2026 (UTC) :'''Delete''' per reasoning. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 22:40, 30 May 2026 (UTC) : '''Delete''' -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 06:48, 31 May 2026 (UTC) i8burdcyulx54wc7scmf6o3wwhet8h6 Saxophone 0 2461 2812877 2707821 2026-06-05T15:53:32Z Kirby - Electrotechnics 3074947 changed to 'category:Music instruments' instead of category:Musical instruments 2812877 wikitext text/x-wiki [[Image:Yamaha YAS-25 20040517.jpg|thumb|Yamaha YAS-25 Alto Saxophone.]] The '''saxophone''', or sax, is a single-reed instrument, which comes in many different sizes: from sopranisimo (soprillo) to sub-contrabass. The four "main" members of the family are the soprano, alto, tenor, and baritone (bari). Although it is almost exclusively made of brass, the sax is a member of the woodwind family. It has fingerings and mouthpiece similar to that of the clarinet, but with a conical bore. The small mouthpiece and short length of the soprano make this member susceptible to changes of the embouchure; therefore the pitch is widely variable and requires control and a well-developed sense of pitch (see [[ear training]]). Conversely, the tenor and bari saxes are somewhat easier to control, but the amount of air needed to produce a tone is much greater. The alto sax is a good beginning instrument because it is widely available, fits into small hands easily (that is, most 9-10 year-old children), and is the most popular. == Musical Styles == The saxophone can be used to play a variety of musical styles. Many modern wind ensembles will include 2 alto saxophone parts, 1 tenor saxophone part, and 1 baritone saxophone part. The lead alto saxophone player may also be asked to play soprano saxophone in more advanced literature. There are many classical pieces have been written for sax quartets, which consist of one tenor, one baritone, one alto and one soprano sax. In a quartet the four musicians usually play with just each other sans accompaniment. In the classical realm of study, the standard instrument is the alto sax, for which such standard pieces as the Creston Sonata have been written. Although the tenor is more commonplace than the soprano, there are quite a number of well-known and prolific artists that write for and perform on the soprano sax. No jazz band would be complete without a sax section, and in a standard jazz big band, the sax section consists of two altos, two tenors and one bari sax. The lead is the first alto, and the bulk of the solos go to the first tenor. For jazz in general, the sax is one of the most popular instruments in the art. Jazz legends such as Charlie Parker, John Coltrane, Ornette Coleman, Cannonball Adderley, Wayne Shorter, Stan Getz, Gerry Mulligan and Paul Desmond are (or were) sax players who had very unique styles of playing their saxes. Saxophones can also be used in many other styles of music such as rock, pop, smooth jazz, adult contemporary, and musical theater. == The Reed == The reed on the saxophone is very similar to that of the clarinet, but somewhat wider. The single reed vibrates against the mouthpiece when air is blown into it. Depending on the mouthpiece, reed, and ligature used, and also on the dimensions of the mouth and embouchure, a wide range of [[timbre]]s (tone colors) can be produced. Reeds come in a wide variety of strengths and cuts, which vary from box to box and from manufacturer to manufacturer. For the beginner, a reed of strength 2 to 2.5 is normal. An intermediary to advanced setup would use a 3.5 (Vandoren) on a Selmer S80 C* (or C for tenor) mouthpiece (This is the "classic" setup, and works for both tenor and alto. It is used by many college institutions, and many private instructors use this). As the [[embouchure]] becomes stronger, higher strength reeds can be utilized. Generally, a stronger reed will be more in-tune throughout the range of the saxophone, and make playing the [[altissimo]] range possible. Using a higher strength reed will also affect the timbre, or tone, and can increase the overall volume. The softer the reed the easier the lower range, but in the upper register the reed may remain closed against the mouthpiece, not allowing the high notes to speak well or at all. The harder the reed the easier the higher notes will pop out but it will be more difficult to get the lower range out. Proper care of your reeds is very important. When not in use, take the reed off and keep it in a protector to prevent warping. A good reed keeper can be commercially acquired, or you can make one out of a glass microscope slide or cut plate glass (careful with the edges) and rubber bands. During periods where the reed will be left on in between playing, use a mouthpiece cap. This protects the reed both from accidental physical damage, keeps it moist for longer, and helps prevent the reed from cracking (by slowing down evaporation). It is important that while playing that the ENTIRE reed maintains its wetness. Before playing, soak the reed for about 2-3 minutes. Remember to wet the stock or back sides of the reed. There are many brands and strengths of reed from the performer to choose from. Players of all abilities are encouraged to try different brands, cuts and strengths. Although a reed may be called a "jazz reed," it could certainly be well suited for classical or rock playing. There are many brands of saxophone reeds sold in music stores that musicians universally recognize as good quality reeds that can be used in a lot of styles. These brands include: Vandoren, Hemke, La Voz, and Rico Royals (not the generic Rico reeds in the orange box). There are other types of reeds available as well. Some players choose to use plastic reeds or plastic covered reeds. These reeds sound and can feel different from plain cane reeds. While some daring, experienced players use artificial reeds with good results, the tone of a good natural reed is usually superior to that of a plastic reed. == The Mouthpiece == The mouthpiece (sometimes abbr. MPC) is one of the most versatile and changeable item that can affect a performer's sound. Mouthpieces are made out of different materials including but not limited to plastic, hard rubber, and metal. Plastic (famously played by Charlie Parker) or hard rubber mouthpieces are more commonly used in classical settings although can be used for jazz, rock and pop style music as well. A metal mouthpiece is usually marketed only for jazz or rock styles, but depending on the opening they can also be suitable for classical playing. In general, the higher the opening of the tip, the softer the reed you must use, and the harder it is to play (correctly, with good tone). Once you have developed decent tone, branch out a bit and try as many mouthpieces as you can. Look at various professional player's setups, and see if you like them. You may want to seek assistance from a private music teacher or your school music teacher when looking for a mouthpiece. But most importantly: Try before you buy! If you are a beginner and have an intermediate or professional saxophone, you may need to look into a different mouthpiece. Consider one of a few standard and widely-available choices as "step up" mouthpieces once the basics have been "mastered": *Selmer S80 C or C* *Yamaha 4CM *Meyer 4M *Rousseau 3R == The Ligature == The ligature, colloquially referred to as a lig, is yet another piece of the mouthpiece setup, and it does deserve some consideration. Because the sound is a result of the reed vibrating, you want to have a ligature that will let the reed vibrate efficiently. The ligature should apply even and symmetrical pressure on the reed. This is so that the reed can vibrate evenly, giving the player the clearest and most balanced possible sound. There are a variety of options to choose from. For most applications, the best, easiest, cheapest (~$12-25), and most widely available are canvas (marketed under the Rovner brand name) and metal ("Selmer style) ligatures. == A Final Note About Reeds, Mouthpieces and Ligatures == Selecting high-quality, hand-selected mouthpieces, ligatures and reeds that fit your playing level, style and mouth is the most important part of your setup (and probably the cheapest, too). A mouthpiece setup will sound similar on many different saxophones, whereas the same sax can sound drastically different with a different MPC/lig/reed setup. Even a beginning student should try several brands and sizes to see what is most comfortable. Buying a MPC/lig/reed setup that costs $150-300 and then buying a $700-1200 (used?) instrument is MUCH better than spending $850-1500 on an instrument and using the "free" mouthpiece that the sax comes with. == The Embouchure == The saxophone embouchure, when formed correctly, should feel natural and comfortable. When forming the embouchure, take the following steps: 1. Cover gently the lower teeth with the lower lip, and press gently against the reed. 2. Place your top teeth on the top (biteplate) of the mouthpiece. Do not bite hard!! 3. "Purse" the corners of your lips, and form an airtight seal around the mouthpiece. 4. Relax and blow easily (think warm, slow air). == Producing a tone == After practicing the embouchure as described above, and the resulting tone is not beautiful, try altering the firmness of your embouchure. Most likely you are biting too hard, especially if it sounds "choked," or you cannot produce a steady, moderate volume, in-tune pitch. The saxophone tone is not overly difficult to develop, but like all instruments improves with practice. Getting with a private instructor is the best method to ensure that your technique is correct, and to learn quickly. == Altissimo Register == It is possible to extend the range by about one octave into what is known as the altissimo register. After a few years of study, one can become very proficient. The notes that can be produced above the fundamental tone are called harmonics. Altissimo notes are harmonics that can be played using alternate / creative fingerings. For example, while playing a low Bb, try using your throat play a middle Bb while maintaining the low Bb fingering (don't pinch!). See how many harmonics you can make while fingering a low Bb. For further study, obtain an altissimo fingering chart. Note that fingerings will be different from one instrument to another, and also vary by the player. Learning altissimo note also helps develop your embouchure. [[User:Mrzonules|Mrzonules]] 20:46, 7 February 2007 (UTC) {{Musical instruments}} [[Category:Music instruments]] n6x4kp862lxoli5v2zxypgfmld75n2r Wedding-Day at Troldhaugen 0 2500 2812953 2373897 2026-06-05T21:39:58Z Kirby - Electrotechnics 3074947 changed to 'category:Music theory' instead of category:Music 2812953 wikitext text/x-wiki This is a rather light sounding piano piece by Edvard Grieg, composed in an A B A format. (He also produced an orchestral version.) When playing this piece, aim for good contrast between staccato and non-staccato notes. == Bars 1 - 20 == In the first 10 bars, the theme is introduced. Note how the first time the theme is stated, the ending chords involve sixth chords. This is present in bar 4. Afterwards, this is nicely resolved the next time in bar 6. Bars 11-20 is a restatement of the theme, but an octave higher. Note how it is marked as pp, as opposed to p. == Bars 31 - 48 == These series of 16th notes form a bridge or transition type series. Observe how Grieg alternates playing the notes on the left and right hand, and how the chords are slightly syncopated from the bar lines. This creates interest in the music. One more thing to observe, are the partial use of whole tone scales. This of course, is characteristic of the era. The scales first start on D flat and increase in tone after the pattern is repeated. Another interesting element are the bars 45-48 - the chords are dissonant with respect to the left hand. The large range of the notes also increases the sense of excitement in the music. ==Listen== [[Image:Troldhaugen.png|thumb|left|600px|"[http://www.mutopiaproject.org/cgibin/piece-info.cgi?id=781 Wedding-day at Troldhaugen]" by [[w:Lyric Pieces|Edvard Grieg]]. Listen to a [[Introduction to GarageBand|GarageBand]]-generated version: {{audio|Troldhaugen.ogg|Troldhaugen.ogg}}. [[w:Wikipedia:Media help (Ogg)|Help]] with ogg audio play.]] [[Category:Music theory]] c9w5k3220q8k19husxvahvx2oq2a828 Template:Welcome and expand 10 4579 2812837 2607987 2026-06-05T12:11:25Z Jtneill 10242 Remove Wikinews (project closed) 2812837 wikitext text/x-wiki <div align="center" class="plainlinks"> {{Robelbox|theme=14|icon=Crystal_Clear_app_kaddressbook.png|iconwidth=48px|title=Please help develop{{#ifeq:{{{cat|no}}}|no|&nbsp;and classify}} this resource|width=60%}} <div align="left" style="{{Robelbox/pad}}"> Learn [[Wikiversity:Editing tutorial|how]] you can [{{fullurl:{{FULLPAGENAME}}|action=edit}} develop this resource]{{#if:{{{1|}}}|&nbsp;by [[User:{{{1}}}|{{{1}}}]]}} to teach [[Wikiversity:Who are Wikiversity participants?|participants]] about '''{{PAGENAME}}'''.{{#ifeq:{{{cat|no}}}|no|&nbsp;Choose a [[Help:Resources by subject|subject]], [[Help:Resources by educational level|education level]], and [[Help:Resources by type|resource type]] to classify the resource by}}. You may learn more about the topic to help you develop{{#ifeq:{{{cat|no}}}|no|&nbsp;and classify}} the resource from: <div style="padding:0px 4em; white-space:nowrap; text-align:center;"> [[File:Wikibooks-logo.svg|24x24px|Wikibooks|link=b:Special:Search/{{{1|{{SUBPAGENAME}}}}}]]&nbsp; [[File:Wikipedia-logo.svg|24x24px|Wikipedia|link=w:Special:Search/{{{1|{{SUBPAGENAME}}}}}]]&nbsp; [[File:Wiktionary-logo.svg|24x24px|Wiktionary|link=wikt:Special:Search/{{{1|{{SUBPAGENAME}}}}}]]&nbsp; [[File:Wikiquote-logo.svg|24x24px|Wikiquote|link=q:Special:Search/{{{1|{{SUBPAGENAME}}}}}]]&nbsp; [[File:Wikisource-logo.svg|24x24px|Wikisource|link=s:Special:Search/{{{1|{{SUBPAGENAME}}}}}]]&nbsp; [[File:Commons-logo.svg|24x24px|Wikimedia Commons|link=commons:Special:Search/{{{1|{{SUBPAGENAME}}}}}]]&nbsp; [[File:Wikispecies-logo.svg|24x24px|Wikispecies|link=species:Special:Search/{{{1|{{SUBPAGENAME}}}}}]]&nbsp; </div> </div> {{Robelbox/close}} </div><noinclude>{{documentation}}</noinclude><includeonly> {{StubDateSort}} [[Category:Resource stubs]] </includeonly> kthzgexjn6ve4b60pov9413jgqwd17o Portal:Music/Beginner 102 4676 2812874 754395 2026-06-05T15:49:42Z Kirby - Electrotechnics 3074947 changed to 'category:Music stubs' instead of category:Music 2812874 wikitext text/x-wiki {{School_of_Music_TOC_SMALL}} *[[Introduction to Music]] *[[Understanding music]] *[[Introduction to Chords]] *[[b:The Physics of Music|The Physics of Music]] *[[b:Triads|Triads]] *[[Beginners Music Theory]] *[[Key Signatures]] *[[Scales]] *[[Intervals]] *[[Aural Training]] [[Category:Pages moved from Wikibooks]] [[Category:Music stubs]] mi489b1cevvzb0obe1rh7mh95cn8i6v Aural Training 0 4692 2813000 2721973 2026-06-06T00:22:44Z Kirby - Electrotechnics 3074947 changed to 'category:Aural theory and ear training' instead of category:Music 2813000 wikitext text/x-wiki {{TOCright}} === Definition === Aural training is training to give the ability to recognize and understand the music one hears. This can take the form of transcribing music into notation by ear or by hearing musical functions. Aural Training could also be used to identify formal structures on both large and small scales. One's ability in aural training varies greatly from person to person. For some it comes rather naturally and for others it is much more difficult, regardless of whether or not they are a musician. For those with perfect (absolute) pitch, some aspects of aural training are certainly made easier, but not all. === Transcription === By focusing one aspect of the music at a time (i.e. rhythm, melody, bass line, etc.) one can, with proper training, transcibe the music they hear, into musical notation. This is a very common practice amongst jazz musicians as they transcribe solos recorded by other jazz musicians so as to develop their own improvisational abilities. === General aspects === In tonal music, different chords have different functions. It is possible to hear these different functions so as to have a better idea of what is happening in the music. Beyond function, even hearing the quality of the chord can be important and be used by musicians. These more general aspects tend to be used by more musicians (classical, jazz, etc.) than transcription. === Formal structures === On a large scale one can train themselves to hear the different formal sections of a piece in any music. This could be through hearing the difference between verse and chorus in a popular song or hearing the difference between the Exposition, Development and subsequent Recapitulation in classical sonata form. On a smaller scale, one could hear the differences between the Main Theme, Transition, and Subordinate Theme or even the structures in a single phrase (i.e. basic idea, contrasting idea, continuation, etc.) === Absolute pitch === The ability to remember the particular quality of pitch classes well enough to so as to recognise them on hearing. It is not a rare ability among musicians. == See also == * [[Musical transcription]] [[Category:Aural theory and ear training]] [[Category:Musicianship]] [[Category:Training]] gzzkia7sjphrgbiqgpe7e02ei6i7w8o Portal:Music/Advanced 102 4699 2812875 230118 2026-06-05T15:50:17Z Kirby - Electrotechnics 3074947 changed to 'category:Music stubs' instead of category:Music 2812875 wikitext text/x-wiki {{School_of_Music_TOC_SMALL}} *[[Secondary Dominants]] *[[Augmented Sixth Chords]] *[[Neapolitan Chords]] *[[Augmented Chords]] *[[Ninth Chords and Beyond]] *[[Other Chromatic Harmony]] *[[Large Forms]] [[Category:Music stubs]] bxfph5c10pgobll6mnktcof63h3bqi1 Euphonium/Jechasteen/Lesson 1 0 5590 2812884 2722178 2026-06-05T15:57:26Z Kirby - Electrotechnics 3074947 changed to 'category:Euphonium' instead of category:Musical instruments 2812884 wikitext text/x-wiki == Embouchure == The single most important factor in beginning to play the Euphonium (hereafter referred to as Euph) without frustration is to get a well-formed embouchure in place. The embouchure, or placement and usage of the lips, is where most of the action in playing takes place. Without a properly formed embouchure, all else fails. Find a mirror so that you can see exactly what your face looks like in place on the mouthpiece. This will help you to see problems with your embouchure. The first step in forming the embouchure is to mouth the letter M, with the lips sealed and the jaw/teeth slightly opened and separated. Next, go through the motion that you would you would use in spitting a very small seed off the very tip of your tongue. Now hold the lips in that position. The result should looks somewhat like this: TODO: Picture example of proper embouchure. Now, with that placement of the lips, use the air in your lungs (more on this in a minute) to create a buzzing noise through your lips. This a part that more or less can not be put into words, so you should go through a process of experimentation until you can create a constant pitch with only you lips. Next, try to make a "siren" sound using only your lips. Todo: Audio sample sans mouthpiece The obvious next step would be to add the mouthpiece to the equation. Pick up your mouthpiece of choice using your non-dominant hand (the hand you do not write with) and place it over the center of your lips, like so: TODO: Picture example of proper mouthpiece placement. Personal opinions abound about the proper placement of the mouthpiece in relation to your lips. Some say that the mouthpiece should be 1/3 on the top and 2/3 on the bottom, while others prefer the mouthpiece to be completely centered on the lips. Once again, I say that this comes down to personal preference and what works best for you. Each person's mouth and teeth are constructed in differing ways, so the placement of the mouthpiece will differ from person to person. The best way to find what is best for you is to experiment with the placement until you find a place on your lips that both feels comfortable and creates the best sound. Now, with the mouthpiece to your lips, form your embouchure in the same way as before and create the same buzzing sound. The technique for creating sound with and without mouthpiece is slightly different, so you will need to experiment in order to find the ideal way to create a good sound. Todo: Audio sample with mouthpiece Creating sound with the mouthpiece to your lips is a different experience than with lips alone. In order to get a good sound out of your mouthpiece, you should experiment within the constrains listed above in order to find '''what works for you'''.<br> Just remember: * Keep the corners (outsides) of your lips firm. The only part of the lips which you use are contained within the rim of the mouthpiece. * Use the muscles of your lower body to move the air. Keep your throat out of the way, so that the air can move freely. * Do what feels right for you, as long as it follows the previous constraints. * Think open. Keep your oral cavity as open as possible. Imagine you are fogging a mirror, then form the embouchure with that same oral space. == Breath == As important as the correct embouchure in creating a good sound in your first playing sessions is the ''correct breath support''. Your air support and embouchure work in tandem and as a team to create tone. Before you even begin to play on the instrument, you should practice taking full, deep breaths. When you breathe in, think about filling up your entire abdomen with air, not just your lungs. Feel your chest, stomach, and lower back expand as you breathe in. When you reach your capacity, breathe out slowly, with your lips shaped as though you are blowing a candle. Keys to good breath: * Normal, shallow breaths won't cut it for a wind instrument. They take a lot of air. * Think of filling your entire abdomen with air. Imagine your lungs filling from top to bottom, with the bottom being at your pelvis. * Keep the throat out of the way. == Exercises == ==== Buzzing ==== The purpose of these exercises is to get the embouchure working on the mouthpiece even before you connect it to the horn. Make sure that you use correct breath support. I would recommend using a piano (or other pitch source) to make sure you have the correct pitch.<br> Long tones: * Match the each pitch with your pitch source for as long as your breath will support. * Chromatically from Low C up to Middle C Sirens: * Glissando from C (below middle C) up to F, slowly. Next, quickly. * From C to C, slowly and quickly. ==== Breathing exercises ==== As you do these exercise, make sure that you keep the motion of air in and out as constant and consistent as possible. Do each of these several times in a row before moving on to another. * Breathe in for 8 counts, breathe out for 8 counts * In for 4 counts, out for 8 counts * In for 2 counts, out for 8 counts * In for 1 count, out for 8 counts * In for 8 counts, out for 4 counts * In for 8 counts, out for 2 counts * In for 8 counts, out for 1 count Advanced: * In for 8, out for 16 * In for 8, out for 12 * In for 4, out for 16 * In for 4, out for 12 * In for 2, out for 16 * In for 2, out for 12 * In for 1, out for 16 * In for 1, out for 12 The purpose and goal of these exercises is to create and even and consistent airflow. The intake and output should be of consistent speed throughout the whole count. |[[Category:Learning activities]] [[Category:Euphonium]] psa9x4tzgz2twurmu6af3ppgljzqkj3 Music Theory I 0 5655 2812872 2696151 2026-06-05T15:48:48Z Kirby - Electrotechnics 3074947 Removed from the main "Category:Music" 2812872 wikitext text/x-wiki {{TOCright}} == Introduction == This is the first Music Theory course. It provides an introduction to basic music theory, pitches, pitch collections (scales), key signatures, rhythmic meters, and modes. You should already know music fundamentals such as clefs, staves, accidentals, dynamics, and rhythmic values. For more fundamental musical theory, see [[Introduction to music|Introduction to Music]] or [[Fundamentals of Music]]. == What is music? == The definition of music itself is elusive. We want to be general enough to include a wide range of styles and cultures and approaches, yet not so wide that the term "music" can mean anything, and therefore, nothing. Here are some attempts: * Music is a series of tones * Music is sound organized by humans * Music is an art-and-science * Music is an abstract communication language * Music is an emotional expression of the soul * Music is made up of the elements of melody, harmony, rhythm, timbre, texture, dynamics, and form * Music is the created order of nature * Music is culturally-defined psychological expression in sound * Music is love All of these definitions have merit and truth. For now, let us define music as an art form expressed through the medium of sound, organized aurally (what we hear) and rhythmically (when we hear it). == What is music theory? == When we say music theory, we usually aren't talking about a universal system of understanding music. Rather, we are focusing specifically on '''Western Tonal Harmonic''' music, or '''"common practice"''' music. This type of music was developed from 1650 to 1900 in Western Europe. We consider it "common practice" not because it is the only good sounding music, but because it was a widely accepted system of analyzing music for three hundred years and is the foundation for much classical, pop, rock, and some aspects of blues and jazz music. Let us define this type of music theory a little further. Music theory is merely a system of describing, notating, communicating, and analyzing music. It is not a "theory" like the theory of relativity, but rather a framework for understanding many types of music. It does ''not'' describe how music should work or what makes music work, but is simply a method by which composers, musicians, educators and anyone else can communicate and understand Western Tonal music. While there are certain advanced elements to music theory (such as tuning and the overtone series) which have a "universal" application due to being based on the physics of vibration, the more common elements of music theory that are taught in universities today are these aspects of Western Tonal Harmony. The music theory we discuss here would not be particularly suited to talking about Indonesian Gamelan music or very modern classical pieces, both of which would require a different method of music theory analysis. == Pitch == All music is based on sound and when we refer to a specific tone we are referring to a pitch. Every key on a piano plays a different pitch. In traditional western music, we focus on twelve pitches to make music. They are as follows: <p style="text-align:center;">C, C♯/D♭, D, D♯/E♭, E, F, F♯/G♭, G, G♯/A♭, A, A♯/B♭, and B</p> It may be helpful to picture a piano keyboard: [[Image:musictheory1_pitches.jpg|frameless|center|The twelve pitches that make up Western Tonal Harmony]] If you've seen a piano before, you'll notice that after we get to B, the whole thing just starts over again. Normally, there are eight C-keys on a piano. Each of those keys plays the same sound (in this case, the tone C) but higher or lower depending on the key. Therefore, we would say that the eight C-keys on a piano are the same '''pitch class''' but different '''pitches'''. If you consult our pitches above, it may appear that seventeen notes have been listed, but this list actually only contains twelve notes. The notes with a slash between them are '''enharmonic''', meaning that they represent the same pitch with two different names. For example, C♯ (C sharp) and D♭ (D flat) are actually the same. To see this, these notes both get played by the same black key on a piano. The '''sharp''' and '''flat''' symbols should be familiar to you by now, but let us have a quick review. A sharp is one pitch higher than the note it is attached to. For instance F♯ (F sharp) is one pitch higher than F. Inversely, a flat is one pitch lower than the note it is attached to. Thus, E♭ (E flat) is one pitch lower than E. You might be tempted to think that any sharp or flat is the same thing as a black key on the piano, but this is not so. For instance, E♯ is enharmonic with F and C♭ is enharmonic with B. While they are not quite as common, there is also a '''double sharp''' (x) and a '''double flat''' (♭♭). These symbols only mean that the tone is ''two'' pitches higher or lower than the note they are attached to. For example, A♭♭ would be enharmonic with G and Dx would be enharmonic with E. The majority of Western music has emerged from the twelve pitch classes listed above. Just about every Western scale and chord (and a great many non-Western scales and chords) are founded from these notes. == Intervals == === The octave === As was briefly mentioned above, all the C-keys on a piano are in the same pitch class, but are different pitches. If you remember, we also said that after we list the twelve pitches, we start over. For example... <p style="text-align:center;">C, C♯/D♭, D, D♯/E♭, E, F, F♯/G♭, G, G♯/A♭, A, A♯/B♭, B, C, C♯/D♭, D, D♯/E♭, E, F</p> ...and so on and so on. If you look at any pitch "C", what is twelve pitches above it? Another C. What is twelve pitches above that? Yet another C. This distance (of twelve pitches) is called an octave. Another way to think of it would be that an octave is the distance between two consecutive pitches in any pitch class. So if our pitch class is "C" we might have the following pitches: C1, C2, C3, C4, C5, C6, C7. The distance between C3 and C4 is one octave. The distance between C2 and C5 is three octaves. === Basic intervals === The octave is a way of measuring musical distance. The distance between any two notes is called an interval. There are twelve basic intervals (corresponding with the twelve pitches) but there are two that particularly important: the '''half step''' (or '''semitone''') and '''whole step''' (or '''whole tone'''). A half step is the distance between two consecutive pitches. For instance, the distance between G and A♭, or F and F♯, or B and C. You could think about a piano; the distance between any key to the keys immediately next to it is one half step. A whole step is two half steps. So the distance between D and E, or A♭ and B♭, or E♭ and F is one whole step. These intervals are listed in order, beginning at the Unison and progressing up to the octave: {| class="wikitable" |- ! Interval Name !! Interval Distance !! Example |- | '''Perfect Unision''' or '''Unison''' or '''P1''' || The distance between one pitch and itself, in other words, no difference. The same pitch twice in a row. || C7 to C7, or F4 to F4 |- | '''Minor Second''' or '''Half Step/Semitone''' or '''m2''' || The distance of one half step. || D to E♭, or G to G♯ |- | '''Major Second''' or '''Whole Step/Whole Tone''' or '''M2''' || The distance of one whole step; the distance of two half steps. || A to B, or E to F♯ |- | '''Minor Third''' or '''m3''' || The distance of one whole step and one half step; the distance of three half steps. || D to F, or G to B♭ |- | '''Major Third''' or '''M3''' || The distance of two whole steps; the distance of four half steps. || E♭ to G, or A to C♯ |- | '''Perfect Fourth''' or '''P4''' || The distance of two whole steps and one half step; the distance of five half steps. || C to F, or A♭ to D♭ |- | '''Tritone''' or '''Augmented Fourth/Diminished Fifth''' or '''+4/°5''' || The distance of three whole steps; the distance of six half steps. || D to G♯, or F to C♭ |- | '''Perfect Fifth''' or '''P5''' || The distance of three whole steps and one half step; the distance of seven half steps. || E to B, or F♯ to C♯ |- | '''Minor Sixth''' or '''m6''' || The distance of four whole steps; the distance of eight half steps. || A♯ to F♯, or E♭ to C♭ |- | '''Major Sixth''' or '''M6''' || The distance of four whole steps and one half step; the distance of nine half steps. || D♯ to B♯, or F to D |- | '''Minor Seventh''' or '''m7''' || The distance of five whole steps; the distance of ten half steps. || G to F, or C to B♭ |- | '''Major Seventh''' or '''M7''' || The distance of five whole steps and one half step; the distance of eleven half steps. || E to D♯, or G♯ to F |- | '''Perfect Octave''' or '''Octave''' or '''P8''' || The distance of six whole steps; the distance of twelve half steps. || E♭4 to E♭5, or D6 to D7 |} The naming conventions come from the following set of rules, which you don't need to learn, but are useful to know about at least. * The unison, fourth, fifth, and octave are perfect. The term perfect doesn't mean that the notes are superior, but comes from the very clear, resolute, final, "perfect" sound they have. * The m3, M3, m6, and M6 are considered consonant intervals because they produce a fairly pleasant, stable sound, although not as stable as the perfect intervals. * The m2, M2, m7, and M7 are considered dissonant intervals because they produce a fairly tension-filled sound that needs to resolve. * A minor interval is smaller than its major counterpart by a semitone. Conversely, a major interval is larger than its minor counterpart by a semitone. * An augmented interval is larger than its perfect/major counterpart by a semitone. * A diminished interval is smaller than its perfect/minor counterpart by a semitone. Although they weren't listed above, there are more intervals. For instance, as stated above, a diminished interval is smaller than its perfect/minor counterpart by one half step. Therefore, a diminished fourth would be a perfect fourth (5 half steps) minus one half step, ending up with four half steps. You may notice that this is identical to the distance of a major third. The difference between the two is how we would write the interval on the page, but there would be no aural difference. Take care to note that you cannot have a minor fifth or a major fourth, for such terms are impossible. If you were to increase the distance in your perfect fifth, you would have an ''augmented'' fifth, and so on. We can name these intervals in a universal way, consisting of the following: <p style="text-align:center;">[Modifying Symbol][Number]</p> where the modifying symbol denotes perfect (P), major (M), minor (m), augmented (+), or diminished (°), and the number corresponds to the name of the interval - so 2 for second, 6 for sixth, etc. Let us talk briefly about enharmonics, for when it comes to intervals they ''do'' make a difference. If you started with C and wanted a minor sixth, you would go up eight half steps until you got to G♯/A♭, however you wouldn't use the note G♯. The distance between C and G♯ is an augmented fifth. If you wanted a minor sixth, you would say C to A♭. The difference here is, again, written but not aural. There is no discernible change from an augmented fifth or a minor sixth, but for our analytical purposes, it is a very important distinction. === Compound intervals === There is a system of naming notes past the octave, used commonly in jazz but also to an extent in modern classical and pop music, called compound intervals. These are ''essentially'' the same as the basic intervals, but expanded by an octave (12 semitones remember). Our naming and notational conventions carry over, but we add seven to the note number. For example, a minor sixth (8 semitones) goes up an octave to become a minor thirteenth (now 20 semitones). Where we notate it before at m6, we now notate with m13. Again, this system is used mostly in jazz. For now, as we study Western Tonal Harmony, we will use our basic intervals. If you were to see a minor thirteenth in a piece of music, we would simplify and call it a minor sixth. == Meter and rhythm == === Introduction to meter === As you listen to most music, you will notice a discernible '''beat''', or a primary pulse. In common practice music, we can either divide this beat into two parts (and multiples of two, such as four), which we call '''simple meters''', or we can divide it into three parts (and multiples of three, such as six), which we call '''compound meter'''. When we say we can divide the beat into two or three, all we are saying is that it is ''easiest'' to divide the beat into this number. We could arbitrarily divide the beat into five or seven, but almost all music naturally divides into two or three. Notice all that we have been talking about is how the beats divide. But beats can also be grouped into larger sections. We could place beats into groups of two (which we call '''duple'''), three (which we call '''triple'''), or four (which we call '''quadruple'''). We could place the beats into as large a group as we want, but for now let us stay with duple/triple/quadruple as they are the most common. These groupings of beats are called '''measures''' or '''bars''' and are shown on sheet music with thin lines to divide the measures. Thus, we could have a piece of music with a simple triple meter. Refer to the terms above. A simple triple meter would be one in which we could put the beats into groups of three and the beats themselves divide easily into two parts. The familiar song "Amazing Grace" is in simple triple meter. Take a look at the chart below: {| class="wikitable" |- ! Beat Division !! Beat Grouping !! # of Beats in a Measure !! Familiar Music |- | Simple || Duple || 2 || ''The Stars and Stripes Forever'' by John Philip Sousa ''Mary Had a Little Lamb'' (Traditional) |- | Simple || Triple || 3 || ''An der schönen blauen Donau (The Blue Danube Waltz)'' by Johann Strauss II ''Amazing Grace'' by John Newton/William Walker ''Caribbean Blue'' by Enya |- | Simple || Quadruple || 4 || ''Don't Stop Believin'' by Journey ''Yesterday'' by The Beatles ''Bad Romance'' by Lady Gaga |- | Compound || Duple || 2 || ''The Washington Post March'' by John Philip Sousa ''We Are The Champions'' by Queen |- | Compound || Triple || 3 || ''Jesus bleibet meine Freude (Jesus, Joy of Man's Desiring)'' from Herz und Mund und Tat und Leben (Heart and Mouth and Deed and Life) by Johann Sebastian Bach ''Walkürenritt (Ride of the Valkyries)'' from Die Walküre (The Valkyrie) by Richard Wagner ''Clair de Lune (Moonlight)'' from Suite Bergamasque by Claude Debussy |- | Compound || Quadruple || 4 || ''Memory'' from Cats by Andrew Lloyd Weber ''Lights'' by Journey |} No doubt you have heard the terms rhythm and meter by now, perhaps interchangeably, however let us separate the two words: '''rhythm''' refers to the length of a pitch in a piece while '''meter''' is the framework of strong and weak beats in which rhythm is placed. === Meter signatures === A '''meter signature''' or a '''time signature''' is located at the beginning of a piece and sets the meter type (Simple/Compound and Duple/Triple/Quadruple) and the '''beat unit'''. The beat unit is the note length which gets one beat. A time signature is set up like this: <p style="text-align:center;">[Beats per Measure]<br> [Beat Unit]</p> The top number will tell us what the meter type is while the bottom number will tell us the beat unit. If the upper number is 2, 3, or 4, it will usually mean that the meter is simple duple, simple triple, or simple quadruple respectively. If the upper number is 6, 9, or 12, the meter will likely be compound duple, compound triple, or compound quadruple. The lower number indicates the following: 2 = half note, 4 = quarter note, 8 = eighth note, 16 = sixteenth note. You can reference this chart for easier understanding: {| class="wikitable" |- ! Meter Signature !! Beats per Measure !! Beat Unit !! Meter Type |- | 2/4 || 2 || Quarter Note || Simple Duple |- | 3/4 || 3 || Quarter Note || Simple Triple |- | 4/4 || 4 || Quarter Note || Simple Quadruple |- | 2/2 || 2 || Half Note || Simple Duple |- | 3/2 || 3 || Half Note || Simple Triple |- | 3/8 || 3 || Eighth Note || Simple Triple |- | 4/8 || 4 || Eighth Note || Simple Quadruple |- | 4/16 || 4 || Sixteenth Note || Simple Quadruple |- | 6/8 || 2 || Dotted Quarter Note || Compound Duple |- | 9/8 || 3 || Dotted Quarter Note || Compound Triple |- | 12/8 || 4 || Dotted Quarter Note || Compound Quadruple |- | 6/4 || 2 || Dotted Half Note || Compound Duple |- | 9/4 || 3 || Dotted Half Note || Compound Triple |- | 6/16 || 2 || Dotted Eighth Note || Compound Duple |- | 9/16 || 3|| Dotted Eighth Note || Compound Triple |} == Pitch collections == The traditional Western music that we are studying does not usually use all twelve of our pitch classes. Instead, we have discovered that it is aurally pleasing to use only some of the twelve pitch classes at any given time. We call this a '''pitch-class collection'''. If we were to select all twelve of the pitch classes we would have a '''chromatic''' collection. A chromatic scale is one in which all of the pitch classes are played. However, we normally use a '''diatonic''' collection ("diatonic" with the root of "tonic", referring to the tones) of seven pitch classes. These diatonic collections usually take the form of the major and minor scales. === Major scale === No doubt you have heard of the '''major scale''' before. As we have said above, the major scale is a diatonic pitch-class collection. The major scale employs seven of the twelve pitch classes and is composed of the unison (P1), the major second (M2), the major third (M3), the perfect fourth (P4), the perfect fifth (P5), the major sixth (M6), and the major seventh (M7) before repeating again. This particular selection of pitches is what gives the major scale its distinct sound and what differentiates it from other pitch-class collections (such as the minor scale). As an example, if we choose C to be the root (the starting point of a pitch-class collection), the Major scale would be: <p style="text-align:center;">C, D, E, F, G, A, B, C, etc.</p> [[Image:musictheory1_cmajor.png|frameless|center|600px]] Note that we repeat the C at the top of the scale. So we use eight notes but only seven pitches (because the C is repeated). Another way to describe the Major scale is to describe the steps between each note of the scale. We know that the distance between each neighboring pitch is one half step. If we use "H" to name a half step and "W" to name a whole step, the Major scale can be described as WWHWWWH. One creates a Major scale by first choosing a certain pitch as the root, or starting point. Because it started on "C", the C Major scale was shown above has a "C" root. Other major scales can be created by starting on other notes and then by following the pattern of WWHWWWH from that root. Any of the twelve pitches may be the basis for the major scale. Some major scales are redundant in the sense that they sound the same as their enharmonic counterparts (e.g. G♭-major sounds the same as F♯-major). Some redundancies are more common while others are not. Generally, if one has to go beyond the use of single sharps or flats, then one has created a redundant scale which can be more simply expressed (e.g. G♭♭-major: G♭♭, A♭♭, B♭♭, C♭♭, D♭♭, E♭♭, F♭, G♭♭ is incredibly redundant compared to F-major: F, G, A, B♭, C, D, E, F). === Minor scale === The '''minor scale''' is another diatonic pitch-class collection. Like the major scale, the minor scale employs only seven notes from the twelve available pitches. There are many types of minor scales but for our purposes, we are going to focus on three. The '''Natural Minor''' scale, the '''Harmonic Minor''' scale, and the '''Melodic Minor''' scale. ==== The natural minor ==== This is the most common form of the minor scale, and can be represented by the pattern WHWWHWW. A common way to create a natural minor scale is to take a major scale a lower scale degree 3,6, and 7 by a semi-tone. The scale could also be thought of in terms of intervals and is composed of the unison (P1), the major second (M2), the minor third (m3), the perfect fourth (P4), the perfect fifth (P5), the minor sixth (m6), and the minor seventh (m7). Using "A" as our root, the scale would be as follows: <p style="text-align:center;">A, B, C, D, E, F, G, A</p> [[Image:musictheory1_anaturalminor.png|frameless|center|600px]] ==== The harmonic minor ==== The Harmonic Minor scale is almost exactly the same as the Natural Minor scale with the exception of one pitch. A common way to create a harmonic minor scale is to take a major scale a lower scale degree 3, 6 by a semi-tone, leaving scale degree 7 as written in its parallel major. It can be represented as WHWWH(W+H)H. Note the large Whole+Half Step (3 semitone) in parenthesis. This augmented second interval between the sixth and seventh pitches is what distinguishes the Harmonic Minor scale and was created to add more tension and interest to the music incorporating it. When A is the root, the scale runs: <p style="text-align:center;">A, B, C, D, E, F, G♯, A</p> [[Image:musictheory1_aharmonicminor.png|frameless|center|600px]] ==== The melodic minor ==== The Melodic Minor is very distinct from the previous scales discussed in that it has two forms, '''ascending''' and '''descending''', which are used respectively according to rise and fall of the scale. The descending form of the scale is the same as the Natural Minor scale: <p style="text-align:center;">A, G, F, E, D, C, B, A</p> [[Image:musictheory1_ameldoicminordescending.png|frameless|center|600px]] Note that the pitches are in reverse order because, obviously, we are descending downwards. The ascending form is a kind of combination between major and minor scales. It can be represented as WHWWWWH or: <p style="text-align:center;">A, B, C, D, E, F♯, G♯, A</p> [[Image:musictheory1_amelodicminorascending.png|frameless|center|600px]] === Modes === Modes are another type of pitch-class collection. There are seven common modes and they are all diatonic. Let's build a major scale on "C": <p style="text-align:center;">C, D, E, F, G, A, B, C</p> We know that this is a major scale but it is also one of the seven modes, called '''Ionian'''. The Ionian Mode and a major scale are one and the same. Let us imagine the same seven pitches (C, D, E, F, G, A, B) but using instead the root of "D". <p style="text-align:center;">D, E, F, G, A, B, C, D</p> This is no major scale, nor is it any minor scale we have learned. It is, however, a mode, called '''Dorian'''. If we continue this pattern, using each of the seven pitches as our root, we will get all of our modes. If we use "E" as the root, the mode would be '''Phrygian'''. If we use "F", we would have '''Lydian''' Mode. If we use "G", we would have '''Mixolydian'''. If we use "A", we would have '''Aeolian''' Mode. Let us take a closer look at Aeolian Mode: <p style="text-align:center;">A, B, C, D, E, F, G, A</p> Look familiar? The Aeolian Mode and the Natural Minor scale are the same thing, just like the Ionian Mode and the major scale. Finally, if we used "B" as the root, we would get '''Locrian''' Mode. Remember that even though we used only C, D, E, F, G, A, B in our example, a mode can be built on any pitch. {| class="wikitable centre" |- ! Mode ! Root relative to major scale ! Root relative to C Major scale ! Interval Sequence ! Example I ! Example II |- | Ionian | I | C | align="center" | WWHWWWH | C, D, E, F, G, A, B, C | C, D, E, F, G, A, B, C |- | Dorian | II | D | align="center" | WHWWWHW | D, E, F, G, A, B, C, D | C, D, E♭, F, G, A, B♭, C |- | Phrygian | III | E | align="center" | HWWWHWW | E, F, G, A, B, C, D, E | C, D♭, E♭, F, G, A♭, B♭, C |- | Lydian | IV | F | align="center" | WWWHWWH | F, G, A, B, C, D, E, F | C, D, E, F♯, G, A, B, C |- | Mixolydian | V | G | align="center" | WWHWWHW | G, A, B, C, D, E, F, G | C, D, E, F, G, A, B♭, C |- | Aeolian | VI | A | align="center" | WHWWHWW | A, B, C, D, E, F, G, A | C, D, E♭, F, G, A♭, B♭, C |- | Locrian | VII | B | align="center" | HWWHWWW | B, C, D, E, F, G, A, B | C, D♭, E♭, F, G♭, A♭, B♭, C |} == Key signatures == A '''key signature''' shows which pitch classes are to be consistently played sharp or flat throughout the piece. It is placed at the beginning of each line of a score. To help understand why key signatures are beneficial, look at this example: [[Image:musictheory1_keysigexample.png|frameless|center|800px]] Even though both images above are of the same music, the second version is much easier to read, analyze, and to play. The key signature, by placing the flat accidentals in a tidy formation, rather than having them dispersed across the page, makes the reading of them much easier. We have eight sharp key signatures and eight flat key signatures. They look like the following:<br> [[Image:musictheory1_keysignatures.png|center|600px]]<br> You may have noticed that each time we add a sharp, we use the pitch a P5 higher. In other words, the order in which we add sharps is: <p style="text-align:center;">F♯, C♯, G♯, D♯, A♯, E♯, B♯</p> After this, another P5 about B♯ is F♯ again so, therefore, that is as many sharps as we can add. You also may have noticed that once we have B♯, all the pitches have the sharp accidental attached and it would be impossible to sharp any more pitches without using double sharps. Each time we add a flat, we use the pitch a P5 lower. The order of flats is: <p style="text-align:center;">B♭, E♭, A♭, D♭, G♭, C♭, F♭</p> Similarly, after F♭, the next pitch would be B♭ again. Many students use a mnemonic to remember the order of sharps and flats. A very common mnemonic would be "Father Charles goes down and ends battle" (the order of sharps), or reversed, "Battle ends and down goes Charles' Father" (the order of flats). You are welcome to use whatever you like but the key signatures are something that should be memorized. Some of the key signatures are enharmonic with each other: D♭ is enharmonic with C♯, G♭ with F♯, and C♭ with B. To better understand this, take a look at the '''circle of fifths''', with shows each of the key signatures, which major or minor key it may represent, the number of sharps or flats it contains, and which keys are enharmonic. [[Image:Circle_of_fifths_deluxe_4.svg|center|frameless|600px]] You may be wondering how we can know what the major or minor key of the piece is from the key signature. Well, the key signature with three flats indicates a major or minor key with B♭, E♭, and A♭. The major key that has such flat notes is E♭ Major, and the minor key is C Minor. However, there are a few tricks that make it easier to determine the key without having to consult sheets of key signatures and major and minor scales. # To determine the major key of a flat key signature, look at the next-to-last flat. The key signature with six flats is G♭ Major. Of course, this does not work with only one flat, but there is only one key signature with one flat: F Major. # To determine the major key of a sharp key signature, look at the last sharp, then move up one half-step. For instance, the key signature with four sharps in E Major. Take care to note that if you move up one half-step but they key signature contains a sharp on that note, you must include the sharp. For instance the key signature with seven sharps is C♯ Major ''not'' C Major, which has no sharps. Be careful that you do not assume that the key signature automatically determines the actual ''key'' of a piece. The key signature might have two flats, but this does not mean that the piece is in B♭ Major. It could be in G Minor, or C Dorian, or E♭ Lydian, or... etc. When determining the key of the piece, we must consider the key signature in combination with the pitch-class collection and the interactions of the intervals within the music. == References == <references/> [[Category:Music theory]] igliu0ojiqvebs6iovd7wx5u14q0tcr Building A Cigar Box Guitar 0 5989 2812956 2811276 2026-06-05T21:44:53Z Kirby - Electrotechnics 3074947 changed to 'category:Guitar' instead of category:Music 2812956 wikitext text/x-wiki From the [[w:Cigar_box_guitar|Wikipedia Article]]: The cigar box guitar is a primitive chordophone whose resonator is a discarded cigar box. Because the instrument is homemade, there is no standard for dimensions, string types or construction techniques. Many early cigar box guitars consisted only of one or two strings that were attached to the ends of a broomstick that was inserted into the cigar box. Other cigar box guitars were more complex, with the builder attempting to simulate a real stringed instrument, such as a guitar, banjo, or fiddle. Currently, there is a resurgence in this humble instrument and musicians are creating many new designs. Some are adding additional strings and necks while others are incorporating active electronics that make them playable through guitar amplifiers. These musicians gather daily in the Cigar Box Guitar Forum [1] to discuss new construction techniques, playing styles and more. There are also several cigar box guitar festivals being organized throughout the United States. If a cigar box is not available, other substitutes have been suggested. Notably, an unused pizza box. == Project Outline == This project is designed to help those in need or want to build a homemade guitar out of a couple of spare parts found inexpensively. For the average price of about $15 (not including necessary tools), you can craft and complete a working instrument, electrified or acoustic, to add a unique flair to performance or casual play. == Materials Needed == *Wooden Cigar Box (the larger, the better) *3 foot long 1x2 (Poplar wood is most affordable for its durability) * Elmer's Wood Glue, or other company's wood glues, or Gorilla glue *Tuning Machines *1/2 pint sealant/stain *Sponge Paint Brush (1/2" to 1" works well) *Sandpaper (220 grain) *Drill w/ multiple sized bits *Hacksaw/Coping Saw *1/2" Woodfile == Getting Started == There are as many ways to construct a Cigar Box Guitar (or CBG) as there are people on this earth, what I am outlining for you is my own method of construction I have come up with through trial and error. To keep things simple, I am going to stick with the most basic techniques. Make sure you have a couple of hours to dedicated to the initial steps of construction, as these require the most care. Give yourself plenty of working space, and depending on where you are working, be prepared to clean up a lot of sawdust and wood scrap. Make sure to envision what you want your CBG to look like before you jump in and start making cuts in your wood. This can be a slow learning process, so just be patient and sooner than you may think, you'll have your very own hand-built CBG to show off and play. ==See also== *[[Introduction to GarageBand]] - If you have a digital recording of your Cigar Box Guitar, please donate it to GarageBand project. [[Category:Craft arts]] [[Category:Guitar]] [[Category:Woodworking]] rdxnrs3p4xnuj471xpbvz2df3hegyyy Basic Blues & Rock 0 7529 2812863 2480320 2026-06-05T15:35:04Z Kirby - Electrotechnics 3074947 Removed from the main "Category:Music" 2812863 wikitext text/x-wiki {{TOCright}} Welcome to the Wikiversity '''Rock & Blues Garage''', part of the [[School:Music and Dance|School of Music]] and home of the [[Talk:Basic Blues & Rock|Garage Band]]. Get ready to Rock! == Department description == Participants study the basic Blues patterns, the elements of Rock & Roll, Hard Rock, Headbanging, Jamming, Establishing a Groove, and other Garage Band traditions. == Department news == * '''Friday 13 October 2006''' - Department founded! * '''02:02, 13 February 2007 (UTC)''' - Guitar lessons being recorded ... tune your [[w:IRC|IRC]] client to [irc://irc.freenode.net/wikiversity-en #wikiversity-en] for details in [[w:real time|real time]]. == [[Portal:Learning Projects|Learning Projects]] == See: [[Wikiversity:Naming conventions#Learning Projects|Learning Projects]] and the [[Wikiversity:Learning]] model. === Portals === '''[[Portal:Music]]''' * [[Introduction to music|Introduction to Music]] * [[Understanding music]] * [[Introduction to Chords]] * [[Beginners Music Theory]] * [[Key Signatures]] * [[Scales]] * [[Intervals]] * [[Aural Training]] * ... Learning materials and [[Portal:Learning Projects|learning projects]] are located in the main Wikiversity namespace. Simply make a [[link]] to the name of the learning project (learning projects are independent pages in the [[Wikiversity:Namespaces|main namespace]]) and start writing! We're going to be using [[Template:Learning project boilerplate|learning project template]], (<nowiki>{{subst:Template:Learning project}}</nowiki>) to build our new: * '''[[Blues and rock|Portal:Music/Blues & Rock]]''' === Collaborations === * [[Wikiversity the Movie/Music|Wikiversity the Movie/music]] - participants make the sound track, including original music, for [[Wikiversity the Movie]]. * [[Music/Software/Creating music with your PC|Creating music with your PC]] - participants make music using their personal computer * [[Portal:Audio engineering|Audio Engineering]] - we're going to try to build a repository of original audio files, through a new project... * [[Portal:Internet audio and video|Internet Audio and Video]] - [[Internet audio|Wikio]] - [[Wikiversity]]'s [[w:Web radio|Web radio]] station, where we may eventially be able to facilitate [[Jamming Online]]. (who knows?) * [[Getting started with sound recording]] - need to do this right away * ... === Lessons === Remember, Wikiversity has adopted the "learning by doing" model for education. Lessons should center on learning activities for Wikiversity participants. We learn by doing. * [[Blues basics]] * [[Rock basics]] * [[Blues & Rock instruments]] * [[Guitar/Lessons|Guitar tuning]] * [[Circle of fifths]] * ''add many!'' === Sessions === We're going to customize [[Wikiversity:Naming conventions#Learning project_codes|Learning project codes]], by adding "sessions": * [[Blues and rock|Portal:Music/Blues & Rock]] * [[Guitar/Lessons|Tuning up]] * [[Blues basics|12-bar Blues Jam]] * [[Rock basics|Learn to Rock]] * [[Forming a band|Forming a Band]] * ... These will probably be made from templates of some sort that work through a [[w:Tablature]] system that we'll have to come up with. The current session will be in the section '''Garage''' below and on [[Talk:Basic Blues & Rock|Topic talk:Basic Blues & Rock]]. If everything goes according to plan, we should be able to produce actual [[Portal:Audio engineering|audio]] files. See [[m:Embed Media#The Future: Native Firefox ogg Support|The Future: Native Firefox ogg Support]] and [[Portal:Internet audio and video|Internet Audio and Video]]. == Garage == Sweeping the [[Topic:Basic Blues & Rock#Garage|garage...]] ''chill for a bit'' Meanwhile, you can read [[GarageBand|Introduction to GarageBand]] == To do == Lots of things to do See: [[Wikiversity:Naming conventions|Naming conventions]] === Chores === [[Wikiversity]] is really new and plenty of mundane chores need to be done: * A lot of things were imported from [[Wikibooks]], and need to be refactored to work well here. * [[Portal:Music]] and [[:Category:Music]] need a major sweep. * [[Wikiversity]] needs better representation at other [[w:Wikipedia:Sister projects]] and we could probably help with that. ''See [[Wikiversity:Wikiversity outreach]]'' * ''add stuff here'' === Lists === We need a list of lists: * things to put in our new [[:Category:Blues & Rock]] * things to include on [[Blues and rock|Portal:Music/Blues & Rock]]. * articles for [[:Category:Blues & Rock]]: #'''genres''' #'''instruments''' #'''artists''' #... We can draw on Wikipedia for most all of this but we need to decide what to include in the Wikiversity main namespace. == Active participants == Active participants in this [[Portal:Learning Projects#Learning Groups|Learning Group]] The histories of Wikiversity pages indicate who the active participants are. Please introduce yourself fully under a new section (use '''+''' and your username as a heading) at [[Talk:Basic Blues & Rock|Topic talk:Basic Blues & Rock]] and enroll below. === Enrolled === # [[User:CQ|CQ]] • no subpage yet # [[User:Exmachina|Ex Machina]] • no subpage yet # [[User:Mirwin|Mirwin]] Music Subpage[[User:Mirwin/musical_activities]] # [[User:TheVividDream|TheVividDream]] • no subpage yet # [[User:Thierry613|Thierry613]] === User subpages === We're going to make this section a bit more accomodating to the Musician ego. We would like to ask participants to create a subpage to their userpage with a format that includes links to instrument you play (or want to learn) &ndash; [[Guitar]], [[Bass guitar|Bass]], [[Drum set|Drums]], [[Saxophone]], etc. Instrument articles should be in the main namespace. If not ''create one!'' You should also indicate styles and influences: [[w:Beatles]], [[w:Pink Floyd]], [[w:Electronica]], [[w:Delta Blues]], etc. If you have your own website, you can of course list that, too, but see the next section. If you are a member of a Music or related [[w:Category:WikiProject Music|WikiProject at Wikipedia]], you can show that also. Once we come up with a format, we'll do a template like <nowiki>{{MusicStudent}}</nowiki> or something. Meanwhile just introduce yourself on the talk page. Someday, we may form the [[Talk:Basic Blues & Rock|Wikiversity Gargage Band]]! == Resources == These are mostly at [[Wikiversity:Sister projects]] (under the [[Wikimedia|Wikimedia Foundation]] umbrella). [[Wikiversity]] resources should be listed above this section. === Meta === [[m:Main Page|Meta]] * [[m:Embed Media]] * ... === Wikibooks === [[b:Main Page|Wikibooks]] These may be imported to Wikiversity later : * [[b:The Physics of Music|The Physics of Music]] * [[b:Triads|Triads]] * ''add other wikibooks here'' === Commons === '''[[commons:Main Page|Commons]] :''' * [[commons:Category:Music]] * [[commons:Category:Wikiversity music]] '''''New!''''' === Wikipedia === '''[[w:Main Page|Wikipedia]] :''' * [[w:Portal:Music]] * [[w:Portal:Rock and Roll]] * ''[[w:Category:Tuning]] START HERE'' * [[w:Category:WikiProject Music]] * [[w:Wikipedia:WikiProject Music terminology]] * [[w:Wikipedia:WikiProject Songs]] * ... === External resources === This section is not for [[w:linkspam|linkspam]]! These should be links to ''relevant resources''. <small>USE COMMON SENSE!: If and only if you are enrolled above, you may include a link to your site below preceded by three tildes. If your sight has innapropriate content, obnoxious advertising ploys, or other crap, its link will be removed!</small> * [http://www.wikia.com/wiki/Category:Music Category:Music @ Wikia.com] * [http://www.music.vt.edu/musicdictionary/ Virginia Tech Multimedia Music Dictionary] * [http://www.ladspa.org/ Linux Audio Developer's Simple Plugin API (LADSPA)] (for [[Portal:Audio engineering|Techs]]) * [http://www.tapeop.com/ TapeOp Magazine] - Creative Music Recording Magazine <small>[[w:Tape Op|Tape Op]]: Free in US and UK but... "''Tape Op Magazine requires you to have a valid email address to maintain and renew a free subscription. If you do not have a valid email address, you can purchase a first class subscription''"</small> * ... [[Category:Blues & rock]] p339mgg26smxd8nbqsu7r6miqvdvci8 Category:Blues & rock 14 7531 2812858 1537887 2026-06-05T15:25:34Z Kirby - Electrotechnics 3074947 changed to 'category:Music genres and styles' instead of category:Music styles 2812858 wikitext text/x-wiki [[Category:Music genres and styles]] qhimz008596xnk5444wagy28xmwcppl History of Ireland through Song 0 8195 2813013 1881086 2026-06-06T00:35:02Z MathXplore 2888076 added [[Category:Songs]] using [[Help:Gadget-HotCat|HotCat]] 2813013 wikitext text/x-wiki :''Part of the Department of [[Portal:European History|European History]]'' {{TOCright}} == Introduction == Ireland's history is rich, and richly recorded in song. The purpose of this project is to provide a listing of songs (and, if possible, a way to obtain them legally, for free), corresponding to various elements of Irish history, complete with commentary and background. The goal is to develop a greater knowledge and understanding of Irish history through the memorable and enjoyable medium of music (particularly Celtic music). ==Tasks== * Create outline of Irish History (this is in the works—see talk page) * Find and add songs relevant to as many aspects of Irish history as possible ====Questions==== (Please discuss on the [[Talk:History of Ireland through Song|talk page]]). * Should we include Scotland, as well, and have a course on the two together, or separately? * Should there be an option to include songs in Gaelic (or even other languages) for ambitious students? ==Participants== ====Coordinator==== * [[User:Jade Knight|The Jade Knight]] ====Contributors==== * [[User:Jade Knight|The Jade Knight]] * [[User:Cormaggio|Cormaggio]] * [[User:Kaycee|Kaycee]] * [[User:Wilkinson|Wilkinson]] * [[User:Geofferybard|<b>{{font|color=#990000|Geo}}{{font|color=#000099|Bard}}</b>]][[User_talk:Geofferybard|{{font|color=#000000|size=smaller|<sup>Rap</sup>}}]] ====Interested Students==== * [[User:Danielle_Ralph|Danielle Ralph]] * [[User:Ioana_med|Ioana]] * [[User:Koala|Koala]] * [[User:Donek|DónalMcK]] 10:55, 26 November 2007 (UTC) [[User:Supersarah1999|Supersarah1999]] 13:09, 8 April 2010 (UTC) * [[User:Lyrasong|Lyrasong]] *[[Mommy Of Four0927|Victoria Butler]] 17:48, 01 December 2011 ==Songs== *"Black and Tans" composed by Dominic Behan. (Full version by StoneRing available [http://www.thestonering.com/media%20kit/01.Black%20and%20Tans%20-%20StoneRing.mp3 here].) *"Bold Fenian Men" written by Peadar Kearney. (2 minute excerpt of the Kellee Brad version available [http://audio.cdbaby.com/mp3lofi/b/r/bradley-11.mp3 here]. Lyrics and midi available [http://www.contemplator.com/ireland/fenian.html here].) *"[[w:The Wind That Shakes the Barley (song)|The Wind That Shakes the Barley]]", composed by Robert Dwyer Joyce. *"Arthur McBride" - [http://www.chivalry.com/cantaria/lyrics/arthur-mcbride.html lyrics and mp3] *[[w:The Sash|"The sash my father wore"]] - [http://sniff.numachi.com/pages/tiSASHFTHR;ttSASHFTHR.html MIDI & notation] *"The Green Fields of America". Lyrics and midi [http://www.mudcat.org/@displaysong.cfm?SongID=6398 here], Original 1852 broadside image [http://bodley24.bodley.ox.ac.uk/cgi-bin/acwwweng/ballads/image.pl?ref=Harding+B+11(1413)&id=02305.gif&seq=1&size=1 here]. *"[[w:Foggy Dew|The Foggy Dew]]", (revolutionary version, 1916) composed by Peadar Kearney or Canon Charles O'Neill. Lyrics and midi [http://www.contemplator.com/ireland/fogydew.html here]. *"The Fields of Athenry" - [http://www.ireland-information.com/irishmusic/thefieldsofathenry.shtml lyrics and midi] *"[[w:The Patriot Game|The Patriot Game]], with lyrics by Dominic Behan. (Full version by the Brobdingnagian Bards available [http://www.thebards.net/music/lyrics/Patriot_Game.shtml here].) *"Banba Óir (in Gaelic), by Clannad, makes reference to ancient Irish history. (Lyrics and translation [http://www.geocities.com/celticlyricscorner/clannad/banbaoir.htm here].) *"Skellig", by Loreena McKennitt; about the monastery on Skellig ([http://www.geocities.com/celticlyricscorner/mckennitt/skellig.htm lyrics]). *"[[w:Bonny Portmore|Bonny Portmore]]" (traditional). ([http://www.geocities.com/celticlyricscorner/mckennitt/bonny.htm lyrics].) *"[[w:From Clare to Here|From Clare to Here]]", by Ralph McTell (about immigration). (Lyrics available [http://www.thestonering.com/toast%20to%20the%20company.htm here].) *"Famine" by Sinead O'Connor, talks about the Irish Famine [http://www.uhb.fr/langues/cei/soconnor.htm lyrics]. *"[[w:The_Wearing_of_the_Green|The Wearing of the Green]]" written at the time of the 1798 Irish rebellion. ==Background== [[Wikibooks:Music Theory/Celtic|Celtic music classifications]] [[Wikibooks:Fiddlers Companion Volume II Celtic|Notes on Celtic tunes]] ==Related resources== *The [[Portal:Irish|Irish Language Department]] [[Category:History of Ireland]] [[Category:History learning projects]] [[Category:Vocal music]] [[Category:Songs]] fwa9x4ngci7jvzm0wf4j6yq7riityno Tabla 0 8221 2812879 2224661 2026-06-05T15:54:19Z Kirby - Electrotechnics 3074947 changed to 'category:Music instruments' instead of category:Musical instruments 2812879 wikitext text/x-wiki ==Introduction== Welcome to the Tabla pages. Somebody who knows more than I do about the Tabla can write this intro. ==The Tabla== [[Image:Bayan.JPG|thumb|right|200px|The Bayan.]] The Tabla is actually a set of two drums, the Tabla and the Bayan (or Dugga). The Bayan is placed on the left under the left hand, and the Tabla goes under the right hand. The drums should touch each other, and this point between the drums should be on your body's centreline. Sit in half-lotus position. While this may be uncomfortable at first, the player will find that this position actually keeps the back straighter than plain crossed legs. It means you can play longer and with less back pain... though your foot may fall asleep. Each drum sits on a cloth ring to keep it stable. The Bayan should be more or less flat, with its head parallel to the ground. The Tabla should be tilted slightly forward, so it is easier for your hand to rest while playing. ==Diagrams for ease in learning== I've put together a couple diagrams of the drum heads to better show where given notes are placed. PLEASE: if you know Devnagari, correct these charts! <table border=1> <CAPTION><EM>The two diagrams</EM></CAPTION> <TR><TH>The Dayan (Tabla)<TD>[[Image:Dayan.jpg]]<TD>The outside rim of the head is woven rawhide, called the Gajara. The black spot in the middle of the drum is called the Syahi. The syahi is a ball of hide glue, ink, and weight that is pressed onto the drum to improve its resonance. <TR><TH>The Bayan (Dugga)<TD>[[Image:bayan.jpg]]<TD>On the outer edge of the head but inside the Gajira is another ring of rawhide called the Chat. This improves the tensile strength of the threading. The skin itself is called a Maidan. The narrow bit of Maidan showing between the Dugga's offset Syahi and the edge of the drum is the Luv. </TABLE> And here are some hand placement diagrams for the specific hand positions. <table border=1> <CAPTION><EM>Bayan hand positions</EM></CAPTION> <TR><TH>Name<TD>Hand position<TD>Devnagari Script<TD>Singhalese Script<TD>Description <TR><TH>Ge<TD><TD>{{font|size=xx-large|ग}}<TD>{{font|size=xx-large|ग}}<TD>Strike the Luv soundly with tips of both the middle and ring fingers. The heel of the hand should come off the drum to allow resonation, and the fingers should be clawed back towards the palm in a single fluid motion after coming off the strike. <TR><TH>Ge-Ge<TD>[[Image:TablaGeGe.jpg]]<TD>{{font|size=xx-large|ग}}<TD>{{font|size=xx-large|ग}}<TD>When doing many Ge strikes in succession, they alternate between the basic above and the index finger strike, done the same as above only with the index finger alone. <TR><TH>Kat/Ki/Ke<TD>[[Image:TablaKat.jpg]]<TD>{{font|size=xx-large|क}}<TD>{{font|size=xx-large|ක}}<TD>Slap the hand down onto the drum. This should not resonate. The palm should cover the Syahi. </TABLE> <table border=1> <CAPTION><EM>Dayan hand positions</EM></CAPTION> <TR><TH>Name<TD>Hand position<TD>Devnagari Script<TD>Singhalese Script<TD>Description <TR><TH>Ti (The)<TD>[[Image:Dayan.jpg]]<TD>{{font|size=xx-large|ती(ट)}}<TD>{{font|size=xx-large|ති(ට)}}<TD>later! <TR><TH>Te (Ri)<TD>[[Image:Dayan.jpg]]<TD>{{font|size=xx-large|ट(र)}}<TD>{{font|size=xx-large|ත(රි)}}<TD>later! <TR><TH>Na (Ta)<TD>[[Image:Dayan.jpg]]<TD>{{font|size=xx-large|ना/ता}}<TD>{{font|size=xx-large|නා(තා)}}<TD>later! <TR><TH>Tin<TD>[[Image:Dayan.jpg]]<TD>{{font|size=xx-large|तीं}}<TD>{{font|size=xx-large|තිං}}<TD>later! <TR><TH>Tat<TD>[[Image:Dayan.jpg]]<TD>{{font|size=xx-large|ता}}<TD>{{font|size=xx-large|තාත්}}<TD>later! <TR><TH>Thun<TD>[[Image:Dayan.jpg]]<TD>{{font|size=xx-large|तूं}}<TD>later!<TD>later! <TR><TH>D<TD>[[Image:Dayan.jpg]]<TD>{{font|size=xx-large|?}}<TD>later!<TD>later! <TR><TH>N<TD>[[Image:Dayan.jpg]]<TD>{{font|size=xx-large|न}}<TD>later!<TD>later! <TR><TH>Dhe<TD>[[Image:Dayan.jpg]]<TD>{{font|size=xx-large|धे}}<TD>later!<TD>later! <TR><TH>R<TD>[[Image:Dayan.jpg]]<TD>{{font|size=xx-large|?}}<TD>later!<TD>later! </TABLE> <table border=1> <CAPTION><EM>Combined hand positions</EM></CAPTION> <TR><TH>Name<TD>Bayan<TD>Dayan<TD>Devnagari Script<TD>Singhalese Script<TD>Description <TR><TH>Dha<TD><TD>[[Image:Dayan.jpg]]<TD>{{font|size=xx-large|धा}}<TD>{{font|size=xx-large|ධ}}<TD>later! <TR><TH>Dhin<TD><TD>[[Image:Dayan.jpg]]<TD>{{font|size=xx-large|धिं}}<TD>{{font|size=xx-large|ධිං}}<TD>later! <TR><TH>Dhi<TD><TD>[[Image:Dayan.jpg]]<TD>{{font|size=xx-large|धि}}<TD>{{font|size=xx-large|ධි}}<TD>later! <TR><TH>Dhit<TD><TD>[[Image:Dayan.jpg]]<TD>{{font|size=xx-large|?}}<TD>{{font|size=xx-large|ධත්}}<TD>later! </TABLE> [[Category:Music instruments]] 2a5a1rc7op1xax05f236xzsmkwukpg2 Trumpet 0 8315 2812878 1480478 2026-06-05T15:53:52Z Kirby - Electrotechnics 3074947 changed to 'category:Music instruments' instead of category:Musical instruments 2812878 wikitext text/x-wiki [[Image:Trumpet 1.jpg|450px|right]] The trumpet is a cylindrical-bore brass instrument that is played by the action of wind upon the lips as it passes from your lungs into the instrument. Pitch is changed both through the action of the lips as well as a system of valves that dynamically changes the length of the instrument as described below. All modern trumpets have at least three valves, and depending on the type of instrument, may have more. The vast majority of the valves found on modern trumpets are piston valves, but rotary-valve horns are still in active use by professional orchestras in America and around the world<ref>ITG Journal, March 2004, pg. 53</ref>, and are still made and produced today by manufacturers such as [http://www.scherzer-trumpets.com Scherzer Trumpets] and [http://www.schagerl.at Schagerl Trumpets]. ==Method of Playing== The playing method is rather simple. Wind is blown through the lips (formed into what is called an ''embouchure'') into the instrument through a mouthpiece. The natural vibration of the lips causes a standing wave to be set up inside the horn, which creates the sound according to the harmonic series (see the corresponding article on [[w:Harmonic_series_(music)|Wikipedia]] for more information on this concept in physics). For a trumpet tuned in Bb, the overtone series(in actual pitch, not written) would be Bb2 (the fundamental), Bb3, F4, Bb4, D5, F5, Ab5, Bb5, C6, D6, F6, and so on. The harmonic series derives from the physics of standing waves in a open tube. The actual pitches that are available are entirely dependent on the length of the tubing. However, for a given tube length, the harmonic series would follow the same pattern. All military bugle calls are achieved with this same series of notes. Modern chromatic trumpets play other notes by use of valves which allow the player to dynamically adjust the tube-length of the trumpet, enabling the use of the full chromatic scale. ===The Valve Pattern=== Most modern trumpets use three valves. Each valve, alone and in combination, modifies a given overtone note according to the following pattern. The first valve, played by the right hand index finger, modifies notes by lowering them two half steps (or a whole). The second valve, played by the right hand middle finger, lowers notes by a single half step. The third valve, played by the right hand ring finger, lowers notes by three half steps. Combinations of these valves simply add their respective modifications. For example, the first valve plus the second valve will lower the note by three half steps. The second valve plus the third will lower the overtone note by four half steps. In this manner, the trumpet is capable of producing every note of the chromatic scale. The following lists the valve fingering combination for the descending C chromatic scale starting on middle C. {| class="wikitable" style="margin: 1em auto 1em auto" |+ '''One octave chromatic scale valve pattern table''' ! '''Note''' || '''Valves''' | colspan = "3" | '''Half steps below each overtone''' |- | || || C5 || G4 || C4 |- | C5 |align="center" | 000 |align="center"| 0 || || |- | B4 |align="center" | 010 |align="center"| 1 || || |- | Bb4 |align="center" | 100 |align="center"| 2 || || |- | A4 |align="center" | 110 |align="center"| 3 || || |- | Ab4 |align="center" | 011 |align="center"| 4 || || |- | G4 |align="center" | 101 or 000 |align="center"| 5 |align="center"| 0 || |- | Gb4 |align="center" | 010 || |align="center"| 1 || |- | F4 |align="center" | 100 || |align="center"| 2 || |- | E4 |align="center" | 110 || |align="center"| 3 || |- | Eb4 |align="center" | 011 || |align="center"| 4 || |- | D4 |align="center" | 101 || |align="center"| 5 || |- | Db4 |align="center" | 111 || |align="center"| 6 || |- | C4 |align="center" | 000 || || |align="center"| 0 |- |} Note that there can be overlap from any given transposition from an overtone series note. These are called "alternate fingerings" and can be played for various musical effects. However, alternate fingerings (and even some standard fingerings) have intonation issues that need to be addressed. ==Notes== {{reflist|1}} ==References== * Burt, Jack. [http://www.trumpetguild.org/pdf/2004journal/200403Rotary.pdf "The Rotary Trumpet: An Introduction."] ITG Journal, March 2004, pp. 52-55 {{Musical instruments}} [[Category:Music instruments]] k91ch6bcobvlrbdzem88erxzphnwe5g Obsolete musical instruments 0 8316 2812881 2472264 2026-06-05T15:55:00Z Kirby - Electrotechnics 3074947 changed to 'category:Music instruments' instead of category:Musical instruments 2812881 wikitext text/x-wiki There are a great variety of other instruments that have been made that are no longer around. Here are some examples: *Basset Horn: a single reed instrument that was often bent in a variety of shapes then sent through a soundbox *Natural horn: an older version of the horn which uses a series of tubing extensions which attach at the mouthpiece hole *Racket: a double-reed instrument that sends the sound through a series of waves to deepen the sound *Saxhorns: Adolphe Sax developed both the woodwind saxophones, and an odd variety of brass instruments called "Saxhorns" *Theremin: an electronic instrument that is played by moving ones hands around *Valve trombone: apparently these instruments exist, but I've never seen one *Viol: an early form of the violin *Wagner Tuba: a tuba-like instrument that is played and built like a horn, mainly of course used by [[w:Richard Wagner|Richard Wagner]] There are a great many others, but this is a start. [[Category:Music instruments]] 9wpa1sr6p9iacib07i5kzqarjjbej68 Music appreciation and history 0 8412 2812980 2621374 2026-06-05T22:18:02Z Kirby - Electrotechnics 3074947 changed to 'category:Music appreciation' instead of category:Music 2812980 wikitext text/x-wiki {{Essay invite}} ==Elements== The first discovered [[musical scale]] was on an instrument called the [[Guqin]]. This instrument represented the [[scale of harmonics]]<ref>http://www.furious.com/perfect/experimentalstringinstruments.html</ref>. ==Middle Ages== The period of European History between 450 and 1400 A.D. is often called the [[w:Middle Ages|Middle Ages]]. The beginning of this era was considered a "dark age" by many Europeans, due to the many migrations, wars, and upheavals therein. The later period of Middle Ages is known for much more cultural growth. Many Romanesque churches and monasteries were established between 1000 and 1150 and, later, the Gothic cathedrals were established (1150-1400). As the cathedrals dominated the medieval landscape and mind, most formal music was used in these churches. Boys received musical education in cathedral schools, while women were not allowed to sing (at least not as part of the church service). Most of medieval music was vocal, however some instruments were in common use. After 1100, instruments became more and more common in church. The [[w:Organ|Organ]] was very prominent, however it was very primitive and quite different from the modern instruments. ===Gregorian Chant=== "Gregorian Chant" is the primary official liturgical music of the medieval Church. The Mass, from the Renaissance Period consists of sacred Latin texts (with the exception of the Kyrie which is written in Greek) set to melody, and is traditionally sung monophonically without instrumental accompaniment (a cappella). Gregorian chants convey the calm, meditative atmosphere of the church prayers. The chants are now usually sung in a free-flowing style with notes of equal length, because scholars were in disagreement about whether or not the notation was intended to represent notes of various durations, although scholars now generally agree that the notation divides into sets of notes of longer and shorter values. Gregorian chant is named after Pope Gregory I. He is known for reorganizing and standardizing the liturgy during the period from 590 to 604. Most of these chants were created between 600 and 1300 CE. The composers of Gregorian chant, like the sculptors who decorated early medieval churches, remain anonymous. Gregorian chant is still used liturgically, and formed the basis for later polyphonic church music. ===Church Modes=== Gregorian Chant developed a system of 8 modes which are linked to 8 "species" of octaves. These "species" are like scales but have no single tonality associated with them as modern scales do, and this is what makes Gregorian modes different from the later Renaissance modes. The octave species were divided into two sets of 4 "authentic" and 4 "plagal" ranges. By the time of the Renaissance, the 8 species developed into scales and 4 new scales were added which are generally called the Renaissance modes. A lot of western folk music follows some of the Renaissance modes, either because as it has similar origins to them or has been influenced by them. One example of such folk music, which is in the Dorian mode, is "What Shall We Do With Drunken Sailor?" A perfect example of Gregorian Chant is a well-known Alleluia. They had 8 modes in total. ===Secular Music in Middle Ages=== In spite of Gregorian Chant domination, secular music was popular too. Most written secular music was composed by [[w:troubadours|troubadours]] between the 12th and 13th centuries. Over 1650 troubadour melodies have survived. They do not have a rhythm, yet they do have regular [[w:Meter (music)|meter]] and definite beat. That's their difference from Gregorian Chant which has no meter at all. During the middle ages wandering minstrels (or jongleurs) performed music and acrobatic tricks in castles, taverns, and town squares. They were lucky enough to work in steady service and ability. ===The development of Polyphony=== For centuries, music was mostly monophonic with a single melody line. But the years of 700 to 900 were revolutionary to music. Monks at monasteries started to add second melody lines to the melodies. The second line of melody duplicated the melody of the first, but with an interval of a few tones. Music of this kind is also known as [[w:Organum|Organum]]. By the year 1100 the second line was allowed to become more independent, not being restricted to note-against-note style. The lower chant tone was sung in very long notes, and the melody was in shorter notes on top. After 1150 the school of Notre Dame introduced the measured rhythm with definite time value and clearly defined meter. Beat notation had yet to be introduced. ===14th Century, the "New Art" in France=== In the years between 1337 and 1453, Europe suffered bubonic plague, which killed one-fourth of European population. As a consequence, both the feudal system and the church were weakened. It is not surprising that secular music became stronger in this period. Composers wrote music which was absolutely independent of Gregorian Chants. The works of that time were the drinking songs and pieces which comprised the bird calls, sounds of nature, sounds of hunting. Music notation evolved, and composers were able to specify the rhythmic pattern. The beats could be subdivided into two as well as three. Syncopation, rarely used earlier, became a more common practice. Changes both in music and in art during this time were so profound that theorists call them the "New Art" or arts nova. [[w:Guillame de Machant|Guillame de Machaut]] is the most prominent figure of this period. ==Renaissance== The 15th and 16th centuries in Europe are known as the "rebirth" or "Renaissance" of human creativity. This period is known for exploration by for example, Christopher Columbus (1492), Vasco da Gama (1498), and Ferdinand Magellan(1519-1522). It was the age of remarkable curiosity and creativity. Leonardo da Vinci (1452-1519) was a painter, sculptor, architect, engineer, scientist and musician as well. During the Renaissance humanism, the dominant intellectual movement, was focused on human life and accomplishment. Being inspired by the pagan cultures and languages of the ancient Romans and Greeks, humanists were concerned by the issues of life, rather than afterlife activities. It was acceptable to picture the beauty of the nude human body. The Catholic church was less powerful in the Renaissance than in the Middle Ages, and no longer monopolized learning. Aristocrats and the upper middle class considered education as a status symbol, and people were hired to teach their children. The invention of printing with movable type (about 1436) accelerated the spread of learning; over 40,000 editions were printed in Europe. As in other forms of the arts, the horizons of music were greatly expanded. Printing widened the circulation of music, and a lot of scores were spread. As a consequence the number of composers and performers increased. The idea of "universal man" implied that everyone should be educated in music. Church choirs grew in size. Polyphonic church music was usually sung by multiple soloists and performed by entire male choirs. Kings and dukes competed for the finest composers and every court had a number of musicians. Women functioned as virtuoso singers during Late Renaissance. *Town musicians played both for civic processions and religious services. *Since humanistic interests in languages and vocal music existed, vocal music became important. Renaissance composers wrote music to enhance the measure of the text, unlike medieval composers. The term 'word paintings' was much in use, which is the musical representation of word specifics. ===Characteristics of the Music in Renaissance=== [[Image:Musica renacimiento.png|thumb|200px|The music of early Renaissance]] The most prominent ways in which the music of the Renaissance is distinguishable from medieval music is a '''much smoother sound''', '''more homogeneous''', '''and less contrast''' The sound is changed as a result of a compositional technique. The new [[w:polyphony|polyphony]] style replaced the highly contrasting lines of medieval [[w:polyphony|polyphony]]. *Imitation is a polyphonic form in which the musical phrase is represented by all musical lines, one after another. As each line enters, previous ones continue, without any overlapping. This strictest kind of imitation is a "round", where all the voices sing exactly the same thing in turn. Only the first few notes are sung by the entering voice, then the whole group continues. ===Texture=== The texture of Renaissance music was chiefly polyphonic. Typical choral pieces had four to five parts. *The bass register is used for the first time, expanding the pitch range for more than four octaves. Bass line led to a good harmony, so the term of chords became more popular. Composers became chord-oriented, yet thought of some melodic lines. *The rhythm of Renaissance music is more a gentle flow than a sharply defined beat. Melodic line is rhythmically independent. It is possible for one singer to start in the middle of the piece. Renaissance music is both for pleasure and challenge. But its pitch patterns are easy to sing. The melody moves in a scale with a few leaps. ===Sacred music in the Renaissance=== ====Motet==== is a polyphonic choral work set to sacred Latin text. ====Renaissance mass==== Also known as [[w:liturgical music|liturgical music]], is a polyphonic choral composition, which comprises five sections: *Kyrie *Gloria *Credo *Sanctus *Agnus Dei Palestrina's music was very widely used in mass music. ===Secular Songs=== ===The High Renaissance=== The unifying the full use of imitation was established. Composers experimented by drawing musical materials from a single source. The source melody usually appeared in tenor voice, but other voices often derived from it as well. *The most versatile and gifted composer of this period was [[w:Josquin Desprez|Josquin Desprez]] (c.1450-1521) He composed music in the three major genres of the Renaissance, which are: Masses, Motets, and secular songs. He brought the imitation to the new heights of clarify and flexibility. The examples are [[w:Pange Lingua Mass|Pange Lingua Mass]], and Pange Lingua Geoli, which is a plainchant hymn. === Late Renaissance === The sixteenth century is known for remarkable achievements, the balance, beauty and exquisite sound of the imitative polyphony was fully explored by all European composers. In addition, the music became more and more homogeneous. There were fewer contrasts between the lines. One technical change of the Renaissance is the sound of the last chord by the end of the sections. The Renaissance composers thought of full chord by the end of piece. The chord was supposed to be tonic, fifth and octave. The distinction between fifth, full, and root chords are apparently heard. ==Baroque== The Baroque period is generally divided into 3 phases: Early (1600-1640), Middle (1640-1680) and Late (1680-1750). ==Classicism== The classical period lasted from 1750 to roughly 1825. One of the primary trends which distinguishes classical era music is the rejection of the complex polyphony of the late baroque period in favor of a simpler, more balanced style. Music in this period is characterized by regular phrases, memorable melodies, and harmonious proportion. The primary texture of the classical period is homophony, the use of one principle melody with harmonic accompaniment. '''The Multi-Movement Cycle''' Another important innovation developed in the classical era is the multi-movement cycle. The multi-movement cycle applies to absolute music (instrumental music with no extra-musical references) starting in the classical period, and its use continued into the romantic era. A typical cycle includes 3 to 4 movements and follows some conventions in terms of the form of the individual movements. Symphonies, chamber music, sonatas, and concertos are genres in the classical period which follow the multi-movment cycle. The first movement of the cycle is typically fast and in sonata allegro form. This form presents a number of contrasting themes (usually 2 or 3) which are later developed and then recapped. The second movement is most often slow and often uses a theme and variations form. The third movement is normally a minuet and trio or scherzo and trio. In either case, it comprises of 2 contrasting dances presented in a ternary or da capo form. The final movement of the cycle is fast and could be in a sonata allegro form or a rondo. Concertos in the classical period only contain three of the four movements described above and follow a fast-slow-fast pattern in terms of the tempo of each movement. Also of note, the first movement of a classical concerto is a blend of the ritornello form prominent in the baroque concerto and the sonata allegro form. '''Opera in the Classical Period''' ==Romanticism== ==National Music Traditions== ===French Impressionism=== French Impressionism (in music) is credited to the French composer Claude Debussy, although Mahler had earlier used some impressionistic idioms in some of his later music. The main other composer in this movement was Maurice Ravel, although his view on music is often considered to be more 'neo-classical'. Impressionism in music is characterised by extensive use of ninth chords and a turn toward ancient church music (e.g. church modes, gregorian chants, etc.) for an almost archaic 'mood'. Overall, Impressionism was meant to, as Debussy said: 'Be supple enough to adapt itself to the lyrical effusions of the soul and the fantasy in dreams'. Basically, although Debussy and the Impressionist composers rejected Romanticism, they were essentially still composing under a romantic idiom. Romanticism is macabre, passionate and active, but still tries to connect music as much as possible with the real world (e.g. tone poems, program music, etc.). Impressionism is quiet, floating and passive, but both still share the same basic goal, they simply express the human soul in different ways. Another strong influence on Impressionism is non-western music Gamelan ensembles. Use of pentatonic scales, chromatic scales and the 'whole-tone scale' are found throughout the harmony and melody of French Impressionism. ===German Expressionism=== '''Arnold Schoenberg (German spelling: 'Schönberg')''' German Expressionism was characterised by extremely wide melodic leaps and experimentation with the extreme registers of musical instruments. One of the main composers in this field was Arnold Schoenberg. Schoenberg was born in Vienna in 1874. Even before his atonal days, Schoenberg's harmonies were very advanced, extending chromaticism to its limit in a Post-Wagnerian manner. His gigantic song-cycle 'Gurrelieder' (also: 'Gurre-Lieder') shows considerable Wagnerian influence in its harmonies, sometimes stretching chromaticism past its tonal limits into a 'pre-atonal feel'. Although the first piece Schoenberg consciously composed using the twelve-tone system is considered to be Op. 19 "Sechs kleine Klavierstücke" ("Six small pieces for Piano"), from 1911 Schoenberg composed some earlier pieces that already had a 'pre-atonal feel' to them. Another example of this is the String Sextet 'Verklärte Nacht' (translated as 'Transfigured Night'). In this context, 'pre-atonal' merely states that those pieces already show Schoenberg's unique rhetoric, although he had not completed his ideas about the twelve-tone system yet. It is important to note that Schoenberg did not use the term "atonality" in the first place. As a matter of fact, he rather liked to speak of his music being "pantonal" (gr. "pan" = "every"), since "atonal" was first used by his critics and can refer to music that literally "lacks tone". '''The twelve-tone System/Atonality''' The twelve-tone system is a method of atonal* composition developed by Schoenberg. The basic underlying rule of this system is in the use of a 'tone row'. A tone row usually includes all twelve chromatic notes, arranged in any conceivable order. Variations of a tone row often use techniques like transposition, retrograde, inversion, augmentation and diminishment. The aim in atonality is to create a music that has no principal tone. Each tone in the chromatic scale is treated equally. This is similar in idea to the Impressionistic idiom of "free[ing] it [i.e. music] from barren traditions that stifle it" (Claude Debussy) (i.e. dissolving the major-minor system that had dominated music since the baroque era). It would be wrong, however, to treat atonality and Impressionism as the same genre because of their role in 'the emancipation of dissonance'. '''The Second Viennese School (Schoenberg, Berg & Webern)''' ===Italian School=== ===Russian School=== ==Modernism== Modernism is a movement in the arts which focusses on experimentation, rejection of tradition, and the search for new forms of expression. Modernism began in the late 19th century and lasted until the middle of the 20th century. Musical genres associated with modernism include impressionism, atonal music, 12-tone music, serialism, neoclassicism, and musique concreté, Compositional techniques employed by the modernists are diverse and may include the use of non-Western scales and harmonies, polyrhythm, frequent meter changes, polytonality, atonality, unusual instruments, playing, and singing techniques, and the use of prerecorded and electronically generated sounds. Notable modernist composers include the German expressionist composers Arnold Schönberg, Anton Webern, and Alban Berg who comprised the 2nd Viennese School and are known for developing a style of atonal music known as 12-tone music. Composers such as Igor Stravinsky and Bela Bartok are known for their neoclassical style, prominent use of folk melodies (primitivism), as well as polytonality and polyrhythm. Other famous modernists include Charles Ives, Milton Babbitt, Pierre Schaeffer, Edgar Varèse, and Pierre Boulez. ==Jazz and Rock== ==References== {{reflist}} ==Bibliography== Music: An Appreciation, by Roger Kamien. The Enjoyment of Music (Ninth Edition), by Joseph Machlis & Kristine Forney ==Call for essays== Please contribute an essay or article on '''{{PAGENAME}}'''. Enter title that is not already on the list below and start your draft. <br>If you wish to change your title, or have any other questions, visit us at ''[[Wikiversity:Colloquium]]''. {{RoundBoxTop|theme=1}}<inputbox> summary= !!!!! NEW INVITED ESSAY !!!!! type=create width=110 buttonlabel=Create subpage break=no prefix={{FULLPAGENAME}}/ placeholder=Title of subpage preload=testing </inputbox>{{RoundBoxBottom}} ;List of contributions {{Special:PrefixIndex/{{FULLPAGENAME}}/}} [[Category:Open essay collections]] ---- [[Category:Music appreciation]] [[Category:Music history]] {{subpagesif}} foe04idxz2m1d2xa4ca63gf8w8zn9vk Oboe d'amore 0 10049 2812876 751573 2026-06-05T15:52:39Z Kirby - Electrotechnics 3074947 changed to 'category:Music instruments' instead of category:Musical instruments 2812876 wikitext text/x-wiki Oboe d'amore is the alto or mezzosoprano member of the [[oboe]] family. A member of the modern oboe family with a soprano/alto range, the range of which sounds from G-sharp below middle C to C-sharp above the treble clef, but is notated a minor third above that. The oboe d'amore was very popular during the Baroque era. Its sound is somewhat gentler than that of the oboe, and its lower notes are dark, full, and rich. The oboe d'amore has almost fallen out of favor today, often being replaced by the oboe and the [[Cor anglais]]. They are part of the wider family of the double reeds which includes the following instruments from the smallest to the largest: *musette *oboe *oboe d'amore *cor anglais, or the English horn *baritone oboe or bass oboe *heckelphone *[[bassoon]] or fagott *contrabassoon or contrafagott. In the baroque time, there were also instruments like oboe da caccia and taille, which are substituted by the english horn in modern performances. In the 18th century, there were also oboes with the same size and the same pitch (A) as the oboe d'amore, but without the pear-shaped bell. These were called just as 'oboe', 'oboe grande' or 'haute-contre'. According to the present knowledge, the oboe d'amore was at the first time used in 1717 by Christoph Graupner in his Cantata Wie wunderbar ist Gottes Güt. The instrument was used by J.S.Bach, G.P.Telemann, the Graun brothers, C.Graupner, G.H.Stölzel, J.M.Böhm, A.Lotti, J.H.Roman and a few others. Possibly the last to write for oboe d'amore in the 18th century was C.D. von Dittersdorf. The oboe d'amore was re-established in the second half of the 19th century and it is used both in presentations of baroque music and in modern compositions. The composers who have used the oboe d'amore after the new rise of the instrument include Debussy, R.Strauss, Ravel, Koechlin, Delius, Henze, Takemitsu and others. Slightly larger than the oboe, the oboe d'amore has a less assertive and more tranquil and serene tone, and is considered the mezzo-soprano or alto of the oboe family. It is a transposing instrument, sounding a minor third lower than it is notated, i.e. in A. The bell is pear-shaped, similar to that of the larger Cor anglais, and it uses a bocal that is similar to it, although shorter. ==Compositions for the Oboe d'amore== *J.S.Bach: Concerto A major, BWV1055 *Claude Debussy: Gigues from Images *Christos Hatzis: Heirmos *G.P.Telemann: Cantata 'Der Herr ist König' and 'Die Donnerode' *G.P.Telemann: Concerto for Flute, Oboe d'amore and Viola d'amore {{Musical instruments}} [[Category:Music instruments]] ch1u46m1qvfag5rryvcldsujav95df2 Film Scoring/Workshop 0 10300 2812955 1674839 2026-06-05T21:43:43Z Kirby - Electrotechnics 3074947 changed to 'category:Music related projects' instead of category:Music 2812955 wikitext text/x-wiki __NOTOC__ <div style="text-align: center;"> {| border=1 cellspacing=0 cellpadding=30 bordercolor="yellow" width="94%" | bgcolor="#ffffe0" | ; [[Music in film|Topic:Music in Film]]<nowiki>:</nowiki> :[[Film Scoring|Mad Max's Course in Film Scoring for Motion Pictures]] :{| cellpadding="4" cellspacing="0" width="350px" | bgcolor="#f0ffff" style="border: 1px #cccccc solid;" |Lesson: [[Film Scoring/Workshop|Film Scoring for a real scene for Television]] |} |} </div> {| cellpadding="20" cellspacing="5" style="width: 95%; background-color: inherit; margin-left: auto; margin-right: auto" | style="background-color: #ffffaa; border: 1px solid #777777; -moz-border-radius-topleft: 8px; -moz-border-radius-bottomleft: 8px; -moz-border-radius-topright: 8px; -moz-border-radius-bottomright: 8px;" colspan="2" | <div style="text-align: center;"><big>Apply what you have learned to create your demo reel</big></div> |- | style="width: 60%; background-color: #ffffe0; 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;" rowspan="2" | ==Score a scene== You have learned the three basic ways to create musical sound effects and musical themes for film scoring. Now you need to create an actual film score and then you need to show it to people. 1. Purchase any of the disks from the [http://www.starmovieshop.com Star Movie Shop] which have edited scenes for you to create a film score. "Blindman's Bluff" and "Home Wrcker" are probably the two best choices. "Blindman's Bluff Production" has two scenes, one short and one long. "Home Wrecker" has only one short scene. 2. Import the edited scene into GarageBand. 3. Decide what you want to do. This will take the most time. Listen to a lot of film scores, Broadway scores, and classical music. While you are trying to decide, create a humming score to test your ideas. 4. Use the three methods you have learned to create musical sound effects. 8. Export the movie with finished score. From this basic form, you can add tracks with other notes. ==Create your demo reel== When you begin looking for a job creating film scores, even if it is just film scores for student films, you will need to show the film's producer what you can do. For this, you need a demo reel. Create film scores for many of the scenes from the [http://www.starmovieshop.com Star Movie Shop] and use iDVD or DVD Studio Pro to create a demo reel. | style="width: 40%; background-color: #f0f0ff; 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;" | ==Pictures to follow shortly...== |} {{Mad Max End of Lesson and TOC}} [[Category:Mad Max film scoring]] [[Category:Filmmaking]] [[Category:Film Scoring Lessons]] [[Category:Music related projects]] 3p5f446conbjhr0kt1h697sgn0giy7a Chords (music) 0 14951 2812977 2547458 2026-06-05T22:12:44Z Kirby - Electrotechnics 3074947 Removed from the main "Category:Music" 2812977 wikitext text/x-wiki {{wikipedia|chord (music)}} A chord is a combination of three or more notes. The most complementary groups of notes make up popular chords. Because of the broad nature of this definition, types of chords have been developed that "sound good" according to the western philosophy of music. While two notes technically constitute harmony, two-note groups are not considered chords. ==Types of chords== The following chords can be played in any key. To understand them, one should have the basic knowledge of a [[Music:Scales_and_Intervals|scale]] first. When considering what quality a chord is (major, minor, etc.), all that matters are the names of the 3 notes; it does not matter whether some notes are doubled or played in different octaves (known as a chord inversion). For example, a C major triad must have a C, an E and a G, but it doesn't matter which is on bottom, nor does it matter if there are 2 E's or 2 G's. Other notes, such as 9ths and 13ths, can be added to a chord, but this learning resource focuses on basic chords and their structure. Depending on the lowest note of the chord and the notes added or removed, formation of a chord is known as an “inversion” or a “voicing.” ===Major Chord=== [[Image:Piano-Cmajor.gif|290px|right|thumb|C Major on the piano]] The major triad is often symbolized with a capital letter ('''C''' | '''C'''maj). It is three notes: the first, third, and fifth note of a scale. Alternatively, you can find this chord by adding a note a major third (4 half-steps) up from the root note and then a minor third (3 half-steps) up from that. If you add the half-steps of a major and minor third you will notice the third note is 7 half-steps up from the root making a perfect fifth. This interval will remain the same in major and minor chords. All primary triads are created by stacking intervals of a third. It is a very whole sounding chord, and is found within most, if not all popular songs. C Major triad: '''C - E - G''' ===Minor Chord=== The minor triad is often symbolized with a lower case letter, or 'min', or just an 'm' ('''c''' | '''C'''min | '''C'''m). It is composed of three notes: the first, flat-third, and fifth. The lowered third represents the third note of a minor scale. Alternatively, you can find this chord by adding a note a minor third (3 half-steps) up from the root note and then a major third (4 half-steps) up from that. The quality of this chord is not as full as the major chord, and has a darker complexion. It also is found in a majority of popular music. C Minor triad: '''C - E♭ - G''' ===Diminished Chord=== The diminished triad can be symbolized with either a '°' or with the abbreviation "dim" ('''C'''° | '''C'''dim). Its components include: the first, flat-third, and flat-fifth. Alternatively, you can find this chord by adding a note a minor third (3 half-steps) up from the root note and then another minor third (3 half-steps) up from that. If you add the half-steps of two minor thirds together you will notice the third note is 6 half-steps up from the root making a diminished fifth, giving the chord its name. Having 2 lowered notes, a diminished chord does not have great euphony. It sounds slightly out of place, but adds a certain musical flavor. C Diminished triad: '''C - E♭ - G♭''' ===Augmented Chord=== The augmented triad is usually symbolized with a '+' or 'aug' ('''C'''+ | '''C'''aug). It is rarely seen in contemporary music (jazz theory, for example, prefers the #5 notation), but is one of the most significant triad structures. It combines the first, third, and a raised fifth. Alternatively, you can find this chord by adding a note a major third (4 half-steps) up from the root note and then another major third (4 half-steps) again. If you add the half-steps of two major thirds together you will notice the third note is 8 half-steps up from the root making an augmented fifth, giving the chord its name. An augmented fifth sounds the same as a minor sixth (note) but is notated differently and consequently will sound different in context. As all the notes of the augmented triad are spaced equally, there are only four augmented chords. C, E, and G# augmented chords all contain the same number of notes. All three notes of the augmented triad are found within the accompanying whole-tone scale. C Augmented triad: '''C - E - G#''' === Seventh Chord === ====Dominant Seventh Chord==== A seventh chord is attached to any other chord by the addition of the number '7' ('''C'''7). It can be attached to pretty much any chord, not just the major chord: ('''C'''7 | dim '''C'''7). Opposed to the addition of the seventh note of the scale, a flat-seventh note is added (in the key of C, this is Bb rather than B). When the third note is flat it is known as a minor 7. Therefore, "dominant 7" refers to a flattened seventh note, while a "minor 7" refers to both a flattened seventh and a flattened third note. These chords, especially the dominant 7, tend to be popular in jazz and blues music, since they lend themselves to the "blues scale," which itself includes the flattened seventh note. Dominant 7: '''C - E - G - B♭''' Minor 7: '''C - E♭ - G - B♭''' ====Major Seventh Chord==== Like a dominant seventh chord, a note is added to the triad to create a major seventh chord. While a '7' represents the dominant seventh chord, the letters "maj" and a '7' represent a major seventh chord ('''C''' maj7). The note added to this chord is the seventh note of the root's major scale. This chord is used to emphasize the major scale or tonality of a phrase, and to clarify that the dominant seventh should be avoided. A sharp fourth (Lydian) note may be acceptable as a grace note over this chord. '''C - E - G - B''' ====Minor (Major Seventh) Chord==== The minor major seventh chord ('''C'''mmaj7) is used in Horace Silver's "Nica's Dream" and Bronislau Kaper's "Invitation." The chord includes both a minor third note and a major seventh note. Although the name of the chord appears to be an oxymoron, the major seventh is an extension of the chord, and therefore can be seen as containing two components: a minor component in the lower part (Eb in the key of C), and a major component (B in the key of C) in the upper part. The scale accompanying this chord is the melodic minor scale. '''C - E♭ - G - B''' ==== Half Diminished Seventh Chord ==== A half diminished 7th chord ('''C'''ø7) adds a flat-seventh note to the diminished chord. This is the diminished 7th chord that is "diatonic" (occurs in the major scale), and can be built using the 7th, 2nd, 4th and 6th notes of the major scale. The "Half-Diminished" part of the chord's name refers to the fact that the note a minor 7th above the chord's root is used rather than the note a diminished 7th above the root. A common substitute notation for this chord in jazz is Cmin7(♭5), since another way to build this chord is to flat the 5th of a minor 7th chord. '''C - E♭ - G♭ - B♭''' ==== Fully Diminished Seventh Chord ==== Like a half-diminished chord, a 7th is added to a diminished chord, but the 7th is flattened twice to create the interval of a diminished 7th (In the key of C, this is B♭♭ (or A) rather than B♭). Fully diminished 7th chords ('''C'''°7) can also be made by adding a note a minor third above the flat 5th of the chord, creating a chord that is a stack of only minor thirds. This chord is not diatonic (does not occur in the major scale), but it does exist in the harmonic minor scale as the 7th chord built off of the raised 7th of the scale (in the key of C harmonic minor, B, D, F, A♭). It is widely considered one of the most dissonant 7th chords, and has a lot of unique harmonic applications due to its repeating structure. '''C - E♭ - G♭ - B♭♭''' or '''C - E♭ - G♭ - A''' ===Sixth Chord=== The sixth chord is a four note chord, adding the sixth note of the root's major scale to a triad. Major and minor triads can become sixths, represented by the addition of a '6' (e.g. '''C'''m6). Major: '''C - E- G - A''' The major sixth is functionally similar to the major seventh but provides a clearer resolution. However, in jazz theory the major seventh is preferred over the major sixth due to its clarification of the major seventh over the dominant seventh. Minor: '''C - E♭ - G - A''' The minor sixth is functionally similar to the minor major seventh and is used on particularly tense sections of ballads as an emotional "climax." ===Suspended (Sustained) Chord=== The suspended chord, also called sustained chord, is symbolized with "sus" or "sus4" ('''C'''sus | '''C'''sus4). In the normal (major) triad in C, the note E is the "third" because is it an interval of a third above C. However, in suspended chords the "third" of the chord is replaced by a fourth. Traditionally the fourth would resolve to the third (E) of the chord, although the sus4 chord can stand on its own and doesn't always resolve to a normal triad. A Csus chord is composed of the notes C, F, G instead of C, E, G (the normal triad). Another version of this chord is symbolized with "sus2" ('''C'''sus2). A sus2 with C as its root would be C, D, G. Sus4: '''C - F - G''' Sus2: '''C - D - G''' ===Add2/Add4 Chord=== This chord is just like a suspended chord in that it uses the second and fourth note. The difference is that, whereas in a suspended chord the third note is omitted, in an "Add2/Add4" chord, the third is kept. Add4: '''C - E - F - G''' Add2: '''C - D - E - G''' <!-- ===Fifth Chord=== The fifth chord or "power chord" is unusual because it only has two notes. The fifth chord is only the first and fifth scale degree of the root note and is represented with a 5 ('''C'''5). It is also called a "power chord" because it produces a very powerful sound. Many musicians use it when they wish the chord to have neither a major nor a minor effect. The sound produced by a so called "power chord" sometimes has an ancient quality to it because it was used in early music. For example, it was used in Gregorian chant and was called parallel organum when a series of such chords were used. The fifth chord was a favorite of Hindemith, nearly all of his compositions end with this chord. '''C - G''' --> ====quiz==== <quiz display=simple> {Which of the following describes a major chord? |type="[]"} - It is composed of the first, flat-third, and fifth. || Incorrect. This describes a minor chord. + It is three notes: the first, third, and fifth note of a scale. || Correct! This is the definition of a major chord. - It combines the first, third, and a raised fifth. || Incorrect. This describes an augmented chord. - Its components include: the first, flat-third, and flat-fifth. || Incorrect. This describes a diminished chord. {Which chord is described as having a darker complexion compared to the major chord? |type="[]"} + Minor Chord || Correct! The minor chord has a darker complexion than the major chord. - Major Chord || Incorrect. The major chord has a fuller and brighter sound. - Diminished Chord || Incorrect. The diminished chord is described as sounding slightly out of place. - Augmented Chord || Incorrect. The augmented chord is one of the most significant triad structures. {Which chord is symbolized with a '°' or with the abbreviation "dim"? |type="[]"} - Major Chord || Incorrect. The major chord is symbolized with a capital letter. - Minor Chord || Incorrect. The minor chord is symbolized with a lower case letter, or 'min', or just an 'm'. + Diminished Chord || Correct! The diminished chord is symbolized with either a '°' or "dim". - Augmented Chord || Incorrect. The augmented chord is usually symbolized with a '+' or 'aug'. {Which chord is made up of the notes C, E♭, and G? |type="[]"} - C Major || Incorrect. C Major is made up of the notes C, E, and G. + C Minor || Correct! C Minor is made up of the notes C, E♭, and G. - C Diminished || Incorrect. C Diminished is made up of the notes C, E♭, and G♭. - C Augmented || Incorrect. C Augmented is made up of the notes C, E, and G#. {Which chord is described as sounding slightly out of place but adds a certain musical flavor? |type="[]"} - Major Chord || Incorrect. The major chord is described as having a very whole sound. - Minor Chord || Incorrect. The minor chord has a darker complexion than the major chord. + Diminished Chord || Correct! The diminished chord is described as sounding slightly out of place. - Augmented Chord || Incorrect. The augmented chord is one of the most significant triad structures. {Which chord replaces the "third" of the chord with a fourth? |type="[]"} - Major Chord || Incorrect. The major chord uses the first, third, and fifth notes of a scale. - Minor Chord || Incorrect. The minor chord uses the first, flat-third, and fifth notes. - Diminished Chord || Incorrect. The diminished chord uses the first, flat-third, and flat-fifth notes. + Suspended (Sustained) Chord || Correct! The suspended chord replaces the "third" of the chord with a fourth. {The dominant 7 chord is popular in which genres of music? |type="[]"} - Classical || Incorrect. While dominant 7 chords can appear in classical music, they are not particularly associated with it. + Jazz and Blues || Correct! Dominant 7 chords are popular in jazz and blues music. - Rock || Incorrect. While dominant 7 chords can appear in rock music, they are not particularly associated with it. - Country || Incorrect. While dominant 7 chords can appear in country music, they are not particularly associated with it. {Which chord is described as having a lot of unique harmonic applications due to its repeating structure? |type="[]"} - Major Chord || Incorrect. The major chord is described as having a very whole sound. - Minor Chord || Incorrect. The minor chord has a darker complexion than the major chord. - Augmented Chord || Incorrect. The augmented chord is one of the most significant triad structures. + Fully Diminished Seventh Chord || Correct! The fully diminished 7th chord has a lot of unique harmonic applications due to its repeating structure. </quiz> [[Category:Music theory]] == Gallery == <gallery> File:Chord Cm.svg|C minor chord File:Chord Cm7.svg|C minor 7 chord File:Major seventh chord on C.png|C major 7 chord File:Minor major seventh chord on C.png|C minor major 7 chord File:Chord Csus2.svg|Csus2 chord File:Chord Csus4.svg|Csus4 chord </gallery> [[Image:C major scale.png|250px|centre|thumb|C Major]] [[Image:C minor scale.png|250px|centre|thumb|C Minor]] == See also== * [[Music production]] * [[Music and Songwriting]] ==External Links== [http://www.lovemusiclovedance.com/chord_substitutions.htm Free tutorial on Chord Substitutions] (dead link) h1hjz6yivhggjeub4vh6qfm0x27l05i Euphonium/Jechasteen/Lesson 2 0 14986 2812885 2146009 2026-06-05T15:57:44Z Kirby - Electrotechnics 3074947 changed to 'category:Euphonium' instead of category:Musical instruments 2812885 wikitext text/x-wiki == First Notes == Now that you know how to form your embouchure and how to control your breath, we will start to learn how to put that learning into practice. First, let's learn a note and how to play it. [[Image:Fbass.jpg|125px]] In bass clef, this is a '''F''' and is fingered using the open fingering (no valves pressed). To play this note, first play it on a piano and match the pitch ''with your voice'' and then attempt matching that pitch with your instrument. [[Category:Learning activities]] [[Category:Euphonium]] 3zuw753mpvnnn2or7j4yeu6me67j83v Category:Jazz 14 18947 2812857 1499492 2026-06-05T15:25:09Z Kirby - Electrotechnics 3074947 changed to 'category:Music genres and styles' instead of category:Music styles 2812857 wikitext text/x-wiki [[Category:Music genres and styles]] qhimz008596xnk5444wagy28xmwcppl Music theory 0 18980 2812871 2799796 2026-06-05T15:48:23Z Kirby - Electrotechnics 3074947 Removed from the main "Category:Music" 2812871 wikitext text/x-wiki Composition is an exciting venue to explore and, here, you can find courses on the elements of music and how to use them as building blocks to write your own music. {{TOCright}} == Courses and resources == === Introductory Courses === * [[Introduction to music]] * [[Fundamentals of Music]] * [[Beginners Music Theory]] * [[Understanding music]] * [[Music Appreciation]] * [[Harmony]] * [[Beginning Composition]] * [[Fixed media composition]] === Music Theory Sequence === * [[Music Theory I]] * [[Music Theory II]] * [[Music Theory III]] * [[Music Theory IV]] ==Readings and learning media== ===Media=== * [https://www.youtube.com/watch?v=PNFXHoL9Zvk Analysis of backing vocals] ===External=== * [https://www.reddit.com/r/musictheory/ Music Theory on Reddit] - Learn about and discuss music theory on Reddit [[Category:Music theory]] 622otkmou8f9ah9g4vc4ales1nkf8yp Experimental music 0 27520 2812860 2777422 2026-06-05T15:27:19Z Kirby - Electrotechnics 3074947 changed to 'category:Music genres and styles' instead of category:Music 2812860 wikitext text/x-wiki '''Experimental Music''' Experimental Music is a music genre that cross boundaries of the genre itself, an interdisciplinary segment involving art, in general (visual arts, dancing, performances), science and classical/popular music. In some cases, Music Theory foundations like scales and tempo might not be considered, either for aesthetically purposes or just to sound as miscellaneous art. Production aspects and sound engineering techniques might be precisely enhanced, but in opposition bad quality recordings can also be classified as experimental. Other artists of the segment also reach and project experimental music into ''pop, rap, electronica'' and ''alternative'', contributing with new tendencies for those styles of music, as well as influencing artists of the respective music genres. == Sub-Genres == In exception to "Left-Field Rap" genre, an experimentative music stream found in popular nowadays music, where the producer is centered from the composition process until the final result,. ''Experimental Music'' will still be focused on the performance of ''the artist'', live, regardless of his background in the music's development. Therefore, due to this comprehensive statement, innovations that go beyond the mixture of multiple genres and don't fit into popular (i.e, not getting popularly labeled as "alternative"), now it can be determined by sub-genres inside the experimental universe. Some examples are: * [[wikipedia:Experimental_pop|Experimental Pop]] * [https://www.allmusic.com/style/left-field-rap-ma0000011909 Experimental Rap (Left-field)] * [[wikipedia:Experimental_rock|Experimental Rock]] * [[wikipedia:Intelligent_dance_music|Experimental Techno]] * [[wikipedia:Intelligent_dance_music|Experimental Electronic]] * [[wikipedia:Ambient_music|Experimental Ambient]] * [[wikipedia:Avant-garde_jazz|Experimental Jazz]] * [[wikipedia:Olivier_Messiaen|Experimental Classical]] ==Resources== *[[w:Experimental music|Experimental music]] at Wikipedia ==External links== * [http://www.coverpop.com/whitney/index.php?var=v5 Music box] [[Category:Music genres and styles]] [[Category:Music stubs]] hupc1y0q4cl07mqi4j2hrm6eoxmrfow African music and dance 0 28353 2812989 2761094 2026-06-05T22:28:26Z Kirby - Electrotechnics 3074947 changed to 'category:Music genres and styles' instead of category:Music styles 2812989 wikitext text/x-wiki Welcome to the '''Department of African Music and Dance''', part of the [[School:Music and Dance|School of Music and Dance]] and the [[School:African studies|School of African Studies]]. __NOTOC__ ==Division description== The scope of this division is * to collect knowledge on African music and dance * to develop teaching materials about African music and dance * to develop research methods for the study of African dance * to provide a scientific basis for the improvement of the coverage and quality of Wikipedia articles related to this subject. == Research topics== * [[African dance as proprioceptive art]] ==Active participants== If you would like to contribute to this, please add a reference to your user page here. * [[User:Nannus]] == References == The following list of references is only the starting point for an extensive anotated bibliography to be developed. *[[w:Hugh Tracey|Tracey, Hugh]]. (1961). The evolution of African music and its function in the present day. Johannesburg: Institute for the Study of Man in Africa. * Lomax, Alan: Folk song style and culture. American Association for the Advancement of Science, Publication No. 88, Washingthon 1968. * Lomax, Alan, Bertenieff, Irmgaard, Paulay, Forrestine: Choreometrics: a method for the study or cross-cultural pattern in film. Research Film, Vol 6, No. 6, Göttingen 1969. * {{cite book |title=Worlds of Music: An Introduction to the Music of the World's Peoples |chapter=Africa/Ghana |author=Koetting, James |pages=67-104 |year=1992 |edition=Second |location=New York |publisher=Schirmer Books |id={{ISBN|0-02-872602-2}} }} * [[w:Gerhard Kubik|Kubik, Gerhard]] [https://www.lit-verlag.de/isbn/978-3-8258-7800-7 Zum Verstehen Afrikanischer Musik, Aufsätze, Reihe: Ethnologie: Forschung und Wissenschaft, Bd. 7, 2., aktualisierte und ergänzte Auflage, 2004, 448 S., ] {{ISBN |978-3-8258-7800-7 }} (in German language) * Simon, Artur (Ed.), "Musik in Afrika", (Staatliche Museen) Berlin 1983 (in German language) * Bender, Wolfgang: Sweet Mother - Moderne afrikanische Musik, 1985, Trickster Verlag, München. {{ISBN |3-923804-10-5 }} (in German language) ==Department news== * '''6 March 2006''' - Department founded! [[Category:Music genres and styles]] [[Category:Dance]] [[Category:Africa|music and dance]] 42oegxkos282a0x1flakct216tumfbc Free license music 0 34624 2813010 558737 2026-06-06T00:34:11Z Kirby - Electrotechnics 3074947 changed to 'category:Music resources' instead of category:Music 2813010 wikitext text/x-wiki Welcome to the Wikiversity learning project for Free license music. This is where Wikiversity participants can learn about sources of music that is licensed under a [[Free content|free]] license. ==Resources== *[http://jamendo.com Jamendo] *Students of musicology may find the [http://www.imslp.org/index.php?title=IMSLP:About International Sheet Music Library Project] to be a helpful source of public domain, and donated scores. *The [http://icking-music-archive.org/index.php Werner-Icking Music Archive] also has a large library of scores available for download. ==See also== *[[Hunter-gatherers project]] [[Category:Hunter-gatherers project]] [[Category:Music resources]] [[Category:Open licensing]] meba8r22n7odne9ihg29wtsff1aamwo Music in the Bible (Psalms) 0 35322 2812982 2472283 2026-06-05T22:22:17Z Kirby - Electrotechnics 3074947 changed to 'category:Music history' instead of category:Music 2812982 wikitext text/x-wiki '''A Brief Study On the Music that Can Be Found In the Psalms of the Bible.''' The Psalms were the hymnbook of the Old Testament Jews. Most of them were written by King David of Israel. Other people who wrote Psalms were Moses, Solomon, etc.<ref>[[Wikipedia:Psalms]]</ref> The Psalms are very poetic. They have a flow to them. You could put a metronome to them and recite them in time. In many Bibles a musical phrase will be written above the chapter. Example: Psalm 61 ''"To the chief musician upon Neginah, A psalm of David"'' Neginah , plural Neginoth , in the Bible, direction for the musical accompaniment of a psalm. Psalms 4, 6, 54, 55, 61, 67, 76. The actual sheet music doesn't exist today. But one can still put music to the psalms today. The method to follow is this. The commas and periods are rest marks. Shape your melody and chords to represent the emotions given by the words. Repeated phrases are found often in music today. They are for emphasis. Pay attention to phrases like "and all Israel cried". This is the call for more voices to join in. == '''Psalms Through the Years''' == Psalms have been used throughout history, including in Christian usage up to today. Eskew and McElrath summarize the use of the Psalms through history this way: ''From one standpoint the entire history of the hymn could be delineated according to its varying relationship to the Scriptures. Generally speaking, the line of evolution in that story, if it were retold, is from the actual singing of parts of the Bible (the psalms, for example) through the strict paraphrasing of extended passages and the dutiful use of biblical allusion, language and figures of speech to the free expression of scriptural thought and teaching in contemporary terms.'' (Harry Eskew and Hugh T. McElrath, Sing With Understanding (Broadman Press, 1980), p. 45.) == '''The Old Testament''' == The history of psalmody actually begins long before the Book of Psalms was ever written. The Israelites sang the Song of the Sea, Exodus 15:1—18, after crossing the Red Sea, and Craigie finds at least six other poetic texts embedded in the prose narrative of the Old Testament. (Peter C. Craigie, Psalms 1-50, Vol. XIX of Word Biblical Commentary (Word Books, 1983), p. 25.) Of course, many of these songs of the Israelites have been recorded in the Book of Psalms itself. Still prior to the birth of the Christian church, the Book of Psalms became in effect the standard “hymnal” for the Jewish nation. On this point scholars are nearly unanimous. (Donald P. Hustad, Jubilate! (Hope, 1981), p. 83.) Before the establishment of the Temple, King David appointed skilled singers and musicians to sing unto the Lord. The Bible makes no specific mention as to what songs were sung, except that they were “joyful songs.” (1 Chronicles 15:16) Perhaps 1 Chronicles 16:14 contains the most revealing detail of what was sung without specifying the Psalms. The passage says that David “appointed Levites to minister before the ark of the LORD, to make petition, to give thanks, and to praise the LORD, the God of Israel.’ This verse enumerates three types of expression: petition, thanks, and praise. Weiser distinguishes these same three types of expression in the book of Psalms. (Artur Weiser, The Psalms (Westminster Press, 1962), p. 52ff.) In 1 Chronicles 25, some of the men are set apart, it says, “for the ministry of prophesying, accompanied by harps, lyres and cymbals” (verse1). The ministry of prophesying coincides with Weiser's subcategory of Wisdom and Didactic Psalms. No doubt songs similar to those in the book of Psalms were in use before the canonic book was compiled. One could assume that many of the songs sung in the Old Testament era eventually found their way into the Psalter. In later years, at the dedication of the Temple, Scripture records that the Levites used the instruments “which King David had made for praising the LORD and which were used when he gave thanks, saying, “His love endures forever.” (2 Chronicles 7:6.) The expression “his love endures forever” is recorded in the first verse of both Psalm 118 and 136. The same words were sung years later at the rededication of the Temple, as recorded in Ezra 3:11. One may assume, then, that psalms, particularly those compiled in the canonic book, were used often by the Israelite priesthood. Many scholars conclude that instrumental music was halted by the Babylonian captivity. When the Temple was destroyed by the Babylonians, very possibly all music was banned as a sign of national mourning. (Many historians have assumed this from historical records and also from Psalm 137:1-4.) After the Jews returned from Babylon and rebuilt the Temple, they again used music in worship. In later years much of the music of the Temple, or music similar to that of the Temple, was transferred to the synagogue. (Eric Werner, The Sacred Bridge (Columbia University Press, 1960), pp. 22-26.) Since music of the Temple consisted largely of psalm singing, (Winfred Douglas, Church Music in History and Practice (New York: Charles Scribners Son, 1957), pp. 13-19) the psalms were also used generously in the synagogue. So it is to this day. (A. Z. Idelsohn, Jewish Music in Its Historical Development (Schocken Books, 1967), pp. 3-336) == '''The New Testament''' == When the Church was born, singing to the Lord entered a new era. This section explores what place the Psalms had in the Church of the New Testament. Even there, the primacy of the Book of Psalms is clear. == '''Psalms, Hymns and Spiritual Songs''' == '''Psalms''' One may assume from James 5:13 and 1 Corinthians 14:15 that the New Testament church was fond of singing, in fact, of singing psalms. James tells his readers, “Is anyone happy? Let him sing songs of praise.” In the Greek, the command is ''psalleto'', to sing psalms. Paul tells the Corinthians, “I will sing with my spirit, but I will also sing with my mind.” Again the word for “sing” here is ''psalo''. Whether these are references to the actual book of Psalms or not is not important ultimately; that they are references to songs at least similar to the book of Psalms is the issue for this study. The apostle Paul also referred to songs in the assembly of the church when he wrote to the Corinthians, “When you come together, everyone has a hymn…(I Cor. 14:26). The actual word used here is not “hymn,” but rather it is ''psalmon''. Some have felt that this was an indication that the Corinthians opened their assemblies with a psalm, since ''psalmon'' begins a list here. Such an assumption may not be correct, but if that congregation did indeed open with a psalm, it would have been in agreement with the pattern for the synagogue service. (Werner, p. 22) Perhaps James 2:2 indicates that services of the early church were patterned after synagogue services. There James speaks of someone coming into their “assembly,” and the word used for “assembly” is synagogue. The book of Psalms is quoted more frequently in the New Testament than any other Old Testament book. (H. B. Swete, An Introduction to the Old Testament in Greek, 2d rev. ed. (Harvard University Press, 1914), pp. 383,384) Jesus Him¬self quoted from the Psalms more than any other book of the Hebrew Scriptures. Briggs finds at least fourteen times when Jesus used the Psalms. (Charles Augustus Briggs and Emilie Grace Briggs, The Book of Psalms, Vol. 1, The International Critical Commentary (T & T Clark, 1907), introduction, pp. ci,cii). '''Hymns''' Jesus and the apostles are recorded as singing only once, that being the hymn after the Last Supper. (Matt. 26:30 and Mark 14:26) Although it is called a hymn, almost all authorities agree that this was in fact the Great Hallel, Psalms 113-118, which was traditionally sung by the Jews after the Fourth Cup of Blessing in the Passover Meal. This being the case, the word “hymn” as used in the New Testament could be defined at least in some cases as a series of psalms linked together, or as a long psalm. Certainly this is at least a part of the definition. After the Last Supper hymn, the next clear reference to hymn singing in the New Testament is found in Acts 16:25. Paul and Silas are in prison “praying and singing hymns.” In light of the definition of the hymn at the Last Supper. it is at least possible that Paul and Silas in fact sang several psalms. Hymn singing is again mentioned in Hebrews 2:12. There the writer quotes a psalm and says, “in the presence of the congregation I will sing your praises.” The actual term is ''hymneso se'', “I will hymn thee”. '''Songs''' Revelation 5:9 mentions a new song to be sung in the Great Throne Room. This verse contains the third word used to describe singing in the New Testament, the Greek word ode. The passage does not necessarily describe what was happening in the early Church, but rather what would happen at the end of the age. Events at the end of the age do not necessarily describe the actual singing of the early Church. '''Psalms, Hymns and Spiritual Songs''' Having made mention of those three words which describe singing in the New Testament, it is well to move on to a discussion of Ephesians 5:19 and Colossians 3:16. These are the most familiar and most often-discussed verses about singing in the New Testament. They read this way: “Speak to one another with psalms, hymns and spiritual songs. Sing and make music in your heart to the Lord, . . (Eph. 5:19), and, Let the word of Christ dwell in you richly as you teach and counsel one another with all wisdom, and as you sing psalms, hymns and spiritual songs with gratitude in your hearts to God (Col. 3:16). Both Ephesians and Colossians use all three terms of psalms, hymns, and spiritual songs. For the purpose of this study, the interest in this passage lies in Paul’s intended definition of these three terms. The word PSALMS, ''psallo'', originally meant to pluck, as in the string of a harp, hence, to sing to the accompaniment of a harp. And then to the making of music in general. Although this was not the original Hebrew title for what became known as the Book of Psalms, the title of “Psalms” came from the Greek translations of the Old Testament. Some have claimed that when Paul referred to psalms he was referring to new Christian compositions patterned after the Old Testament book, but most scholars agree that this term probably referred to the Old Testament book of Psalms itself. Without a doubt, the term psalms could mean the book of Psalms. The word HYMNS, ''humnos'', is a little more difficult to define. It seems that the emphasis of a hymn was not on musical accompaniment, as with the term but more on praise of God. (Joseph Henry Thayer, ed., Greek-English Lexicon of the New Testament (Grand Rapids: Zondervan, 1978), s.v. “''humnos''”, p. 637) This word could mean, as mentioned earlier, a series of psalms linked together, as in the Last Supper passage. Some scholars label as hymns the canonical Canticles or so-called “hymn fragments” of the New Testament. This is an interesting thought; however, since none of the “canticles” or “hymn fragments" of the New Testament actually claim to be music, it is difficult to be dogmatic on that definition. Some scholars have suggested that the term could be applied to poetry in general. Their suggestion would resolve the dilemma. There is some support for their view in that there is a strong relationship be¬tween poetry and music in Greece, and these letters were written to churches that no doubt had many converts who would be a part of Greek culture. Ambrose’s famous definition of a hymn sheds light on the post—apostolic view. He says that a hymn is a song containing praise of God. If you praise God, but without song, you do not have a hymn. If you praise anything which does not pertain to the glory of God, even if you sing it, you do not have a hymn. Hence, a hymn contains the three elements: song and praise of God [underline added]. (Werner, p. 207) The term “hymn” at least could mean the book of Psalms. The term SPIRITUAL SONGS is the most difficult of the three to define precisely. The Greek word is ode, and it is a generic word for “song.” In the New Testament ode is found only in the passages in Ephesians and Colossians and three times in the book of Revelation. (Revelation 5:9, 14:3, and 15:3) The Revelation passages are not very helpful in defining ode because in two of the three cases it is modified by the term anew,” “a new song.” In any case they are not very helpful because they are set in a future dispensation. The passages from Ephesians and Colossians are difficult because ode is modified by the term “spiritual” (spiritual songs). What does Paul mean when he specifies spiritual songs? Some have suggested that the word “spiritual” here refers to a song that is inspired by the Holy Spirit, in the sense that Scripture is inspired or God-breathed. Some have, in light of I Corinthians 14:15, seen this to mean outbursts of speaking in tongues, although that view is generally discounted by scholars. Could they be songs of the spiritual life? Could they be songs composed by spiritual men? Could they be freely composed hymns, inspired in the modern sense of being highly creative? Some have asserted that they might have been chants without words, melodies sung on just one syllable, called a melisma. The explanation that this author leans toward is that “song,” being a generic term, is qualified by the term ‘spiritual’ to clarify that the song is not to be just any pagan song, but a godly one. Possibly the term “spiritual songs” could also be referring to the Old Testament book of Psalms. The Ephesian and Colossian congregations could possibly have used some kind of hymn book. Their being familiar with a common hymn book would explain some of the so-called hymn fragments which Paul uses elsewhere. Louis Benson believes that Paul used all three terms, “psalms, hymns and spiritual songs,” to connote an actual hymn book already being used by the churches. (Louis F. Benson, The Hymnody of the Christian Church (Richmond, Va: John Knox Press, 1956), p. 45-48) Such a hymnal would not be likely to be so well known so early, however. All three terms could as easily refer to the book of Psalms as they could to a common hymnal. Jerome, in his “Commentary on the Epistle of Paul to the Ephesians,” assumes that all three terms refer to different aspects of the canonical Psalms. He states that those who sing hymns “declare the power and majesty of the Lord and continually praise his works and favors.” (Oliver Strunk, Source Readings in Music History (W..W. Norton & Co., 1950), p. 72) Hymns, then, according to Jerome, are Psalms of declarative worship. Spiritual songs, Jerome says, “properly affect the seat of ethos, so that we know what ought and what ought not be done.” (ibid) He claims that the psalm, use of which implies instrumental accompaniment, is directed toward the body, while the spiritual song is directed toward the mind. He concludes, “We ought, then, to sing and make melody and praise the Lord more with heart than with voice.” (p.72) Other scholars more recently have also asserted that all three terms refer to the Psalms. Edward Robson documents well his view that “all three terms can only mean psalms, hymns, and spiritual songs of the Old Testament Book of Psalms.” (Edward J. Robson, “An Exposition of the Psalms, Hymns, and Spiritual Songs of Eph. 5:19 and Col. 3:16,” The Biblical Doctrine of Worship (n.p., The Reformed Presbyterian Church of North America, 1974), p. 197) He goes into great detail to prove this point by observing syntax and discussing use of the word “spiritual” in the context. He demonstrates that the syntax of “psalms and hymns and spiritual songs” is the same parallel structure as in Matthew 28:19, in which baptizing is in the name of the Father and Son and Holy Spirit. His claim is that psalms, hymns and spiritual songs are as much a three-in-one as the Trinity, and thus must mean the same thing: the Psalms. Others have made some convincinq arguments that all three terms refer to the Book of Psalms. Frank Frazer points out that the command is for Christians to sing psalms, hymns and spiritual songs, not to make them. (Frank D. Frazer, “Psalms and Hymns and Spiritual Songs,” The Biblical Doctrine of Worship, p. 334) He feels that the Church should sing the old Psalms which are already written instead of creating new compositions. Perhaps the best argument which would allow all three terms to mean the Book of Psalms is put forth by G. I. Williamson: …We are in the habit of using the terms “hymns” and “songs” for those compositions that are not Psalms. But Paul and the Christians at Ephesus and Colossae used these terms as the Bible itself uses them, namely, as a title for the various Psalms in the Old Testament Psalter. (G. I. Williamson, “The Singing of Psalms in Worship,” The Biblical Doctrine of Worship, p. 321) Earlier he had explained: …For the fact is that all three of these terms are used in the Bible to designate various selections contained in the Old Testament Psalter. In the Greek version of the Old Testament familiar to the Ephesians and Colossians the entire Psalter is entitled “Psalms.” In sixty-seven of the titles within the book the word “psalm” is used. However, in six titles the word “hymn” is used, rather than the “psalm,” and in thirty— five the word “song," appears. Even more important, twelve titles use both “psalm” and “song,” and two have “psalm” and “hymn.” Psalm seventy-six is designated “psalm, hymn and song.” And at the end of the first seventy-two Psalms we read this, “the hymns of David the son of Jesse are ended. This present study is not making some proscription to use only the Psalms in worship, as these Reformed scholars would do. It is, however, calling the church to a more scriptural balance. If new compositions are to be written, they should reflect a similar sentiment and balance as did the Old Testament Book of Psalms itself. In the next chapter, this balance of the primacy of worship will be established, and that is the end result of all of this argument. == '''The Early Church''' == In Jesus’ day, the Jews described their entire Bible in three words. It was called the Law, the Prophets and the Writings. (Briggs, p. xix) Jesus used these three terms in Luke 24:44. The Law, of course, was the Pentateuch and the books of history. The Prophets were the prophetic writings, both major prophets and minor. The Writings were the Psalms and other books of poetry. Jesus left out mentioning the Writings when He said that He had come not to abolish the Law and the Prophets but to fulfill them. Could it be that the Writings were not fulfilled? The New Testament contains no section of poetry. Both Old and New Testaments have sections of history. Both have sections of prophecy. Only the Old Testament, though, has a section of poetry. The Psalms transcended the New Testament into the early church. What happened to the Psalms after the canon was closed? Were second-century Christians bound to the Psalter? What did they sing? Benson gives his theory: These first Christians are described as in that state of spiritual elation out of which song springs as naturally as flowers blossom. And plainly they felt perfectly free to add new songs to the old, which the more gifted among them did from the beginning. (30) He goes on to say, To most students that early atmosphere seems to embody a spirituality of the creative sort, of expansion rather than compulsory restriction. It appears to have been a divine providence rather than a divine prescription that laid the Psalter ready to the Church s hand, and as though its contents rather than the urgency of its rubrics recommended its use to the first Christians. After the close of the New Testament canon, however, the information on the singing of the Church is sparse. Delling explains that Philo writes that in addition to the Psalms early Christians created songs and hymns in various meters and melodies in traditional Hebrew styles. (Gerhard Delling, Worship in the New Testament, trans. Percy Scott (Philadelphia: Westminter Press, 1962),p. 85) Some of the apocryphal writings contain new songs. The Acts of John supposedly has the hymn which Jesus and the apostles sang at the Last Supper. The Acts of Thomas contains several songs which begin to reflect some Gnostic theology. Clement of Alexandria wrote at least one formal hymn, and Ambrose wrote several intricate hymns. (Robert Payne, The Christian Centuries (New York: W.W. Norton, 1966), pp. 112-128) Other Christians, several of whom were martyred, are known to have written songs. Among them are Ignatius, Athenogenes, Hippolytus, and Nepos. (Benson 60) whether or not the apocryphal writings are genuine and whether any of these contain Gnostic thoughts is irrelevant for this chapter. The Christian community wrote some original compositions early in its history. This confirms Benson s thought that there was no early proscription against adding to the Psalter. Werner states that the Church of the first millennium used to its best advantage the one book of the Old Testament that came to be the backbone of most liturgies, the Psalter. . . . Whoever evaluates psalmody and its part in daily life approaches the spirit of the Middle Ages.’ (128, 129) He pursues his thought further, linking the association of the church with the Jewish synagogue and Temple practices. The single most noticeable common denominator between the Jewish and early Christian practice of worship is the Psalms. Apel also traces this early development. (Willi Apel, Gregorian Chant (Indiana University Press, 1973) pp. 33-83) Not long after the Church began, however, heretics began to discover the power of these new compositions. The Gnostics began to gain popularity through their songs. The heretic Anus also began to write songs. He said, “Let me make a people’s songs and I care not who makes their laws.” (Williamson, p. 325) John Chrysostom countered the heretics in his city by having the believers sing some popular songs with orthodox theology while marching in procession. (Benson, p. 67) This worked with some success for the church there. Others, such as Augustine, reacted to these new compositions with suspicion and legalism. In A.D. 430, he testified, The Donatists make it a matter of reproach against us, that, in the Church, we sing with sobriety the divine songs . . . whereas they inflame the intoxication of their minds by singing psalms of human composition. (Williamson, p. 325) In an effort to carefully preserve the orthodox faith, the Church began to limit all non-scriptural compositions. In fact, as early as A.D. 343, the Synod of Laodicea forbade “the singing of uninspired hymns in Church” (Williamson, 325) as it also forbade “the reading of uncanonical books of Scripture.” (ibid) The proscription was reaffirmed at the Council of Chalcedon in A.D. 451. During these early centuries, the liturgies of the Church used what became known as Gregorian Chant. Willi Apel notes that the Psalms played a major role in the formation of this body of song: Not without justification has the Book of Psalms been called the most influential single source of texts in all music history. Indeed it is by far the most important textual source in Gregorian Chant. -Nearly all the chants of the Gregorian repertory have a psalmodic background, the main exceptions being the Antiphons, the Resonsories, and the Hymns. (Apel 87) Evidently, the Church leaders thought that if no one wrote or read or sang anything that was not directly Scripture, they would eliminate doctrinal error in song. As time went on this suspicion of any unscriptural composition faded, and composers moved away from writing only Psalm settings. What had begun in apparent freedom went through a period of control and suspicion, and then slowly returned to freedom again. The problem was that in the Church's loosening of restrictions on the songs of her services, she also loosened her grip on orthodox belief. These were the Dark Ages, and the Reformation grew out of this period. == '''The Reformation and Beyond''' == During the Reformation the Protestant Church broke away from the Catholic Church in Rome. With the new—found freedom from Roman control, a scramble began for a new authority, both in theology and in music. Each new group that began had its own view of music. By and large, the history of church music can be summarized by looking at a few of the major leaders who helped to shape Protestant thought. The present study now turns to examine some of these leaders. '''Luther''' The most prominent figure of the early Reformation is Martin Luther. The Hussites had already re-invented the vernacular hymn before him. Luther also used and wrote vernacular hymns. He believed in the priesthood of all believers, and as such he wanted the congregation singing instead of spectating as the professional clergy sang for them. Luther had no reservations about adding to the Psalter and he actively sought to find new hymns which would be appropriate for his people. He wrote some of them himself. His choice was to call them “spiritual songs” (Benson 76) to distinguish them from the Latin Psalms and to reflect their inward direction. Luther, however, was far from wanting to abandon the Psalms altogether. He said, “Let the entire Psalter, distributed into parts, remain in use at the morn¬ing and evening service,” (Benson, 76,77) and he was quite willing to have them sung in their then-customary Latin. Luther believed that when Paul used “psalms, hymns and spiritual songs" instead of just “psalms” he gave the Church permission to freely compose its own new songs. Benson says, he calls “the songs of holy writ to witness that patriarchs and prophets composed original hymns,” and therefore a modern reformer and his friends who do likewise should “not be looked upon as innovators” as following in the train of these ancient worthies. (78) '''Calvin''' John Calvin was the next prominent figure to rise in the history of Church music. Unlike Luther, Calvin did not desire to reform the church; he desired to restore the primitive church as it had originally been. This difference in basic viewpoints created the difference in their attitude toward the Psalms. Calvin, being more conservative in his approach, concluded that the Old Testament Psalms were the only permissible source for church singing. Benson describes Calvin's thinking by summarizing Calvin’s preface of 1543 like this: First: To find songs not only pure but holy. Second: But none can write them save he who has received the power from God Himself. Third: “When we have searched all around, here and there, we shall find no songs better or more suitable than the Psalms of David which the Holy Spirit dictated and gave to him.” Fourth: “And therefore, when we sing them, we are as sure that God hath put words into our mouths as if lie Himself sang with us to exalt His glory.” (83) Calvin differed from the Catholic Church in that he pre¬scribed that the congregation be encouraged to sing, and that he also prescribed that the Psalms be sung in the vernacular instead of in Latin. Calvin also avoided those Roman chants not directly based on the Psalms. Calvin's idea of bringing the Psalms to the people was very far-reaching in congregational song. To begin with, it created the Metrical Psalm, a psalm set to vernacular poetry. Setting the Psalms to verse helped the Calvinists sing in meter. It also helped the Church move away from singing strict Scripture and move toward freedom of expression, even though it was not nearly as free as Luther’s concept of song. English Psalmody grew from this limited freedom of Calvin, and some two hundred years later the changes of Isaac Watts came from English Psalmody. '''Watts''' The English psalm was already changing before the time of Isaac Watts. Writers had begun to feel more free to add personal applications and thoughts to the strict prose text of the Psalms. Sternhold and Hopkins completed The Whole Book of Psalms in 1562, and this psalter remained popular until Watts entered the scene. Isaac Watts’ perspective on singing the Psalms was different than Calvin’s perspective. According to Benson, Watts “denied in general that we are under the call, either of God or of Christian prudence, to sing the Bible.” (Benson, 88) In fact, Watts believed that the church’s hymns should not simply be the Word of God, but should be the Church’s response to the Word. Secondly, Watts denied that the Book of Psalms was God’s intended hymn book for Christians. it was Jewish, not Christian, and some of the messages are far from New Covenant messages. So Watts set out, with his Psalms of David Imitated in the Language of the New Testament, and Apply’d to the Christian State and Worship, to remedy that flaw. Thirdly, Watts denied that the metrical psalm was the pure Word of God in the first place. If the Church is to sing Scripture only, then human versions of Scripture manipulated to rhyme and meter are not sufficient. Watts went about creating hymns, many of which were based on the Psalms but which reflected the joy and truths of the new creation in Christ. Watts’s ideas were radical in the early eighteenth century, but before long they became widely accepted. Watts, as a Calvinist, seldom extended an invitation for salvation in his hymns. Later generations would soon change that as well. '''Wesleys''' John and Charles Wesley represent the next major shift in Christian song. The Wesleys were Arminians, who, unlike the Calvinists, believed in the free will of man with regard to salvation. This theological position, coupled with freedom from strict Psalmody, helped to create the first “invitation” songs. The Wesleys approached music in much the same way as Luther had. They wanted people singing their songs more than they wanted to win the traditional Church’s approval. They, therefore, had little concern with singing Scriptural paraphrase, and even began to neglect the practice of listing a Scriptural reference at the head of a hymn. The Wesleys were out to shake the sleepy Church of England into revival, and the old forms which came from the Church were no longer adequate to carry the power of their call. Their cry was that a person was not a Christian just by being in the Church. That person must be in Christ, and the only way to have assurance of being in Christ was through personal experience. The Wesleys created music for the people. According to Hustad, the Wesleyan songs “covered every conceivable aspect of Christian devotional experience and may be said to be the progenitors of the modern gospel songs.” (Hustad 127) The Wesleys' music accompanied England's Great Revival. In America, the Great Awakening and later revivals shared a similar kind of music as that of the Wesleys. Church music had taken another major step away from the Psalms and toward the gospel song. '''Sankey and Bliss''' The latter half of the nineteenth century in America was the era of great mass evangelistic meetings. During that era, the gospel song became a popular vehicle for presenting the Good News about Jesus. Gospel songs were the next great step away from the Psalms. Many wrote and com¬piled gospel songs during this time, but the music of Ira Sankey and Phillip Bliss seems to have been a rallying point for the rest of evangelical Christian music. Gospel songs are subjective songs emphasizing human experience and testimony. (Donald P. Elsworth, Christian Music in Contemporary Witness (Grand Rapids: Baker Book House, 1979), p. 93) They are usually directed to the unbeliever and press for a decision from him. Their music is consciously the simple, singable music of the world. Usually they have a refrain. The texts, far removed from any attempt to imitate the Psalms, basically are strongly evangelistic sermons set to rhyme. Gospel songs were found to be quite effective in bringing about mass evangelism in the late nineteenth cen¬tury. Even today they are quite popular among evangelicals. They are seldom used strictly for mass evangelism anymore, for the edge of their effectiveness has long ago worn dull. Nonetheless, congregations who have become used to gospel songs still sing them, even as part of their worship on Sunday mornings. '''Recent Trends''' More recent trends are perhaps too close to categorize yet, until history can put them into perspective. The trend seems to indicate that modern Christian music up through the 1970s has continued on the same path as the last century. Personal experience and evangelistic invitation have been the major themes. The music of John W. Peterson dominated the 1960s, and that of he and Ralph Carmichael dominated most of the 1970s. Almost all of the music of these two men was written in a popular, palatable style, full of testimonies and mild invitations. By the 1980s there seems to have been a heartening trend back to an emphasis on the Psalms and on music of worship, especially represented by the writing of Don Moen, Twila Paris, Michael W. Smith and others. The decade of the 1990s brought an even greater renewal of worship music, including touring “worship concerts” and many “worship” recordings by such groups as Sonic Flood and Delirious?. Martin Smith and Matt Redman from the UK, and Darlene Zschech from Australia joined US writers from Integrity and Vineyard churches to create intimate songs that were designed for the common use of both believers and seekers alike. Historians will have to see if these trends continue. == Summary == The Psalms in the Church have made a full swing. The Church began with free composition but heavy reliance on Scripture, almost solely the Psalms. It changed to an attitude of prohibition and control to sing only the Psalms and a few other texts. Luther advocated a whole new freedom from such limitations. Calvin preferred rather strict limitations to the use of metrical psalms only, but once he allowed the Scripture to become metered, human creativity was set loose. Watts applied the Psalms to the Christian state, and then went on to write free hymns. Wesley introduced the invitation song and more of an element of personal experience to the free hymn. Sankey and Bliss- then added the gospel song. Benson summarizes the flow of Christian song this way: The era of “Psalms and Hymns” gradually merged into an era of “Hymns.” As the books labeled on their backs, “Psalms and Hymns” had replaced the Psalms in Meter, so the “Hymnal” came to replace the “Psalms and Hymns.” (92) Benson made his observation early in this century. If he were to update it he might add: “hymns and spiritual songs” have now replaced the “Hymnal." The history of Christian song could also be summarized this way: it began with psalms, hymns and spiritual songs, which were all Psalms or similar to the Psalms. Calvin limited the Church to using psalms only. Eventually Luther and Watts changed the Church to sing psalms and hymns. Finally evangelicals like Wesley came to use hymns and spiritual songs, then Sankey and Bliss used only spiritual songs. The trend away from psalms has continued to today. A modern problem is that hymnal compilers, in an effort to have a “balanced” collection, are gathering material from Watts forward in time. What compilers consider “balance” is to have a nearly equal number of older hymns and of recent evangelistic spiritual songs. Psalms have been neglected. The Church would do well to return to the psalms, hymns and spiritual songs of its primitive days. The Psalter is not the only proper source for Christian song. Surely within the Church there is room for original composition of any topic and of any style of music. If the Church is going to look for an objective standard for her song today, however, the most logical source would be the Old Testament Book of Psalms. Sadly enough, the Psalms have been almost totally neglected in modern evangelical hymnals. Even more sadly, the very spirit of the Psalms has been abandoned for pragmatism. == References == {{Reflist}} [[Category:Music history]] [[Category:Christian Studies]] pvbbdmgv6hn23liumsrzj97vxa40y1m Musical direction/Introduction 0 35668 2813008 2707596 2026-06-06T00:32:51Z Kirby - Electrotechnics 3074947 Removed from main Category:Music. This Page needs to be deleted, as the page it stems from has already been deleted. Seen at https://en.wikiversity.org/wiki/Musical_direction 2813008 wikitext text/x-wiki {| cellspacing="0" cellpadding="0" style="margin:0em 0em 1em 0em; width:100%" | colspan="1" style="width:100%; vertical-align:top; border:1px solid Sienna; background-color:Cornsilk;" | <div style="border-bottom:1px solid Sienna; background-color:Wheat; padding:0.2em 0.5em 0.2em 0.5em; font-size:130%; font-weight:bold;">Introduction</div> <div style="padding:2em 5em 0em 3em;"> <!--Note: Top, right, bottom, left --> <!-- PICTURE ON SIDE OF HEADER SECTION -----> [[Image:Nuvola apps korganizer.svg|right]] <!-- FOR MORE ICONS, GO TO [[Help:Icons]] --> ==Introduction- Why?- The Immortal Question Answered== :As a Musical Director for numerous Musical Theatre companies and productions, I have often attempted to research into techniques, tactics, and knowledge on the subject of Musical Directing in Musical Theatre. Of course, this being a very specific field, there is little useful knowledge around. This is partly because of it being a very specific field, as fore-mentioned, but this lack of educational resources for Musical Theatre Music Directors (we'll refer to them as MD's) may also be caused by the wide range of knowledge needed to MD (musically direct) a full-throttle, leg-kickin' musical. The MD must be fluent in not only Musical knowledge (reading the score, interpreting the score, etc.) but also in the theatrical aspect. In fact, many MDs have been actors/actresses at one point or another, or have worked as Directors or Assistant Directors of theatrical productions. This cannot be stressed enough. A familiarity in theatre is completely necesarry in order to succeed as an MD in musical theatre. :This course will attempt to train MDs in Musical Theatre with all the knowledge needed. However, some basic knowledge is needed before the start of this course. You must be fluent in basic music theory, and must be able to, at the least, play notes on the piano. You also should have general theatre experience, however this is not required. :Most of all.. HAVE FUN! This course is intended to be fun and exciting! </div> |} {{center top}} [[Image:Música_romanticismo.png|Click To Proceed]] {{center bottom}} {{CourseCat}} 7oxy3xs8topl2ixymkfzp7sehu1cs5s Sound synthesis 0 39247 2812997 2723407 2026-06-06T00:20:03Z Kirby - Electrotechnics 3074947 changed to 'category:Music technology' instead of category:Music 2812997 wikitext text/x-wiki ''Sound synthesis'' is the process of altering frequencies and voltage-controlled electrical pulses in order to develop sounds and music. It was first intended to emulate existing acoustic and electro-acoustic instruments, but through years of research it defined a new segment. Along with ''musique concrete, sound synthesis'' contributed for the dissemination of ''electronic music.'' There are many different types of synthesis in common use today: * Subtractive Synthesis (otherwise knows as Analogue Synthesis) * Modular Synthesis * Additive Synthesis * Frequency Modulation (FM) Synthesis * Phase Distortion Synthesis * Granular Synthesis * Wavetable Synthesis * Physical Modelling Synthesis (a fairly recent development which does enable realistic recreation of acoustic intruments) == External resources == [http://www.wikiaudio.org WikiAudio - Encyclopedia & database for all things audio and sound related] [[Category:Music technology]] [[Category:Audio engineering]] [[Category:Sound|synthesis]] 63zw1l93gw8sru2sr3vta5kjs7qdvac Pipe and tabor 0 39877 2812880 1819432 2026-06-05T15:54:38Z Kirby - Electrotechnics 3074947 changed to 'category:Music instruments' instead of category:Musical instruments 2812880 wikitext text/x-wiki [[Image:Thegaloubet.png|thumb|right|125px|An 18th Century pipe and tabor player (tambourinaire).]] The pipe and tabor are two instruments played by a single player. The first instrument is the tabor, which is a small drum, and this is played with a drumstick in one hand. The other instrument is the pipe, which is a three-holed flute with a whistle mouthpiece of the same sort as on a tinwhistle or a recorder. Playing the pipe and tabor might be seen as similar to playing a keyboard instrument, in that one hand plays the melody, while the other provides the accompaniment. In English, the player of pipe and tabor may be called a taborer (''tă'-bər-ər''). The French equivalent name is ''tambourinaire,'' and the Spanish equivalent is ''tamborilero''. ==History== [[Image:Hobbyhorsemorrispaint.png|thumb|left|17th Century English pipe and taborer with dancers.]] Medieval art provides ample illustration of the popularity of one-man pipe and tabor playing during that period. The taborer's pipe was played in the British Isles and many parts of Western Europe. Sometimes another instrument was played in place of the tabor. In France and Spain a special psaltery was tuned as drones and beaten with a stick as a sort of stringed-tabor. Others played bell, triangle, or even a second pipe in place of the tabor. Pipe and tabor was used to accompany clown acts, dances, puppet plays, religious processions, and any outdoor occasion calling for music by a single musician. Pipe and tabor also was seen in bands, along with other instruments, from the Mediaeval period into the 19th Century. The pipe and tabor waned in popularity as bands became more accessible, but never entirely died out. In the 20th Century the regional forms in Spain, Britain, and France resurged. In the late 19th and early 20th Century, British folklorist Cecil Sharp helped to repopularize a national dance called morris dancing. The pipe and tabor, having been a traditional accompaniment to these dances, with the drum providing audible cues to the dancers, was revived by many morris dance sides (troupes). Some of the new taborers looked to the continent and adopted a larger form of pipe, adapting it by adding a fourth hole, so they would not need to partially cover the bottom of the instrument to get an extra low note. Others stayed with the smaller three-hole pipes. Metal versions of both were produced, the latter being made by the makers of Generation tinwhistles, and widely available inexpensively. ==Method== This treatise will focus on the playing of the English pipe and tabor, though much of the information may apply to other species of the pipe and tabor, as well. Aside from regional musical traditions, other forms of pipe vary in their scales, and other forms of tabor vary in their dimensions and in how they are held. ===Playing the tabor=== [[Image:Taborerstainedglass.png|thumb|left|The tabor is suspended from the arm or, if small and light enough, may be suspended from a finger.]] The tabor is often suspended from the arm of the same hand that holds the pipe, and it is beaten with the other hand. Some players use a deeper drum, and hang it from a strap around the neck or attach it to a belt. If it has a snare, the player beats on the snare side, with the snare giving a distinctive buzzing sound. The light shell and rope-tensioned natural heads give the traditional tabor a relatively low pitch. The tabor should be affixed to the arm in such a way that it will not give problems with swinging or spinning too much, but it should not be held firmly against the body, because this will mute the sound and render an inferior tone. It is important to practice the tabor. The job of the drumming-hand is to maintain a steady beat. Many players practice by beating with a drumstick to about any music they hear. Playing with recordings of professional musicians, regardless of the style, will teach the aspiring taborer proper rhythm. Some styles of tabor playing are little more than beating the drum like one would clap one's hand with music. Others incorporate rolls and enhanced rhythmic patterns. It is important to listen to other drummers in order to understand what can be done with the instrument. [[Image:Taborroll.gif|thumb]] ====Drum rolls==== It is quite possible to do rolls on the tabor when using an ordinary drumstick. This is accomplished by using a fulcrum effect, bouncing the head of the stick on the drumhead and the part near the rim of the tabor off the rim. ===Playing the pipe=== [[Image:Taborpipegrip.png|thumb|How the pipe is held.]] The pipe is narrow, enabling the taborer to play all of his scales using the upper registers. Most woodwind instruments play a scale beginning with the lowest possible note, and then add more octaves by overblowing. However, as the taborer's pipe has only three holes, it is impossible to obtain the notes between the lowest four notes and the second octave. Therefore, the pipe begins its first octave at the first overblow. The pipe is gripped, below the lowest finger hole, between the ring finger on front and the little finger on back, which leaves two fingers and a thumb completely free for playing the instrument. The following chart shows the fingering of the pipe. Filled circles represent covered holes. Open circles represent holes left uncovered. Plus signs (+) represent overblown registers. {{audio|Englishpipefingerchart.mid|Play scale}} [[Image:Englishpipefingerchart.png]] ===Lessons=== Some players recommend that a beginner first practice blowing the pipe at different pressures, and then learn the fingerings once comfortable with these. With the above fingerings one can begin playing tunes. The following will provide the beginning piper with some material to practice. ====Lesson One: A Carol==== [[Image:Pipetaborchristmasminstrel.png|thumb|120px|A pipe and taborer at Chrismas]] Christmas is a time when pipe and tabor is often used, due to its merry and traditional nature. It is thus fitting to begin by learning to play a familiar carol. It may help to sing the words in your mind as you play. {{audio|Isaw3shipspipe.mid|Play}} [[Image:Isaw3shipspipe.png]] ====Lesson Two: A Jig==== It is important to play a tune through enough times to memorize it well. This tune is one that is used by morris dancers in England. {{audio|Noroomforthecuckoo.mid|Play}} [[Image:Noroomforthecuckoo.png]] ====Lesson Three: A Reel==== A reel is a traditional type of tune in two parts. The A part is played a few times and then the B part a few times, alternating. {{audio|Pipingsoldiersjoy.mid|Play}} [[Image:Pipingsoldiersjoy.png]] ==Off-site resources== {{wikipedia}} * [http://www.archive.org/details/morrisdancetunes00shar Morris Dance Tunes] compiled by Cecil Sharp * [http://chrisbrady.itgo.com/pipntab/pipntab.htm Address to a Society of Morris Dancers, 1914] by Sir Francis Darwin [[Category:Music instruments]] orky9g7dqsnyqina04ro51fbi5x4xay Introduction to music/Temporary 0 43810 2813018 1801397 2026-06-06T00:36:18Z Kirby - Electrotechnics 3074947 Removed from the main "Category:Music" 2813018 wikitext text/x-wiki {{wv-doomed}} {{Wikiversity:School_of_Music_TOC_SMALL}} == Introduction to music == What is music? Humans have been making music since the Dawn of Time. Some of our strongest emotions may be brought on by listening to a piece of music. In this modern age, We hear music around us almost all of our waking hours, in one form or another. Most of us listen to recorded music or go to performances regularly, and some of us play a musical instrument. Prior to modern audio recording technology, music was available only in the presence of a musician, or to those who play an instrument or sing. === Musical concepts === A basic definition of what music really is (in the Western World), is the chronological organisation of sounds, that is, making certain sounds at certain times, which for the composer (writer) make sense. The first, most basic concept, is keeping the sounds "in time". This leads us to some of the first few musical concepts; beat, rhythm and duration. Duration is the length of a note or sound. Rhythm is a combination of different durations. If one duration is repeated (as with a ticking clock), the continuous repetition is known as the beat. Music is also the relationship between sound and silence. Duration and rhythm apply to silence in the same manner as they apply to sound. We perceive music as horizontal and vertical patterns. We hear melodies as a horizontal pattern. The notes (and silences) are heard one after the other over a period of time. We hear chords (groups of notes played simultaneously) in a vertical pattern. A mixture of one or all of these: melody, rhythm, chords, and silence form musical patterns. === Rhythm === Rhythm is the most basic concept of music. In all cultures worldwide, the most simple and basic forms of music are purely rhythms. A rhythm is a pulse; a repetition of sounds in a pattern. Simple rhythms can be recognised straight away. Tapping ''rhythmically'' at a drum constitutes tapping it at timed intervals in a pattern. The most common rhythmic pattern in modern-day Western music is <math>4/4</math> time (pronounced four-four time). This is where four pulses come one after the other, with the first of each four being given emphasis (known as an accent). Try this exercise: *Say the words "one, two, three, four, one, two, three, four..." etc. continously, and at even time intervals. *Now each time you say "one", say it slightly louder: "'''one''', two, three, four, '''one''', two, three, four..." etc. *You have just been saying the words "one", "two", "three" and "four" in <math>4/4</math> time. === Melody === In music, '''pitch''' is used to describe how high or low a note sounds. Anyone would know the difference between a high-pitched scratching of fingernails across a blackboard, and the low, rumbling growl of thunder. When Maria tells the children in ''The Sound Of Music'' that "the first few notes just happen to be...do re mi", she was referring to pitch. Using pitch, together with rhythm, we can start to construct '''melodies'''. Without rhythm, notes would just be long, sustained sounds. Without pitch, each sound would be the same (for example, the beating of a single drum). === Dynamics === In music, we use the word '''dynamics''' to describe how loudly or softly a note is played. Dynamics falls under the wider category of '''expressive techniques''', which are instructions for the performer to play loudly of softly, smoothly or detached, and many other effects. === Timbre (tone colour) === In music, the "colour" of a sound being produced is referred to as '''timbre''', or '''tone colour'''. Timbre is the difference between the harsh, scratchy sound of an electric guitar with distortion; the glassy, rounded sound of a piano; and the bird-like whispering sound of a flute. All these instruments could play exactly the same note, yet anyone would be able to recognise instantly an electric guitar from a piano from a flute. === Structure === Musical structure is usually defined by several things including [[w:Scale_(music)|scales]] and/or [[w:Arpeggio|arpeggios]] , [[w:Rhythm|rhythm]], [[w:Key_signature|key signature]], melodic patterns, variations Etc. To keep this particular paragraph concise, all or some of the elements of [[w:Music|music theory]] can be used in the structure of a musical piece. Many [[w:List_of_composers|composers]] that listen to [[w:Bach|Bach]] are fascinated by the [[w:palindromes|palindromes]], variations and inversions of patterns contained in its compositional structure, most of which will never be noticed unless you plan on going through the [[[w:Musical_notation|notation]]] with a magnifying glass. --Subnote, Colhsh: Musical Structure also refers to the overall layout of a musical work as a whole, these come in several forms the simplest form is Binary form, in which there is one section of music "A" which is then juxtaposed against a contrasting "B" section which finishes it, giving the piece an "A-B" structure. An expansion on this is ternary form which is the same as binary, except that the "A" section is repeated, making the overall structure "A-B-A" Another musical form that was popular during the classical era is that of the Rondo form, in which there is an "A" section that is repeated throughout the work, but is interrupted by contrasting episodes, making any work in rondo form typically have the structure of "A-B-A-C-A-D-Etc." Larger structures include "Sonata form," which was developed in the Classical Period. The "Sonata form" often is the structure of the first movement of a Sonata, Symphony, and Concerto. The Sonata form is comprised of four sections. *Exposition - Introduces a main theme in the tonic key, and a subordinate theme in a related key - often the dominant, or if in a minor key - the relative major. *Development - Develops and elaborates the themes and explores new and exciting key centers. *Recapitulation - Returns to the tonic key and states the main theme and subordinate theme. The subordinate theme is often reworked to stay in the tonic. *Coda - Concludes the piece. === Texture === Texture refers to the layering of sounds on top of each other. For example, someone might be playing chords on the guitar, while singing a song over the top. Someone else might be playing the drums. Here we can observe '''three''' "layers of sound"; the '''melody''' (the voice), the '''accompaniment''' (the guitar chords) and '''rhycompaniment''' (the drum kit). anndzlfiz6zbw5oeibc9cjxz3l33utk User:Jtneill/Wikiversity 2 56061 2813030 2812409 2026-06-06T08:28:35Z Jtneill 10242 /* Diffs */ + Textdiff 2813030 wikitext text/x-wiki {{TOCright}} [[:Category:Wikiversitans|Wikiversitan]] since March, 2008 ''A loose, personal (i.e., somewhat idiosynchratic) organisation of Wikiversity-related how-tos and links.'' ==To sort== {|style="background:transparent;" |valign=top| * [http://tools.wikimedia.de/~magnus/commonshelper.php commonshelper] * [[User:Jtneill/Wikification|Wikification]] * [[w:Help:Interwiki_linking#Project_titles_and_shortcuts|Interwiki linking]] * [[Wikiversity:Activity bars]] * [[Wikiversity:Percent complete]] |valign=top| * [[Wikiversity:Import|import]] * [[Wikiversity:Maintenance]] * [[Wikiversity:Namespaces]] * [[Wikiversity:Naming conventions]] |valign=top| * [[Wikiversity:Participants]] * [[Wikiversity:Peer review]] * [[Wikiversity:Review board]] * [[Wikiversity:Searching]] * [[How to be a Wikimedia sysop]] |} ==Anchor== * [[Template:Anchor]], e.g., [[#test]] will go to <code><nowiki>{{anchor|test}}</nowiki></code> or <code><nowiki>{{anchor|anchor=test}}</nowiki></code> (should go to end of page) ==Archiving== * Example of autoarchiving: [[User talk:Terra]] ==Blogging== * [[Wikiversity Blog howto]] ==Boxes== [[User:Jtneill/Sandbox/Tables and boxes]] The simplest of boxes {| class="messagebox" |- | ABC XYZ |} <blockquote style="padding-left:1.0em; padding-right:1.0em; background-color:#eaf8f4;"> Its good that it works in practice, because it certainly doesn’t work in theory[https://blogs.ch.cam.ac.uk/pmr/2007/10/14/the-thing-about-wikipedia-is-that-it-only-works-in-practice-in-theory-it-can-never-work/] </blockquote> ==Categories== It is possible to change the order in which a page’s categories are displayed. By default, categories are displayed in the order they appear in the wikitext. Wikis with a consensus to do so can [[m:Special:MyLanguage/Requesting wiki configuration changes|request]] a configuration change to display them in alphabetical order. [https://phabricator.wikimedia.org/T373480] Using titleparts <nowiki>[[Category:{{#titleparts:{{PAGENAME}}|1}}]]</nowiki> ==[[/Centering/]]== {{User:Jtneill/Wikiversity/Centering}} ==Chat== * [[irc:wikiversity-en|#wikiversity-en]] ==Citations and referencing== * [[w:Help:Citation tools|Citation tools]] * [[:Category:Citation templates]] * [[mw:Help:Cite]] * [[Template:Citation]] * [[WV:REF]] * Example: Outward Bound Process Model<ref>Walsh, V., & Golins, G. L. (1976). ''[http://wilderdom.com/theory/OutwardBoundProcessModel.html The exploration of the Outward Bound process]''. Denver, CO: Colorado Outward Bound School.</ref> ;References {{reflist|1}} ==Collapse boxes== {{collapse top|Mary had a little lamb}} Mary had a little lamb, Little lamb, little lamb, Mary had a little lamb, Its fleece was white as snow And everywhere that Mary went, Mary went, Mary went, Everywhere that Mary went The lamb was sure to go It followed her to school one day School one day, school one day It followed her to school one day Which was against the rules. It made the children laugh and play, Laugh and play, laugh and play, It made the children laugh and play To see a lamb at school And so the teacher turned it out, Turned it out, turned it out, And so the teacher turned it out, But still it lingered near And waited patiently about, Patiently about, patiently about, And waited patiently about Till Mary did appear "Why does the lamb love Mary so?" Love Mary so? Love Mary so? "Why does the lamb love Mary so?" The eager children cry "Why, Mary loves the lamb, you know." Loves the lamb, you know, loves the lamb, you know "Why, Mary loves the lamb, you know." The teacher did reply {{collapse bottom}} ==Colour== * [[Wikiversity web page colors|Color tables]] | [[Wikiversity:Color names|Color names]] * e.g., Font: {{font|color=green|Green}}, Background: <span style="background:hotpink; color:white;">Pink</span> ==Columns== ===Column breaks=== {| |- | Works on all browsers (col-begin/break/end): {{col-begin}} {{col-break}} * Col1 {{col-break}} * Col2 {{col-break}} * Col3 {{col-end}} Works on all browsers (col/break/colend): {{col}} {{break}} * Col1 {{break}} * Col2 {{break}} * Col3 {{col/end}} |} ===Moz-column=== Easier to use, but doesn't work on all browsers: <div style="column-count:3;-moz-column-count:3;-webkit-column-count:3"> * Ant * Bee * Buzzard * Cat * Dog * Egret * Elephant * Tiger * Whale * Worm </div> ==Conversions== ===HTML=== * [[w:Wikipedia:Tools/Editing_tools#From_HTML]] * [http://www.ebruni.it/en/software/os/i_love_wiki/index.mpl i love wiki] * {{tick}} [http://diberri.dyndns.org/wikipedia/html2wiki/index.html HTML::WikiConverter] * {{tick}} [http://openfacts2.berlios.de/html2wiki/index.php HTML::WikiConverter]] Add URL ==CSS== * [[MediaWiki:Common.css]] ==Custodianship== * [[Wikiversity:Custodianship]] ** [[Wikiversity:Candidates for Custodianship]] ** [[Wikiversity:Notices for custodians]] ** [[Wikiversity:Request custodian action]] ** [[:Category:Wikiversity custodians]] ==Diffs== Some ways of showing diffs: * [[Wikipedia:User:NguoiDungKhongDinhDanh/FormattedEditRequest|Proposed edits side by side]] (using script) {{Textdiff|[[file:Question book-new.svg|50x40px|alt=icon]]|[[file:Question book-new.svg|50x40px|alt=icon|link=]]}} {| class="diff" ! Before ! After |- | class="diff-deletedline" | <div> The course page should list all enrolled students.<br> Student names should be updated weekly. </div> | class="diff-addedline" | <div> The course page should identify participating student editors.<br> Student editor names may be added as appropriate. </div> |} <syntaxhighlight lang="diff"> -The course page should list all enrolled students. +The course page should identify participating student editors. </syntaxhighlight> ==Edit page== Create an internal link to the edit source page using: [[Special:EditPage/{{FULLPAGENAME}}|Edit source]] <nowiki> {{edit page}} </nowiki> gives: {{edit page}} <nowiki> {{edit page box}} </nowiki> gives: {{edit page box}} ==Extensions== * [[Special:Version#Extensions]] * [[/CategoryTree|CategoryTree]] * [http://www.sandboxserver.org/wiki/index.php?title=Testing_Mediawiki_extensions Sandbox server - testing extensions] * [[User:Jtneill/WYSIWIG|WYSIWIG]] ==Font== <p>{{font|face="courier"|size=medium|courier size 3}}</p> <p>{{font|face="verdana"|size=large|verdana size 4}}</p> <p>{{font|face="arial"|size=x-large|arial size 5}}</p> <p>{{font|face="times new roman"|size=xx-large|times new roman size 6}}</p> <p><b>{{font|face="verdana"|size=xx-large|verdana bold size 6}}</b></p> <p>{{font|face="lucida calligraphy"|size=xx-large|lucida calligraphy size 7}}</p> ==Formatting== ===Justification=== <div style="text-align: justify"> This text is right justified (but it doesn't look like unless the paragraph is long enough to go over one line on the page, so this is intentionally a particularly and unnecessarily long sentence in order to demonstrate right justification using <nowiki><div style="text-align: justify">...</div></nowiki>).</div> ==Line height== {{center top}}<p style="line-height: 36px;"> <big><big><big><big>This uses a<br>line height of 36px</big></big></big></big></p> <pre><p style="line-height: 36px;">...</p></pre> {{center bottom}} ===Mouse-over=== * [[Help:Mouse-over]] * [[Template:H:title]] ==Getting started== * [[Wikiversity:Guided tour|Guided tour]] * [[Wikiversity:Introduction|Introduction]] (Wikiversity) * [[/Introduction|Introduction]] (Jtneill) * [[/Welcome|Welcome]] (Jtneill) * [[Introduction to Wiki]] - [[Wiki 101]] * [[How to use wiki technology as a free learner]] * [[:Image:Short.ogg|Wikiversity - short intro]] (10 sec. video) * [[:Image:Editing_tutorial-large.ogg|Wikiversity editing tutorial]] (2 min video) * [[Wikiversity:Community Portal]] * [[Wikiversity:Content development]] * [[Help:Edit summary]] * [[Making links]] ==Good design== * [[User:Jtneill/Good design]] ==Icons== * [[Help:Icons]] * [[User:McCormack/icons]] ==Images== ===[[Template:Gallery|Gallery]]=== {{Gallery |title=Gallery of images |footer=Uses this [[Template:Gallery|template]] |width=150 |lines=2 ||Comment |File:Wikiversity-logo-Snorky.svg|[[Help:Contents/Links|Links]] can be put in captions. |File:Wikiversity-logo-Snorky.svg|Full [[MediaWiki]]<br />[[syntax]] may be used… |File:Wikiversity-logo-Snorky.svg| }} <!-- Fixed image in bottom right which is linked --> <div id="template-navbar" style="position: fixed; left:1; right:0; bottom:0; padding:0; font-size:122%;">[[Image:Happy.png|right|50px|link=en:Happiness|Happiness]]</div> ===ImageMap=== * [[mw:Extension:ImageMap|Extension ImageMap]] e.g., {{center top}} <imagemap>File:Treasurchest.svg|center|80px default [[Special:Random/|Random Wikiversity mainspace page]] desc none</imagemap>Click the treasure box to go to a random [[Wikiversity]] page{{center bottom}} ;Explanation The ImageMap extension allows, among other things, an image to link directly to a page e.g., as an internal link: <imagemap> File:Treasurchest.svg|center|150px|alt=Alt text default [[Motivation and emotion/Book/2015|Motivation and emotion Book - 2015]] </imagemap> The syntax is: <pre style="overflow:auto"> <imagemap> File:Treasurchest.svg|center|150px|alt=Alt text default [[Motivation and emotion/Book/2015|Motivation and emotion Book - 2015]] </imagemap> </pre> or as an external link: <imagemap> File:Treasurchest.svg|center|150px|alt=Alt text default [https://www.psychologytoday.com/basics/motivation Motivation (Psychology Today)] </imagemap> The syntax is: <pre style="overflow:auto"> <imagemap> File:Treasurchest.svg|center|150px|alt=Alt text default [https://www.psychologytoday.com/basics/motivation Motivation (Psychology Today)] </imagemap> </pre> ==Integrations== I'm interested to explore possible connections between WV and: * [http://archive.org Archive.org] * [[w:Citizendium|Citizendium]] * [[w:Google Groups]] * [[Moodle]] * [[Open University]] * [http://openlearn.open.ac.uk/course/view.php?name=Cohere Cohere] * [[WikiMedia Sister Projects]], particularly: ** [[Wikibooks]] ** [[Wikipedia]] ** [[Simple Wikipedia]] ==Licensing== * My teaching materials are licensed under [[Wikiversity:License tags#Free licenses|creative commons attribution 2.5]] and hosted either on http://wilderdom.com or http://ucspace.canberra.edu.au. I am thinking I should be dual licensing, but am still coming to grips with trying to understand the licensing similarities, differences, and issues. * I plan to gradually transfer most of my teaching materials to the various [[w:WikiMedia Foundation|WikiMedia Foundation]] wiki projects, particularly wikiversity. [[m:Polls|Let's just hope Jimbo doesn't put adds on these sites]], otherwise I will be transferring the materials somewhere else (again). * [http://beta.wikiversity.org/wiki/Wikiversity:IRC_meeting:New_licence_for_Wikiversity_Beta New_licence_for_Wikiversity_Beta] * {{tl|db-copyvio}} * {{tl|hangon}} * [[:Category:Astronomy Images]] ==Links== * Plain links: e.g., <span class="plainlinks">[http://archive.org http://archive.org]</span>: <br><nowiki><span class="plainlinks"> ... </span></nowiki> * [[mw:Manual:Opening external links in a new window]] ==Long page warning== * [[MediaWiki:Longpagewarning]] ==[[Main page]]== * [[:Category:Main page templates]] * [[Main Page/Layout 0.5]] * <span class="plainlinks">[http://en.wikiversity.org/w/index.php?title=Wikiversity:Main_Page&oldid=209253 Main page]</span> (old) ==Map== <mapframe latitude="-28.420391" longitude="136.757813" zoom="2" width="200" height="109" align="right">{ "type": "FeatureCollection", "features": [ { "type": "Feature", "properties": {}, "geometry": { "type": "Point", "coordinates": [ 149.12419, -35.308275 ] } } ] }</mapframe> ==Namespaces== * [[Special:NamespaceInfo]] ==Navigation== {{nav|User:Jtneill}} * [[Template:nav]] ==Notes== Small e.g., {{attention}} <small>For calendar due dates, see unit outline.</small> Notice templates {{Notice|{{tl|Notice}}}} {{Note|{{tl|Note}}}} ==Notifications== * [[Help:Notifications]] ==Pages== * [[Special:AllPages]] * Number of pages in category: <nowiki>{{PAGESINCATEGORY:User:Jtneill}}</nowiki> * {{hitcounter}} - <nowiki>{{hitcounter}}</nowiki> ==[[Project:Participants|Participants]]== *[[Wikiversity:Support staff]] ===Users=== *{{Participant|CQ}} - see Person of the Hour script *{{Participant|Donek}} *{{Participant|Dan Polansky}} ==Pedagogy== * [[Learning by doing]] * [[Wikiversity:Project incubator]] ==Policy== * [[w:Wikipedia:Contributing_FAQ#Is_there_a_minimum_age_requirement_to_contribute_or_register.3F|Is there a minimum age requirement?]] {{Official policies}} {{Proposed policies}} ==Project boxes== * [[Help:Resource attribution]] ==Purge== To purge the cache for a given page, append this to the URL: ?action=purge [[mw:Manual:Purge]] ==Quotes== * [[Template:Quote]] * ==[[Quizzes]]== * [[Help:Quiz-Simple]] * [http://www.qedoc.org/en/index.php?title=User:Jtneill My Qedoc user page] ** [http://eduforge.org/forum/forum.php?forum_id=1138 Qedoc now exports quizzes to Wikiversity] ==Referencing== * [[meta:WMDE Technical Wishes/Sub-referencing]] ==Sandbox== * http://www.sandboxserver.org/ * [[Wikiversity:Sandbox Server]] * [[Topic:Sandbox Server 0.5]] * [http://scratchpad.wikia.com/wiki/Scratchpad_Wiki_Labs Scratchpad] * [[../Sandbox]] ==Searching== * [[Help:Google]] * [[Wikiversity:Colloquium/archives/April 2008#Google search|Google search]] - <nowiki>[[google:wikiversity]]</nowiki> [[google:wikiversity]] * Use a + instead of a space ==Search multiple categories== ;Dual category search including one category with subcategories Search for chapters which [[Template:Clarification templates|need clarification]]: <inputbox> type=search width=33 default=incategory:"Resources needing clarification" namespaces=Main** prefix=Motivation and emotion/Book searchbuttonlabel=Search book chapters bgcolor=transparent break=no </inputbox> ==Sitenotice== * [[MediaWiki:Sitenotice]] * [[MediaWiki:Sitenotice id]] ==Size== ===Big/small=== * Use <code><nowiki><big>...</big> - could be also <big><big>...</big></big> etc. and also <small>...</small></nowiki></code> ===CSS=== <div style="font-size: 200%">200% text</div><code><nowiki><div style="font-size: 200%">200% text</div></nowiki></code> <div style="font-size: 150%">150% text</div><code><nowiki><div style="font-size: 150%">150% text</div></nowiki></code> ==Special== * [[Special:SpecialPages]] * Abuse ** [[Special:AbuseFilter]] ** [[Special:AbuseLog]] * [[Special:AccountSecurity]] * [[Special:Allpages]] * [https://auth.wikimedia.org/enwikiversity/wiki/Special:CreateAccount Create account] * [[meta:Special:GlobalWatchlist]] * [[Special:ListGroupRights]] * [[Special:PermanentLink]] * [[Random]] - [[Special:Random]] - [[Wikiversity:Random]] * [[Special:ShortPages]] * [[Special:Version#Installed extensions]] ==Strategy== * [[Wikiversity:Publicity]] * [[Wikiversity:Vision]] * [[Wikiversity:Vision 2009]] ==Statistics== * [[Wikiversity:Statistics]] * [[Google Search and Wikiversity]] * [http://wikistics.falsikon.de/latest/wikiversity/en/ Monthly page hits for wikiversity.en] * [http://gtools.org/tool/wikipedia-edit-counter/?str=jtneill&project=en.wikiversity Jtneill edit count] * https://xtools.wmcloud.org/pageinfo/en.wikiversity.org/ * [[Special:Impact]] - [[w:Special:Impact]] ==Sub-pages== * [[Special:Prefixindex/User:Jtneill]] * Transclude: ** <code><nowiki>{{Special:Prefixindex/User:Jtneill}}</nowiki></code> ** <code><nowiki>{{Special:Prefixindex/{{NAMESPACE}}:{{PAGENAME}}}}</nowiki></code> ==Stubs== * [[:Category:Stub templates]] ==Structure== * [[Wikiversity:Browse/Concept]] ==Symbols== 🟨🟡⭐💛🟥⭕️❌🟦🔵🟩🟢✅ * [[User:VeronicaJeanAnderson]] ==System messages== * [[Special:AllMessages]] * [[#Sitenotice|Site notice]] ==Style== * [[MoS]] * [[MediaWiki:Common.css]] ==Tables== * [[Help:Table]] * [[User:Jtneill/Sandbox/Tables and boxes]] ==Tagging/notification== * <nowiki>@[[User:UserName|UserName]]</nowiki> * <nowiki>{{ping|UserName}}</nowiki> ==Templates== ===Page development=== * {{tl|welcome and expand}} - {{tl|we}} * {{tl|main welcome}} * {{tl|search}} * {{tl|draft}} * {{tl|underconstruction}} * {{tl|Learning project boilerplate}} * {{tl|info}} * {{tl|note}} * {{tl|notice}} * {{tl|Nutshell}} * <nowiki>{{notice|{{findsources}}}}</nowiki> ===Page navigation=== * [[Template:EasyNavBar]] * [[Template:Recovery psychology]] (example) * [[Workshop for Australian education policy]] (example) ===Sister projects=== * [[Template:Sisterprojectsearch]] * [[Template:Wikibooks]] * [[Template:Wikipedia]] * [[Template:Wikiversity]] ===User talk=== * {{tl|Welcomeip}} * {{tl|Welcome}} * {{tl|Talk header}} * [[:Category:User warning templates]] ===Administrative=== * [[Template:Category redirect]] * [[Template:Warning]] ==Theory== * [[Learning by engagement]] * [[User:JWSchmidt/Wiki Scholar]] ==Thoughts== * [[Red link]]s are doorways to the infinite library ([[w:The Library of Babel|Library of Babel]]) ==Tooltips== {{Tooltip|Tooltips allow additional text to be displayed when cursor hovers over|Pretty cool, eh?}} ==User== * [[w:Special:GlobalRenameRequest]] * [[Special:UserGroupRights]] * [[Special:UserRights]] * [[m:Steward requests/Permissions]] * [[meta:Help:Two-factor authentication]] ==Usability== * [[Wikiversity:Usability]] * http://usability.wikimedia.org - [http://usability.wikimedia.org/wiki/User:Jtneill Jtneill] ==Video== * .ogg files can be uploaded and embedded * See [[/Video]] for examples ==wikEd== * [[w:User_talk:Cacycle/wikEd]] ==Wiki2Reveal== * [[Wiki2Reveal]] (slides on the fly from MediaWiki page) ==x Test anchor== <!-- Test anchor - don't delete! --> {{anchor|test}} ==See also== * [[User:Jade Knight/Tools]] g19aw075qqhj02limp14fxwex2xy1yx Template:Documentation subpage 10 57631 2813129 2526485 2025-10-14T17:59:49Z Anomie 27260 The CC BY-SA license on [[File:Edit-copy green.svg]] requires a link to the file description page for attribution and notice of license. See [[MOS:PDI]] for more info. If someone would like to create a fresh derivative of [[File:Edit-copy.svg]] that maintains the public domain licensing of that source, that'd be useful. 2813129 wikitext text/x-wiki <includeonly><!-- -->{{#ifeq:{{lc:{{SUBPAGENAME}}}} |{{{override|doc}}} | <!--(this template has been transcluded on a /doc or /{{{override}}} page)--> </includeonly><!-- -->{{#ifeq:{{{doc-notice|show}}} |show | {{Mbox | type = notice | style = margin-bottom:1.0em; | image = [[File:Edit-copy green.svg|40px|alt=icon]] | text = {{strong|This is a [[Wikipedia:Template documentation|documentation]] [[Wikipedia:Subpages|subpage]]}} for {{terminate sentence|{{{1|[[:{{SUBJECTSPACE}}:{{BASEPAGENAME}}]]}}}}}<br />It may contain usage information, [[Wikipedia:Categorization|categories]] and other content that is not part of the original {{#if:{{{text2|}}} |{{{text2}}} |{{#if:{{{text1|}}} |{{{text1}}} |{{#ifeq:{{SUBJECTSPACE}} |{{ns:User}} |{{lc:{{SUBJECTSPACE}}}} template page |{{#if:{{SUBJECTSPACE}} |{{lc:{{SUBJECTSPACE}}}} page|article}}}}}}}}. }} }}<!-- -->{{DEFAULTSORT:{{{defaultsort|{{PAGENAME}}}}}}}<!-- -->{{#if:{{{inhibit|}}} |<!--(don't categorize)--> | <includeonly><!-- -->{{#ifexist:{{NAMESPACE}}:{{BASEPAGENAME}} | [[Category:{{#switch:{{SUBJECTSPACE}} |Template=Template |Module=Module |User=User |#default=Wikipedia}} documentation pages]] | [[Category:Documentation subpages without corresponding pages]] }}<!-- --></includeonly> }}<!-- (completing initial #ifeq: at start of template:) --><includeonly> | <!--(this template has not been transcluded on a /doc or /{{{override}}} page)--> }}<!-- --></includeonly><noinclude>{{Documentation}}</noinclude> 17qbz5a9qk3g761gr43dx5nl4pc1l84 2813130 2813129 2025-11-04T00:25:49Z Andrybak 1869413 add parameter "nocat" for a more standardized [[WP:NOCAT|category suppression]] in addition to the existing parameter "inhibit" 2813130 wikitext text/x-wiki <includeonly><!-- -->{{#ifeq:{{lc:{{SUBPAGENAME}}}} |{{{override|doc}}} | <!--(this template has been transcluded on a /doc or /{{{override}}} page)--> </includeonly><!-- -->{{#ifeq:{{{doc-notice|show}}} |show | {{Mbox | type = notice | style = margin-bottom:1.0em; | image = [[File:Edit-copy green.svg|40px|alt=icon]] | text = {{strong|This is a [[Wikipedia:Template documentation|documentation]] [[Wikipedia:Subpages|subpage]]}} for {{terminate sentence|{{{1|[[:{{SUBJECTSPACE}}:{{BASEPAGENAME}}]]}}}}}<br />It may contain usage information, [[Wikipedia:Categorization|categories]] and other content that is not part of the original {{#if:{{{text2|}}} |{{{text2}}} |{{#if:{{{text1|}}} |{{{text1}}} |{{#ifeq:{{SUBJECTSPACE}} |{{ns:User}} |{{lc:{{SUBJECTSPACE}}}} template page |{{#if:{{SUBJECTSPACE}} |{{lc:{{SUBJECTSPACE}}}} page|article}}}}}}}}. }} }}<!-- -->{{DEFAULTSORT:{{{defaultsort|{{PAGENAME}}}}}}}<!-- -->{{Category handler |nocat={{{nocat|{{{inhibit|}}}}}} |all=<includeonly><!-- -->{{#ifexist:{{NAMESPACE}}:{{BASEPAGENAME}} | [[Category:{{#switch:{{SUBJECTSPACE}} |Template=Template |Module=Module |User=User |#default=Wikipedia}} documentation pages]] | [[Category:Documentation subpages without corresponding pages]] }}<!-- --></includeonly> }}<!-- (completing initial #ifeq: at start of template:) --><includeonly> | <!--(this template has not been transcluded on a /doc or /{{{override}}} page)--> }}<!-- --></includeonly><noinclude> {{Documentation}} </noinclude> ts6noosbz7c9lll2pblw35mx6b8wufa 2813131 2813130 2025-11-04T01:38:19Z Andrybak 1869413 remove usage of [[Template:Category handler]] – because of pattern '^User talk:UBX%f[/\0]' at [[Module:Category handler/blacklist]] (originated at [[Special:Diff/454139417]]), the previous edit broke categorization of pages like [[User:UBX/Sample/doc]] 2813131 wikitext text/x-wiki <includeonly><!-- -->{{#ifeq:{{lc:{{SUBPAGENAME}}}} |{{{override|doc}}} | <!--(this template has been transcluded on a /doc or /{{{override}}} page)--> </includeonly><!-- -->{{#ifeq:{{{doc-notice|show}}} |show | {{Mbox | type = notice | style = margin-bottom:1.0em; | image = [[File:Edit-copy green.svg|40px|alt=icon]] | text = {{strong|This is a [[Wikipedia:Template documentation|documentation]] [[Wikipedia:Subpages|subpage]]}} for {{terminate sentence|{{{1|[[:{{SUBJECTSPACE}}:{{BASEPAGENAME}}]]}}}}}<br />It may contain usage information, [[Wikipedia:Categorization|categories]] and other content that is not part of the original {{#if:{{{text2|}}} |{{{text2}}} |{{#if:{{{text1|}}} |{{{text1}}} |{{#ifeq:{{SUBJECTSPACE}} |{{ns:User}} |{{lc:{{SUBJECTSPACE}}}} template page |{{#if:{{SUBJECTSPACE}} |{{lc:{{SUBJECTSPACE}}}} page|article}}}}}}}}. }} }}<!-- -->{{DEFAULTSORT:{{{defaultsort|{{PAGENAME}}}}}}}<!-- -->{{#if:{{{nocat|}}}{{{inhibit|}}}|<!--(don't categorize)--> |<includeonly><!-- -->{{#ifexist:{{NAMESPACE}}:{{BASEPAGENAME}} | [[Category:{{#switch:{{SUBJECTSPACE}} |Template=Template |Module=Module |User=User |#default=Wikipedia}} documentation pages]] | [[Category:Documentation subpages without corresponding pages]] }}<!-- --></includeonly> }}<!-- (completing initial #ifeq: at start of template:) --><includeonly> | <!--(this template has not been transcluded on a /doc or /{{{override}}} page)--> }}<!-- --></includeonly><noinclude>{{Documentation}}</noinclude> eqi6b0pjsizfmg2rq1t66pwtl185hlq 2813132 2813131 2025-12-20T16:57:59Z Pppery 2799023 Fix link 2813132 wikitext text/x-wiki <includeonly><!-- -->{{#ifeq:{{lc:{{SUBPAGENAME}}}} |{{{override|doc}}} | <!--(this template has been transcluded on a /doc or /{{{override}}} page)--> </includeonly><!-- -->{{#ifeq:{{{doc-notice|show}}} |show | {{Mbox | type = notice | style = margin-bottom:1.0em; | image = [[File:Edit-copy green.svg|40px|alt=icon]] | text = {{strong|This is a [[Wikipedia:Template documentation|documentation subpage]]}} for {{terminate sentence|{{{1|[[:{{SUBJECTSPACE}}:{{BASEPAGENAME}}]]}}}}}<br />It may contain usage information, [[Wikipedia:Categorization|categories]] and other content that is not part of the original {{#if:{{{text2|}}} |{{{text2}}} |{{#if:{{{text1|}}} |{{{text1}}} |{{#ifeq:{{SUBJECTSPACE}} |{{ns:User}} |{{lc:{{SUBJECTSPACE}}}} template page |{{#if:{{SUBJECTSPACE}} |{{lc:{{SUBJECTSPACE}}}} page|article}}}}}}}}. }} }}<!-- -->{{DEFAULTSORT:{{{defaultsort|{{PAGENAME}}}}}}}<!-- -->{{#if:{{{nocat|}}}{{{inhibit|}}}|<!--(don't categorize)--> |<includeonly><!-- -->{{#ifexist:{{NAMESPACE}}:{{BASEPAGENAME}} | [[Category:{{#switch:{{SUBJECTSPACE}} |Template=Template |Module=Module |User=User |#default=Wikipedia}} documentation pages]] | [[Category:Documentation subpages without corresponding pages]] }}<!-- --></includeonly> }}<!-- (completing initial #ifeq: at start of template:) --><includeonly> | <!--(this template has not been transcluded on a /doc or /{{{override}}} page)--> }}<!-- --></includeonly><noinclude>{{Documentation}}</noinclude> ngbz78vpmpnny1xbfvkmebdalzs512j 2813133 2813132 2026-06-06T08:30:17Z Jtneill 10242 4 revisions imported from [[:w:Template:Documentation_subpage]] 2813132 wikitext text/x-wiki <includeonly><!-- -->{{#ifeq:{{lc:{{SUBPAGENAME}}}} |{{{override|doc}}} | <!--(this template has been transcluded on a /doc or /{{{override}}} page)--> </includeonly><!-- -->{{#ifeq:{{{doc-notice|show}}} |show | {{Mbox | type = notice | style = margin-bottom:1.0em; | image = [[File:Edit-copy green.svg|40px|alt=icon]] | text = {{strong|This is a [[Wikipedia:Template documentation|documentation subpage]]}} for {{terminate sentence|{{{1|[[:{{SUBJECTSPACE}}:{{BASEPAGENAME}}]]}}}}}<br />It may contain usage information, [[Wikipedia:Categorization|categories]] and other content that is not part of the original {{#if:{{{text2|}}} |{{{text2}}} |{{#if:{{{text1|}}} |{{{text1}}} |{{#ifeq:{{SUBJECTSPACE}} |{{ns:User}} |{{lc:{{SUBJECTSPACE}}}} template page |{{#if:{{SUBJECTSPACE}} |{{lc:{{SUBJECTSPACE}}}} page|article}}}}}}}}. }} }}<!-- -->{{DEFAULTSORT:{{{defaultsort|{{PAGENAME}}}}}}}<!-- -->{{#if:{{{nocat|}}}{{{inhibit|}}}|<!--(don't categorize)--> |<includeonly><!-- -->{{#ifexist:{{NAMESPACE}}:{{BASEPAGENAME}} | [[Category:{{#switch:{{SUBJECTSPACE}} |Template=Template |Module=Module |User=User |#default=Wikipedia}} documentation pages]] | [[Category:Documentation subpages without corresponding pages]] }}<!-- --></includeonly> }}<!-- (completing initial #ifeq: at start of template:) --><includeonly> | <!--(this template has not been transcluded on a /doc or /{{{override}}} page)--> }}<!-- --></includeonly><noinclude>{{Documentation}}</noinclude> ngbz78vpmpnny1xbfvkmebdalzs512j Median 0 63371 2813025 2112892 2026-06-06T04:44:29Z OzmoOzmo 3055305 ce 2813025 wikitext text/x-wiki {{Statistics}} In Statistics, median is the number in the middle of a set of numbers. The median of a series of numbers is obtained by arranging the numbers in ascending order and then choosing the number in the 'middle'. If there are ''two'' middle numbers, the median is the average (mean) of these two numbers. == Examples and application == 1. <u>Find the median of the following set of numbers:</u> <p style="text-align:center;">5, 4, 10, 3, 3, 4, 7, 4, 6, 5 </p> So, first of all, let's arrange the numbers in ascending order: <p style="text-align:center;">3, 3, 4, 4, 4, 5, 5, 6, 7, 10 </p> Then you can use 'striking out' to find the median easily and without mixing as I did here: <s>3, 3, 4, 4,</s> 4, 5,<s>5, 6, 7, 10</s> Here, what I did was that I began canceling for the ends of the set: cancel the 3 with 10, the second three with 7 and so on.. So, now we found out that the median was two numbers: 4 and 5. In this case we will add them and then divide them by two, just as if we are finding their mean. So 4+5=9, and then 9\2 is 4.5, so the median here is 4.5. 560vgjsgll1k862mjszbxvd3h8risl1 Electronic Music and DJing 0 74818 2812957 2722036 2026-06-05T21:46:10Z Kirby - Electrotechnics 3074947 changed to 'category:Electronic Music' instead of category:Music 2812957 wikitext text/x-wiki {{TOCright}} [[Electronics/Resistors|Electronic]] Music and DJing is a proposed 2 credit [[Olin_College_Student_Courses|Student Designed Course at Olin College]] in Boston, MA. The current plan is to have it centered around the musical aspects of DJing Electronic Music, with studying done on working the crowd and music selection. It will probably involve studying the styles of various DJs, comparing them to each other, practicing/recording your own mixes, and critiquing the rest of the class. Due to the limited set of equipment, enrollment will probably be limited to the folks most interested in the subject as determined by the equipment's owner. We will probably be working under the advisement of Diana Dabby to make this course a reality. == Course Background, Description, and Expectations == Electronic Music and DJing is proposed as a 2 credit student organized class during Spring 2009. The students will be learning about the musical aspects of DJing electronic music through both studying other DJs and learning to DJ themselves. These analysis will be quite brief and will be centered around the overall impressions of the mix. This means that each student will have around 6 hours of DJing and listening as a time commitment every week. Additionally, compiling a list of what you learn over the course of the semester would help demonstrate improvement, and learning. The primary reason for taking this course would be because you have a passion to learn and become an electronic music DJ (or you would like to become a better one). === Time commitment === As a 2 credit course, 6 hours of work a week are expected, though this course will very likely take longer than that. Proposed for now is that every week following equipment training, a 1-2 hour mix of music should be recorded by each student, and then listened to and analyzed by the student and at least one other student. === Culture and history === In addition to practical aspects of DJing, the course will also cover the history of DJing and electronic music. There will be an emphasis on understanding the various subcultures surrounding electronic music (underground, commercial, and radio, for example). === Music === We would like to investigate the various subcultures surrounding various subgenres of electronic music (trance, electro-house, breaks, psytrance, etc.). In addition, basic traditional music theory principles will be explored in relation to electronic music - everything from understanding modern remixes of classical music (Tiesto, etc.) to understanding how key, tempo, and rhythmic structures affect the musical experience. === Deliverables === '''These are proposed deliverable ideas, and not solidified''' Weekly deliverables should include a 1-2 hour mix of music created by each student. Each student will also be expected to listen to the mix of another person in the class and provide analysis and feedback. The final course deliverable will include a mix and a paper explaining the decisions made in the production of the mix, the style of the music included, and the musical theory underlying the individual pieces and the entire mix. == Competencies == Olin College employs a competency system. The following competencies will possibly be addressed by the Electronic Music and DJing course: * Life Long Learning * Communication * Context * Diagnosis == How to Join == Send an e-mail to Matt Wootton including the following: why you would like to take the course, any prior DJ experience, and what you hope to gain out of the course. Due to the equipment's location and the fact that the meetings would likely be in the evenings, it would be best to limit enrollment to Olin students for access reasons. Depending on how the course description works out, it will also be likely that the number of students will be limited in order to accommodate the limited amount of equipment. [[Category:Electronic Music]] 8kxj3by5fqszj5o03tcmw9daqmd2xy7 Wikiversity:Request custodian action 4 75745 2812895 2811659 2026-06-05T17:37:26Z Codename Noreste 2969951 /* ~2026-28792-52 */ archive to [[Wikiversity:Request custodian action/Archive/30#~2026-28792-52]] ([[mw:c:Special:MyLanguage/User:JWBTH/CD|CD]]) 2812895 wikitext text/x-wiki {{/Header}} == New User Exceeded New Page Limit == Hello, my action of creating a Portal for Banjo learning (Portal:Banjo) was denied as I am a user whose had my account for a while but not written before today. My actions are constructive, but if you would rather me wait and let the system work as it intends to then that's okay too. --[[User:Kirby - Electrotechnics|Kirby - Electrotechnics]] ([[User talk:Kirby - Electrotechnics|discuss]] • [[Special:Contributions/Kirby - Electrotechnics|contribs]]) 01:13, 17 May 2026 (UTC) :That is standard and the rate limit will fall off as you stay and edit; it has some pretty easy barriers to cross. I can create a blank [[Portal:Banjo]] if you want, but to be clear, portal pages are usually much broader topics that can help orient you to specific pages. Do you think there will be that many pages about the topic of banjo playing? If not, then you can just continue editing [[Banjo]]. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 03:36, 17 May 2026 (UTC) == Disable [[Special:AbuseFilter/3]] == Please disable [[Special:AbuseFilter/3]] because it is redundant with [[m:Special:AbuseFilter/104|a global filter]] that disallows the same type of spam that filter 3 would have caught. Thank you. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 01:06, 19 May 2026 (UTC) : [[Special:AbuseFilter/history/3/diff/prev/454|Done]] -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 02:15, 19 May 2026 (UTC) == [[One man's look at concept]] == Please semi-protect that page indefinitely due to vandalism from an unregistered vandal. Thank you. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 18:52, 19 May 2026 (UTC) :I've protected the page, but I've decided to keep it to a year just because the vandalism seemed to have started in late 2025, and the longest protection imposed was just about 3 months. If vandalism persists after the protection, I'd be more than happy to protect it indefinitely. Thanks. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 21:24, 19 May 2026 (UTC) ==Review changes to [[Special:AbuseFilter/4]]== {{ping|Codename Noreste}} Could you review changes I made to this filter to help prevent profanity spam using obfuscated spellings? -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 03:44, 25 May 2026 (UTC) : I've adjusted the filter to prevent any potential false positives, see its conditions. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 18:10, 26 May 2026 (UTC) b50i5si20p0jh5qos5jgv6xkq0ogd99 2812897 2812895 2026-06-05T17:38:28Z Codename Noreste 2969951 /* New User Exceeded New Page Limit */ archive to [[Wikiversity:Request custodian action/Archive/30#New User Exceeded New Page Limit]] ([[mw:c:Special:MyLanguage/User:JWBTH/CD|CD]]) 2812897 wikitext text/x-wiki {{/Header}} == Disable [[Special:AbuseFilter/3]] == Please disable [[Special:AbuseFilter/3]] because it is redundant with [[m:Special:AbuseFilter/104|a global filter]] that disallows the same type of spam that filter 3 would have caught. Thank you. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 01:06, 19 May 2026 (UTC) : [[Special:AbuseFilter/history/3/diff/prev/454|Done]] -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 02:15, 19 May 2026 (UTC) == [[One man's look at concept]] == Please semi-protect that page indefinitely due to vandalism from an unregistered vandal. Thank you. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 18:52, 19 May 2026 (UTC) :I've protected the page, but I've decided to keep it to a year just because the vandalism seemed to have started in late 2025, and the longest protection imposed was just about 3 months. If vandalism persists after the protection, I'd be more than happy to protect it indefinitely. Thanks. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 21:24, 19 May 2026 (UTC) ==Review changes to [[Special:AbuseFilter/4]]== {{ping|Codename Noreste}} Could you review changes I made to this filter to help prevent profanity spam using obfuscated spellings? -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 03:44, 25 May 2026 (UTC) : I've adjusted the filter to prevent any potential false positives, see its conditions. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 18:10, 26 May 2026 (UTC) atjh3ts387933jd8kusnbe4vfr0niqi 2812899 2812897 2026-06-05T17:38:40Z Codename Noreste 2969951 /* Disable Special:AbuseFilter/3 */ archive to [[Wikiversity:Request custodian action/Archive/30#Disable Special:AbuseFilter/3]] ([[mw:c:Special:MyLanguage/User:JWBTH/CD|CD]]) 2812899 wikitext text/x-wiki {{/Header}} == [[One man's look at concept]] == Please semi-protect that page indefinitely due to vandalism from an unregistered vandal. Thank you. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 18:52, 19 May 2026 (UTC) :I've protected the page, but I've decided to keep it to a year just because the vandalism seemed to have started in late 2025, and the longest protection imposed was just about 3 months. If vandalism persists after the protection, I'd be more than happy to protect it indefinitely. Thanks. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 21:24, 19 May 2026 (UTC) ==Review changes to [[Special:AbuseFilter/4]]== {{ping|Codename Noreste}} Could you review changes I made to this filter to help prevent profanity spam using obfuscated spellings? -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 03:44, 25 May 2026 (UTC) : I've adjusted the filter to prevent any potential false positives, see its conditions. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 18:10, 26 May 2026 (UTC) 8s2qxiowdmdrhrbig8zcxty8zfri6n9 2812909 2812899 2026-06-05T18:13:40Z Codename Noreste 2969951 /* One man's look at concept */ archive to [[Wikiversity:Request custodian action/Archive/30#One man's look at concept]] ([[mw:c:Special:MyLanguage/User:JWBTH/CD|CD]]) 2812909 wikitext text/x-wiki {{/Header}} ==Review changes to [[Special:AbuseFilter/4]]== {{ping|Codename Noreste}} Could you review changes I made to this filter to help prevent profanity spam using obfuscated spellings? -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 03:44, 25 May 2026 (UTC) : I've adjusted the filter to prevent any potential false positives, see its conditions. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 18:10, 26 May 2026 (UTC) 5fozupxwfakjor80pqv5j9ibifxy0o3 Music Learning Project 0 79232 2812966 2528929 2026-06-05T21:56:51Z Kirby - Electrotechnics 3074947 changed to 'category:Music education' instead of category:Music 2812966 wikitext text/x-wiki This is a project to produce a 15 minute introduction to music, with an emphasis on the Romantic, Baroque and Classical periods. Possibly for intro/leadin: Gregorian Chants http://www.christusrex.org/www2/cantgreg/missa_h4qua.html ==The Formal Presentation== '''This lesson plan was discussed by Historybuff and Mona, written by Mona, and published by Historybuff''' *Play Beethoven’s 5th symphony as we prepare for the presentation *Stop the music; say hello and introduce names. Our topic tonight is Classical Music, and our agenda is: **What is Classical Music? **Three historical periods **Evaluation **Wrap Up *Icebreaker: **Ask: Who heard before the piece we just played? **Ask: Do you know who wrote it? **Show: Beethoven’s picture **Say: Beethoven’s music belongs to a historical period called “Classical.” Classical either refers to the broad genre in which many different historical periods belong to, or it refers to one period among other periods. **Show as slide with a definition of Classical Music: (say that C.M. can have more than one definition but this is probably the most related to our presentation) **Classical Music is a traditional genre of Western Music, which conformed to established rules (as opposed to popular music which was made “composed” without strict rules.) http://wordnetweb.princeton.edu/perl/webwn?s=classical%20music **Say: as for the specific period, these are the dates: **Show a slide with the timeline for this Classical Period. ***Classical 1750-1825 *Body: **There are other musical periods or eras. In tonight’s presentation, we will cover two more periods: the Baroque and the Romantic. **Ask: anyone knows which period of the two comes first? **Show a slide with the three periods: Baroque: 1600-1750/Classical 1750-1825/Romantic 1825-1900 **Ask: any guesses who were the composers belonging to these periods? ***Possible answers: Mozart, Haydn, Dvoark, Handel, etc. **Say: we will play two musical pieces and we want you to guess the periods. (Play Bach and Chopin) **Ask: what did you think about the music you heard? Have you heard them before? Which one you liked better? Any features you found appealing? Do you know who wrote these pieces? **Say: each musical period had distinct features in relation to melody, harmony, and rhythm among other factors. **Show a slide that compares the three periods **Ask: why is it important to know music history? [possible answers: because music related to other historic events? Or because it help us appreciate the different styles?] **Ask “what if” classical music … (I have no idea yet what to write here!) *Conclusion: **Give a short evaluation form. (We can use the form I usually use): ==Handout== ===A Snapshot of Musical History=== [[Image:Music_notation.svg]] Classical Music Defined: Classical Music is a traditional genre of Western Music, which conforms to established rules (as opposed to popular music which is composed without strict rules.) [[Image:Johann Sebastian Bach.jpg|thumb|Bach]] Today's session covered three periods: *Baroque (1600-1750) *Classical (1750-1825) *Romantic (1825-1900) The Baroque period saw the beginning of modern musical notation and the birth of operas. Music was performed by small ensembles, The musical style was characterized by counterpoint. Examples of this style are the Fugue, Invention, Sonata and Concerto. Harpsichord and the pipe organ were key instruments of this time, while the violin evolved into the form we are familiar with today. Composers in this period include [[w:Johann_Sebastian_Bach|Bach]] and [[w:George_Frideric_Handel|Handel]]. The Classical period saw the birth of the Symphony. The musical style was characterized by melody with accompaniment, also knows as homophonic texture. The violin emerged as a key instrument, and single-reed instruments like the clarinet started to come into their own. Composers in this period include [[w:Ludwig_van_Beethoven|Beethoven]], [[w:Wolfgang_Amadeus_Mozart|Mozart]] and [[w:Joseph_Haydn|Haydn]]. [[Image:Fryderyk Chopin.jpg|thumb|left|Chopin]] The Romantic period saw the enlargement of the orchestra, and a larger role for brass instruments with the invention of the rotary valve.. The musical style was much more dramatic and expressive, and towards the end of the period, compositions became longer and more complex. Composers in this period include [[w:Franz_Schubert|Schubert]], [[w:Frédéric_Chopin|Chopin]] and [[w:Hector_Berlioz|Berlioz]] ===Suggested web references=== "Essentials of Music" at http://www.essentialsofmusic.com/ [[w:Music]] is an excellent starting point. http://en.wikiversity.org/wiki/Music_Learning_Project#Handout [[Category:Music education]] dlbwwua6fx2zk65jnpx3bbs5fym9ibv Musical modes 0 79377 2813005 2703503 2026-06-06T00:28:55Z Kirby - Electrotechnics 3074947 Removed from the main "Category:Music" 2813005 wikitext text/x-wiki {{TOCright}} ''This article is about modes as used in music. For other uses, see mode.'' == Definition == Mode (from Latin modus, "measure, standard, manner, way") is a term from Western music theory having three senses : (1) the rhythmic relationship between long and short values in the late medieval period; (2) in early medieval theory, interval; and, (3) most commonly, a concept involving scale and melody type (Powers 2001). Beginning at the end of the eighteenth century, the term began to be used in ethnomusicological contexts to describe pitch structures in non-European musical cultures, sometimes with doubtful compatibility (Powers 2001 §V,1). == Explanations == === Mode and scale === In music, a "scale" is an ordered series of intervals, which, along with the key or tonic, define the pitches. However, "mode" is usually used in the sense of "scale" applied only to the specific diatonic scales found below.[citation needed] ==== Diatonic Modes ==== A mode can be viewed as an alternative ordering of an existing collection of pitches. For example, the 7 pitches contained in a key with 0 flats and 0 sharps (C major) can be ordered with C as the root to create the Ionian mode (C D E F G A B C), or with D as the root to create the Dorian mode (D E F G A B C D). In total, there are 7 different modes of the diatonic collection, which can be viewed as the major scale with the root changed to different scale degrees. The Ionian mode is more commonly known as the major scale, and the Aeolian mode is more commonly known as the minor scale (or the natural minor scale, as opposed to the harmonic and melodic minor scale, which are not modes of the major scale). {| class="wikitable" |+ !Name !Scale Degree (relative to the major scale) !Pitch Relationship !Example (from C major) !Parallel Major Scale (for comparison) |- |Ionian |1 |W-W-H-W-W-W-H |C D E F G A B C |C D E F G A B C |- |Dorian |2 |W-H-W-W-W-H-W |D E F G A B C D |D E F# G A B C# D |- |Phrygian |3 |H-W-W-W-H-W-W |E F G A B C D E |E F# G# A B C# D# E |- |Lydian |4 |W-W-W-H-W-W-H |F G A B C D E F |F G A Bb C D E F |- |Mixolydian |5 |W-W-H-W-W-H-W |G A B C D E F G |G A B C D E F# G |- |Aeolian |6 |W-H-W-W-H-W-W |A B C D E F G A |A B C# D E F# G# A |- |Locrian |7 |H-W-W-H-W-W-W |B C D E F G A B |B C# D# E F# G# A# B |} ==== Non-Diatonic Modes ==== It is also possible to create modes out of scales that are non-diatonic, such as the ascending melodic minor scale or the harmonic minor scale. Especially in jazz, it's common to use the modes of melodic minor in junction with certain chord extensions/progressions that do not exist in the diatonic collection (for example, the Lydian Dominant mode is used frequently with a V7(#11) chord). Listed below are some of the more common non-diatonic modes, however modes can be made from any parent scale. {| class="wikitable" |+ !Name !Scale Degrees and Alterations (from parallel major scale) !Scale and Mode Number (of the parent scale) !Notable Characteristics |- |Lydian Dominant |1, 2, 3, #4, 5, 6, b7 |4th mode of ascending melodic minor |The V chord in this mode is a Vmin(maj7) chord, and the root is a I7 chord |- |Melodic Major |1, 2, 3, 4, 5, b6, b7 |5th mode of ascending melodic minor |This mode allows for I-iv and I-iiø7 progressions without changing keys |- |Altered Dominant/Super Locrian |1, b2, b3, b4, b5, b6, b7 |7th mode of ascending melodic minor |The b4 is en-harmonic to the 3rd of the scale, meaning the tonic is a I7(b5/#5) chord |- |Harmonic Major |1, b2, 3, 4, 5, b6, b7 |5th mode of harmonic minor |I-bII progression becomes possible (as opposed to i-bII in Phrydgian) |} === Approaches to Learning Diatonic Modes === ==== Relative Major Approach ==== A very common approach to learning modes in American music education is to learn modes as they relate to their relative major scale, the scale that shares the same pitches (i.e C Major and D Dorian). In this approach modes are constructed by taking the major scale and selecting different scale degrees as the root of the mode (i.e selecting the 2nd scale degree of the major scale as the root creates the Dorian mode, in C major the 2nd scale degree is D, so D Dorian has the same pitches as C major). This approach can be very useful when used in junction with chord scale theory, as the mode built off of a given scale degree is usually the mode paired with the chord built off of that same scale degree (i.e in C major, the ii chord is D minor, D Dorian (the 2nd mode of C major) is the mode that is commonly paired with that chord). This approach allows students who already know their major scales to learn other modes quickly, since each mode can be constructed using existing major scales (for example, to find B locrian, you find which major scale has B as the 7th scale degree (C Major) and order the pitches of that collection starting on B). It also helps students make connections between chords and modes (I-Ionian, ii-Dorian, V7-Mixolydian) when applying chord-scale theory, especially jazz students first learning to solo over chord changes. ==== Parallel Major Approach ==== Another approach is to learn modes as they relate to their parallel major scale, the scale that has the same root (i.e C Major and C Dorian). In this approach, modes are created by raising/lowering the scale degrees of the major scale, similar to the way that harmonic minor is created by raising the 7th scale degree of natural minor (note that harmonic minor is not a mode of the natural minor scale, it is its own scale entirely because that collection of intervals (W-H-W-W-H-m3-H) does not exist in any diatonic scale). A benefit to this approach is that it's easier to see the differences between Ionian/Aeolian and other modes to recognize their unique characteristics. Students can extrapolate changes to melodies and harmony by raising/lowering scale degrees from major. For example the third of a V chord is the 7th scale degree of a scale, in major the V chord is major (in C major, G B D), but in Mixolydian the v chord is minor (in C mixolydian, G Bb D) because the 7th scale degree is lowered by a half step. This is a useful approach when modes are utilized the same way keys are, where the harmony is built off of the scale and the 1st scale degree is largely tonicized, however it can also be a useful approach when modes are paired with chords, such as in chord-scale theory. All 7 of the Ionian modes can be expressed as alterations to scale degrees of the major scale, as shown below. {| class="wikitable" |+ !Mode !Scale Degree Altered (from parallel major scale) !Notable and changed chords/progressions !Unique Characteristics |- |Lydian |#4 |Vmaj7, Imaj7#11, I-II |There is a tritone from the root of the scale to the 4th scale degree |- |Ionian |none |V7-I |The V chord is dominant, and can resolve by half step to tonic chord tones |- |Mixolydian |b7 |I-bVII, I-Vmin7, I7 |The I chord is dominant, and both the bVII and I chord are major |- |Dorian |b7, b3 |i-IV |The i chord is minor but the IV is major, creating a brighter minor scale sound |- |Aeolian |b7, b3, b6 |i-iv, iiø-v |The 6th scale degree is a half step above the 5th scale degree |- |Phrygian |b7, b3, b6, b2 |bII, i-bII-bVII-i |The 2nd scale degree is a half step above the tonic |- |Locrian |b7, b3, b6, b2, b5 |iø |There is no note perfect 5th above the root of the scale |} Something potentially useful to keep in mind is that the change in pitches between modes (from Lydian to Ionian, from Ionian to Mixolydian, etc.) follows the circle of 5ths. Going through the circle clockwise raises pitches in modes (C lydian contains the same notes as G major, F is raised by a half step to F#) and going through the circle counterclockwise lowers pitches in modes (C Mixolydian contains the same notes as F major, B is lowered by a half step to Bb). If you know the order that pitches are raised/lowered in the circle of 5ths, you can find out which pitches in a given mode change. This also works for larger "leaps" between modes, (for example, C Dorian has 2 notes lowered from C Ioanian, the 2 notes that change when you move twice counterclockwise in the circle of 5ths are Bb and Eb, so C Dorian contains both Eb and Bb). This method may be helpful for students who have already learned their major scales through the circle of 5ths, and are moving on to learning diatonic modes. === Modality === "Modality" refers to the pitch relationships found in music using modes and contrasted with later tonality. The use of more than one mode makes music polymodal, such as with polymodal chromaticism. While all tonal music may technically be described as modal, music that is called modal often has less diatonic functionality and changes key less often than other music.[citation needed] [[Category:Music theory]] srlo98qir94a4ue5ont74rt20n54cxo Template:Para 10 88663 2813044 2007-08-18T02:56:57Z ClaudioMB 3089799 [[WP:AES|←]]Created page with '<!-- Template:Fb p staff (Football - parameter - staff) Parameters p : parameter o : optional ut : use together dut : don't use together d : description ...' 2813044 wikitext text/x-wiki <!-- Template:Fb p staff (Football - parameter - staff) Parameters p : parameter o : optional ut : use together dut : don't use together d : description -->|- |{{{p}}} |{{{{{o}}}}} |{{{ut}}} |{{{dut}}}<!-- --><noinclude>[[Category:Fb_templates]]</noinclude> rh2dhi1djo1k7zt3rc9xgws1d3t3iqe 2813045 2813044 2007-08-18T02:57:23Z ClaudioMB 3089799 moved [[Template:Fb p staff]] to [[Template:Fb p]] 2813044 wikitext text/x-wiki <!-- Template:Fb p staff (Football - 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--><noinclude>[[Category:Fb_templates]]</noinclude> ccn395jni1rn91szr9d4ehjhbu5lqkb 2813048 2813047 2007-08-18T03:09:15Z ClaudioMB 3089799 2813048 wikitext text/x-wiki <!-- Template:Fb p staff (Football - parameter - staff) Parameters p : parameter o : optional ut : use together dut : don't use together d : description -->|- |<code>{{{p}}}</code> |{{{{{o}}}}} |{{{ut}}} |{{{dut}}}<!-- -->{{#if:{{{dut2|}}}|{{bar}}{{bar}}{{{dut2}}}}} |align=left|{{{d}}}<!-- --><noinclude>[[Category:Fb_templates]]</noinclude> k6vgjri3yrgcvy4d87usbr5fianala2 2813049 2813048 2007-08-18T03:13:19Z ClaudioMB 3089799 2813049 wikitext text/x-wiki <!-- Template:Fb p staff (Football - parameter - staff) Parameters p : parameter o : optional ut : use together dut : don't use together d : description -->|- |<code>{{{p}}}</code> |{{{{{o}}}}} |{{{ut}}} |{{{dut}}}<!-- -->{{#ifeq:{{{dut2}}}|{{non existant|dut2}}| |{{bar}}{{bar}}{{{dut2}}}}} |align=left|{{{d}}}<!-- --><noinclude>[[Category:Fb_templates]]</noinclude> 96qmdirpmtlz36yrozzinuonb2bqy7y 2813050 2813049 2007-08-18T03:14:41Z ClaudioMB 3089799 2813050 wikitext text/x-wiki <!-- Template:Fb p staff (Football - parameter - staff) Parameters p : parameter o : optional ut : use together dut : don't use together d : description -->|- |<code>{{{p}}}</code> |{{{{{o}}}}} |{{Fb p color|ut}} |{{Fb p color|dut}}<!-- -->{{#ifeq:{{{dut2}}}|{{non existant|dut2}}| |{{bar}}{{bar}}{{Fb p color|dut2}}}} |align=left|{{{d}}}<!-- --><noinclude>[[Category:Fb_templates]]</noinclude> glews4sg6ke3j5349u2lcrvzk8imk68 2813051 2813050 2007-08-18T03:25:20Z ClaudioMB 3089799 2813051 wikitext text/x-wiki <!-- Template:Fb p staff (Football - parameter - staff) Parameters p : parameter o : optional ut : use together dut : don't use together d : description -->|- |<code>{{{p}}}</code> |{{{{{o}}}}} |{{Fb p color|{{{ut}}}}} |{{Fb p color|{{{dut}}}}}<!-- -->{{#ifeq:{{{dut2}}}|{{non existant|dut2}}| |{{bar}}{{bar}}{{Fb p color|{{{dut2}}}}}}} |align=left|{{{d}}}<!-- --><noinclude>[[Category:Fb_templates]]</noinclude> j366qj2btw9wqjco9b7snz0koa6hm1p 2813052 2813051 2007-08-18T03:34:09Z ClaudioMB 3089799 2813052 wikitext text/x-wiki <!-- Template:Fb p staff (Football - parameter - staff) Parameters p : parameter o : optional ut : use together dut : don't use together d : description v : values -->|- |<code>{{{p}}}</code> |{{{{{o}}}}} |{{Fb p color|{{{ut}}}}} |{{Fb p color|{{{dut}}}}}<!-- -->{{#ifeq:{{{dut2}}}|{{non existant|dut2}}| |{{bar}}{{bar}}{{Fb p color|{{{dut2}}}}}}} |align=left|{{{d}}} |align=left|{{{v}}}<!-- --><noinclude>[[Category:Fb_templates]]</noinclude> 66wndf256g90usu0h02kocm83arcb5j 2813053 2813052 2007-08-19T17:30:39Z ClaudioMB 3089799 2813053 wikitext text/x-wiki <!-- Template:Parameter Parameters p : parameter o : optional ut : use together dut : don't use together d : description v : values -->|- |<code>{{{p}}}</code> |{{{{{o}}}}} |{{Fb p color|{{{ut}}}}} |{{Fb p color|{{{dut}}}}}<!-- -->{{#ifeq:{{{dut2}}}|{{non existant|dut2}}| |{{bar}}{{bar}}{{Fb p color|{{{dut2}}}}}}} |align=left|{{{d}}} |align=left|{{{v}}}<!-- --><noinclude>[[Category:Fb_templates]]</noinclude> dw8eacb1gpz4zt1919mopum19fh4g0u 2813054 2813053 2007-08-19T17:30:51Z ClaudioMB 3089799 moved [[Template:Fb p]] to [[Template:Parameter]] 2813053 wikitext text/x-wiki <!-- Template:Parameter Parameters p : parameter o : optional ut : use together dut : don't use together d : description v : values -->|- |<code>{{{p}}}</code> |{{{{{o}}}}} |{{Fb p color|{{{ut}}}}} |{{Fb p color|{{{dut}}}}}<!-- -->{{#ifeq:{{{dut2}}}|{{non existant|dut2}}| |{{bar}}{{bar}}{{Fb p color|{{{dut2}}}}}}} |align=left|{{{d}}} |align=left|{{{v}}}<!-- --><noinclude>[[Category:Fb_templates]]</noinclude> dw8eacb1gpz4zt1919mopum19fh4g0u 2813055 2813054 2007-08-19T17:31:31Z ClaudioMB 3089799 2813055 wikitext text/x-wiki <!-- Template:Parameter Parameters p : parameter o : optional ut : use together dut : don't use together d : description v : values -->|- |<code>{{{p}}}</code> |{{{{{o}}}}} |{{Fb p color|{{{ut}}}}} |{{Fb p color|{{{dut}}}}}<!-- -->{{#ifeq:{{{dut2}}}|{{non existant|dut2}}| |{{bar}}{{bar}}{{Fb p color|{{{dut2}}}}}}} |align=left|{{{d}}} |align=left|{{{v}}}<!-- --><noinclude>[[Category:Templates used by fb templates]]</noinclude> dd4u1ntwmq8svqah1f5kogjzdul7nei 2813056 2813055 2007-09-09T20:39:06Z ClaudioMB 3089799 2813056 wikitext text/x-wiki <!-- Template:Parameter Parameters p : parameter o : optional ut : use together dut : don't use together d : description v : values -->|- |<code>{{{p}}}</code> |{{{{{o}}}}} |{{Fb p color|{{{ut}}}}} |{{Fb p color|{{{dut}}}}}<!-- -->{{#ifeq:{{{dut2}}}|{{non existing|dut2}}| |{{bar}}{{bar}}{{Fb p color|{{{dut2}}}}}}} |align=left|{{{d}}} |align=left|{{{v}}}<!-- --><noinclude>[[Category:Templates used by fb templates]]</noinclude> jeizs3nyev1hjzcmnpvek7ixk2opr1e 2813057 501749 2008-04-17T03:37:57Z ClaudioMB 3089799 2813057 wikitext text/x-wiki |- |<code>{{{p}}}</code> |{{{{{o}}}}} |{{Fb p color|{{{ut}}}}} |{{Fb p color|{{{dut}}}}}<!-- -->{{#ifeq:{{{dut2}}}|{{non existing|dut2}}| |{{!}}{{!}}{{Fb p color|{{{dut2}}}}}}} |align=left|{{{d}}} |align=left|{{{v}}}<!-- --><noinclude>[[Category:Templates used by fb templates]]</noinclude> ilr9u3zlp6exxi8147xc4p65kxm40xx 2813058 2813057 2008-04-17T03:38:31Z ClaudioMB 3089799 2813058 wikitext text/x-wiki <!-- Template:Parameter Parameters p : parameter o : optional ut : use together dut : don't use together d : description v : values -->|- |<code>{{{p}}}</code> |{{{{{o}}}}} |{{Fb p color|{{{ut}}}}} |{{Fb p color|{{{dut}}}}}<!-- -->{{#ifeq:{{{dut2}}}|{{non existing|dut2}}| |{{!}}{{!}}{{Fb p color|{{{dut2}}}}}}} |align=left|{{{d}}} |align=left|{{{v}}}<!-- --><noinclude>[[Category:Templates used by fb templates]]</noinclude> 58i0txdr2ml30gsag0kr3kidrqx9kaq 2813059 2813058 2008-05-06T03:08:54Z ClaudioMB 3089799 2813059 wikitext text/x-wiki |- |<code>{{{p}}}</code> |{{{{{r}}}}} |{{Fb p color|{{{ut}}}}} |{{Fb p color|{{{dut}}}}}<!-- -->{{#ifeq:{{{dut2}}}|{{non existing|dut2}}| |{{!}}{{!}}{{Fb p color|{{{dut2}}}}}}} |align=left|{{{d}}} |align=left|{{{v}}}<!-- --><noinclude>[[Category:Templates used by fb templates]]</noinclude> q5vx9io6kfymqi82kxuo6bp19moz9p2 2813060 2813059 2008-05-06T03:09:09Z ClaudioMB 3089799 2813060 wikitext text/x-wiki <!-- Template:Parameter Parameters p : parameter r : required ut : use together dut : don't use together d : description v : values -->|- |<code>{{{p}}}</code> |{{{{{r}}}}} |{{Fb p color|{{{ut}}}}} |{{Fb p color|{{{dut}}}}}<!-- -->{{#ifeq:{{{dut2}}}|{{non existing|dut2}}| |{{!}}{{!}}{{Fb p color|{{{dut2}}}}}}} |align=left|{{{d}}} |align=left|{{{v}}}<!-- --><noinclude>[[Category:Templates used by fb templates]]</noinclude> lj8tvon9hmjzs3b66b15hqgzf1jyypv 2813061 1062380 2012-07-29T16:22:57Z EmausBot 998234 r2.7.3) (Robot: Adding [[ru:Шаблон:Parameter]] 2813061 wikitext text/x-wiki <!-- Template:Parameter Parameters p : parameter r : required ut : use together dut : don't use together d : description v : values -->|- |<code>{{{p}}}</code> |{{{{{r}}}}} |{{Fb p color|{{{ut}}}}} |{{Fb p color|{{{dut}}}}}<!-- -->{{#ifeq:{{{dut2}}}|{{non existing|dut2}}| |{{!}}{{!}}{{Fb p color|{{{dut2}}}}}}} |align=left|{{{d}}} |align=left|{{{v}}}<!-- --><noinclude>[[Category:Templates used by fb templates]][[ru:Шаблон:Parameter]] </noinclude> qh123o6q4d6jo6f78t94j4dv6mi9n6y 2813062 2813061 2012-09-04T14:16:56Z Khan27 311712 2813062 wikitext text/x-wiki <!-- Template:Parameter Parameters p : parameter r : required ut : use together dut : don't use together d : description v : values -->|- |<code>{{{p}}}</code> |{{{{{r}}}}} |{{Fb p color|{{{ut}}}}} |{{Fb p color|{{{dut}}}}}<!-- -->{{#ifeq:{{{dut2}}}|{{non existing|dut2}}| |{{!}}{{!}}{{Fb p color|{{{dut2}}}}}}} |align=left|{{{d}}} |align=left|{{{v}}}<!-- --><noinclude>[[Category:Templates used by fb templates]] [[az:Şablon:Parameter]] [[ru:Шаблон:Parameter]] </noinclude> d7725l5ranj7fc80jf1g1rrb6xrcksf 2813063 2813062 2012-12-15T04:30:51Z Sorakara023 326232 add ja 2813063 wikitext text/x-wiki <!-- Template:Parameter Parameters p : parameter r : required ut : use together dut : don't use together d : description v : values -->|- |<code>{{{p}}}</code> |{{{{{r}}}}} |{{Fb p color|{{{ut}}}}} |{{Fb p color|{{{dut}}}}}<!-- -->{{#ifeq:{{{dut2}}}|{{non existing|dut2}}| |{{!}}{{!}}{{Fb p color|{{{dut2}}}}}}} |align=left|{{{d}}} |align=left|{{{v}}}<!-- --><noinclude>[[Category:Templates used by fb templates]] [[az:Şablon:Parameter]] [[ja:Template:Parameter]] [[ru:Шаблон:Parameter]] </noinclude> 8uugl80aufqdupguldkwxwd3bjzsiuz 2813064 1062381 2013-03-17T11:05:12Z Addbot 2879960 [[User:Addbot|Bot:]] Migrating 2 interwiki links, now provided by [[Wikipedia:Wikidata|Wikidata]] on [[d:q6307866]] 2813064 wikitext text/x-wiki <!-- Template:Parameter Parameters p : parameter r : required ut : use together dut : don't use together d : description v : values -->|- |<code>{{{p}}}</code> |{{{{{r}}}}} |{{Fb p color|{{{ut}}}}} |{{Fb p color|{{{dut}}}}}<!-- -->{{#ifeq:{{{dut2}}}|{{non existing|dut2}}| |{{!}}{{!}}{{Fb p color|{{{dut2}}}}}}} |align=left|{{{d}}} |align=left|{{{v}}}<!-- --><noinclude>[[Category:Templates used by fb templates]] [[az:Şablon:Parameter]] </noinclude> pkpb5pvm268cdjq0z7wny30kwg3fveq 2813065 1062382 2013-11-30T21:42:05Z Hyacinth 2086319 ==See also== *{{tl|Para}} 2813065 wikitext text/x-wiki <!-- Template:Parameter Parameters p : parameter r : required ut : use together dut : don't use together d : description v : values -->|- |<code>{{{p}}}</code> |{{{{{r}}}}} |{{Fb p color|{{{ut}}}}} |{{Fb p color|{{{dut}}}}}<!-- -->{{#ifeq:{{{dut2}}}|{{non existing|dut2}}| |{{!}}{{!}}{{Fb p color|{{{dut2}}}}}}} |align=left|{{{d}}} |align=left|{{{v}}}<!-- --><noinclude> ==See also== *{{tl|Para}} [[Category:Templates used by fb templates]] [[az:Şablon:Parameter]] </noinclude> qw84v5gbbxqguo91lp4dogx4ebx7rxu 2813066 2813065 2014-02-20T14:27:31Z 176.250.73.19 2813066 wikitext text/x-wiki <noinclude>{{update}}{{incomplete}}</noinclude> <!-- Template:Parameter Parameters p : parameter r : required ut : use together dut : don't use together d : description v : values -->|- |<code>{{{p}}}</code> |{{{{{r}}}}} |{{Fb p color|{{{ut}}}}} |{{Fb p color|{{{dut}}}}}<!-- -->{{#ifeq:{{{dut2}}}|{{non existing|dut2}}| |{{!}}{{!}}{{Fb p color|{{{dut2}}}}}}} |align=left|{{{d}}} |align=left|{{{v}}}<!-- --><noinclude> {{documentation}} ==See also== *{{tl|Para}} [[Category:Templates used by fb templates]] [[az:Şablon:Parameter]] </noinclude> 1qh6vypycbwrjg3hjz5qqjshew0jrxb 2813067 2813066 2014-02-20T14:27:54Z 176.250.73.19 2813067 wikitext text/x-wiki <noinclude>{{update}}{{incomplete}}</noinclude> <!-- Template:Parameter Parameters p : parameter r : required ut : use together dut : don't use together d : description v : values -->|- |<code>{{{p}}}</code> |{{{{{r}}}}} |{{Fb p color|{{{ut}}}}} |{{Fb p color|{{{dut}}}}}<!-- -->{{#ifeq:{{{dut2}}}|{{non existing|dut2}}| |{{!}}{{!}}{{Fb p color|{{{dut2}}}}}}} |align=left|{{{d}}} |align=left|{{{v}}}<!-- --><noinclude> {{documentation}} ==See also== *{{tl|Para}} [[Category:Templates used by fb templates]] </noinclude> 9ug7tyqvkls5b2ds3ll659qj907x87m 2813068 1775537 2015-01-03T08:27:54Z WOSlinker 351635 tidy style for better HTML5 compatibility 2813068 wikitext text/x-wiki <noinclude>{{update}}{{incomplete}}</noinclude> <!-- Template:Parameter Parameters p : parameter r : required ut : use together dut : don't use together d : description v : values -->|- |<code>{{{p}}}</code> |{{{{{r}}}}} |{{Fb p color|{{{ut}}}}} |{{Fb p color|{{{dut}}}}}<!-- -->{{#ifeq:{{{dut2}}}|{{non existing|dut2}}| |{{!}}{{!}}{{Fb p color|{{{dut2}}}}}}} | style="text-align:left;" |{{{d}}} | style="text-align:left;" |{{{v}}}<!-- --><noinclude> {{documentation}} ==See also== *{{tl|Para}} [[Category:Templates used by fb templates]] </noinclude> gdtzbq5yvhte9t0ved2pb0mh8vr61y0 2813069 2813068 2015-01-03T08:29:16Z WOSlinker 351635 tidy more 2813069 wikitext text/x-wiki <noinclude>{{update}}{{incomplete}}</noinclude> <!-- Template:Parameter Parameters p : parameter r : required ut : use together dut : don't use together d : description v : values -->|- |<code>{{{p}}}</code> |{{{{{r}}}}} |bgcolor={{#switch:{{{ut}}} |1 = lightgreen |2 = lightblue |3 = orange }}|{{Unicode| }}<!-- --> |bgcolor={{#switch:{{{dut}}} |1 = lightgreen |2 = lightblue |3 = orange }}|{{Unicode| }}<!-- --><!-- -->{{#ifeq:{{{dut2}}}|{{non existing|dut2}}| |{{!}}{{!}}{{Fb p color|{{{dut2}}}}}}} | style="text-align:left;" |{{{d}}} | style="text-align:left;" |{{{v}}}<!-- --><noinclude> {{documentation}} ==See also== *{{tl|Para}} [[Category:Templates used by fb templates]] </noinclude> ila81icouft8sogxz2e8q58ogbx2rs2 2813070 2813069 2015-01-03T08:30:50Z WOSlinker 351635 tidy style for better HTML5 compatibility 2813070 wikitext text/x-wiki <noinclude>{{update}}{{incomplete}}</noinclude> <!-- Template:Parameter Parameters p : parameter r : required ut : use together dut : don't use together d : description v : values -->|- |<code>{{{p}}}</code> |{{{{{r}}}}} |style="background-color:{{#switch:{{{ut}}} |1 = lightgreen |2 = lightblue |3 = orange }}"|{{Unicode| }} |style="background-color:{{#switch:{{{dut}}} |1 = lightgreen |2 = lightblue |3 = orange }}|{{Unicode| }}<!-- -->{{#ifeq:{{{dut2}}}|{{non existing|dut2}}| |{{!}}{{!}}style="background-color:{{#switch:{{{dut2}}} |1 = lightgreen |2 = lightblue |3 = orange }}|{{Unicode| }}<!-- -->}} | style="text-align:left;" |{{{d}}} | style="text-align:left;" |{{{v}}}<!-- --><noinclude> {{documentation}} ==See also== *{{tl|Para}} [[Category:Templates used by fb templates]] </noinclude> g68go2on79ig27gj9w868kp0dlsmiuc 2813071 2813070 2015-01-03T08:31:58Z WOSlinker 351635 tidy style for better HTML5 compatibility 2813071 wikitext text/x-wiki <noinclude>{{update}}{{incomplete}}</noinclude> <!-- Template:Parameter Parameters p : parameter r : required ut : use together dut : don't use together d : description v : values -->|- |<code>{{{p}}}</code> |{{{{{r}}}}} |style="background-color:{{#switch:{{{ut}}} |1 = lightgreen |2 = lightblue |3 = orange }}"|{{Unicode| }} |style="background-color:{{#switch:{{{dut}}} |1 = lightgreen |2 = lightblue |3 = orange }}"|{{Unicode| }}<!-- -->{{#ifeq:{{{dut2}}}|{{non existing|dut2}}| |{{!}}{{!}}style="background-color:{{#switch:{{{dut2}}} |1 = lightgreen |2 = lightblue |3 = orange }}"|{{Unicode| }}<!-- -->}} | style="text-align:left;" |{{{d}}} | style="text-align:left;" |{{{v}}}<!-- --><noinclude> {{documentation}} ==See also== *{{tl|Para}} [[Category:Templates used by fb templates]] </noinclude> 562w5tlxifuhd62bg3xhso8faxf31gf 2813072 1775539 2015-07-14T15:59:36Z Funandtrvl 47226 fx 2813072 wikitext text/x-wiki <noinclude>{{update}}{{incomplete}}</noinclude> <!-- Template:Parameter Parameters p : parameter r : required ut : use together dut : don't use together d : description v : values -->|- |<code>{{{p}}}</code> |{{{{{r}}}}} |style="background-color:{{#switch:{{{ut}}} |1 = lightgreen |2 = lightblue |3 = orange }}"|{{Unicode| }} |style="background-color:{{#switch:{{{dut}}} |1 = lightgreen |2 = lightblue |3 = orange }}"|{{Unicode| }}<!-- -->{{#ifeq:{{{dut2}}}|{{non existing|dut2}}| |{{!}}{{!}}style="background-color:{{#switch:{{{dut2}}} |1 = lightgreen |2 = lightblue |3 = orange }}"|{{Unicode| }}<!-- -->}} | style="text-align:left;" |{{{d}}} | style="text-align:left;" |{{{v}}}<!-- --><noinclude> {{documentation|content= ==See also== *{{tl|Para}} [[Category:Templates used by fb templates]] }}</noinclude> 5tchk99hkg7o6b79wn524oeulv4ruoz 2813073 1775540 2016-04-19T16:56:45Z AnomieBOT 656185 [[User:AnomieBOT/docs/TemplateSubster|Substing templates]]: {{Unicode}}. See [[User:AnomieBOT/docs/TemplateSubster]] for info. 2813073 wikitext text/x-wiki <noinclude>{{update}}{{incomplete}}</noinclude> <!-- Template:Parameter Parameters p : parameter r : required ut : use together dut : don't use together d : description v : values -->|- |<code>{{{p}}}</code> |{{{{{r}}}}} |style="background-color:{{#switch:{{{ut}}} |1 = lightgreen |2 = lightblue |3 = orange }}"| |style="background-color:{{#switch:{{{dut}}} |1 = lightgreen |2 = lightblue |3 = orange }}"|<!-- -->{{#ifeq:{{{dut2}}}|{{non existing|dut2}}| |{{!}}{{!}}style="background-color:{{#switch:{{{dut2}}} |1 = lightgreen |2 = lightblue |3 = orange }}"|<!-- -->}} | style="text-align:left;" |{{{d}}} | style="text-align:left;" |{{{v}}}<!-- --><noinclude> {{documentation|content= ==See also== *{{tl|Para}} [[Category:Templates used by fb templates]] }}</noinclude> 0fzajq1butdkhjnq6mejfpq5k2u91e1 2813074 2813073 2016-10-15T10:47:22Z 95.49.124.30 Added content 2813074 wikitext text/x-wiki <noinclude><!-- Please do not remove or change this AfD message until the issue is settled --> {{Article for deletion/dated|page=Parameter|timestamp=20161015104708|year=2016|month=October|day=15|substed=yes}} <!-- Once discussion is closed, please place on talk page: {{Old AfD multi|page=Parameter|date=15 October 2016|result='''keep'''}} --> <!-- End of AfD message, feel free to edit beyond this point -->{{update}}{{incomplete}}</noinclude> <!-- Template:Parameter Parameters p : parameter r : required ut : use together dut : don't use together d : description v : values -->|- |<code>{{{p}}}</code> |{{{{{r}}}}} |style="background-color:{{#switch:{{{ut}}} |1 = lightgreen |2 = lightblue |3 = orange }}"| |style="background-color:{{#switch:{{{dut}}} |1 = lightgreen |2 = lightblue |3 = orange }}"|<!-- -->{{#ifeq:{{{dut2}}}|{{non existing|dut2}}| |{{!}}{{!}}style="background-color:{{#switch:{{{dut2}}} |1 = lightgreen |2 = lightblue |3 = orange }}"|<!-- -->}} | style="text-align:left;" |{{{d}}} | style="text-align:left;" |{{{v}}}<!-- --><noinclude> {{documentation|content= ==See also== *{{tl|Para}} [[Category:Templates used by fb templates]] }}</noinclude> bbks5gl4kisjdwc40wcfz0kt4mbaz66 2813075 2813074 2016-10-15T11:33:14Z Favonian 279753 Reverted edits by [[Special:Contribs/95.49.124.30|95.49.124.30]] ([[User talk:95.49.124.30|talk]]) to last version by AnomieBOT 2813075 wikitext text/x-wiki <noinclude>{{update}}{{incomplete}}</noinclude> <!-- Template:Parameter Parameters p : parameter r : required ut : use together dut : don't use together d : description v : values -->|- |<code>{{{p}}}</code> |{{{{{r}}}}} |style="background-color:{{#switch:{{{ut}}} |1 = lightgreen |2 = lightblue |3 = orange }}"| |style="background-color:{{#switch:{{{dut}}} |1 = lightgreen |2 = lightblue |3 = orange }}"|<!-- -->{{#ifeq:{{{dut2}}}|{{non existing|dut2}}| |{{!}}{{!}}style="background-color:{{#switch:{{{dut2}}} |1 = lightgreen |2 = lightblue |3 = orange }}"|<!-- -->}} | style="text-align:left;" |{{{d}}} | style="text-align:left;" |{{{v}}}<!-- --><noinclude> {{documentation|content= ==See also== *{{tl|Para}} [[Category:Templates used by fb templates]] }}</noinclude> 0fzajq1butdkhjnq6mejfpq5k2u91e1 2813076 1870330 2018-10-11T15:40:56Z AnomieBOT 656185 [[User:AnomieBOT/docs/TemplateSubster|Substing templates]]: {{Incomplete}}. See [[User:AnomieBOT/docs/TemplateSubster]] for info. 2813076 wikitext text/x-wiki <noinclude>{{update}}</noinclude> <!-- Template:Parameter Parameters p : parameter r : required ut : use together dut : don't use together d : description v : values -->|- |<code>{{{p}}}</code> |{{{{{r}}}}} |style="background-color:{{#switch:{{{ut}}} |1 = lightgreen |2 = lightblue |3 = orange }}"| |style="background-color:{{#switch:{{{dut}}} |1 = lightgreen |2 = lightblue |3 = orange }}"|<!-- -->{{#ifeq:{{{dut2}}}|{{non existing|dut2}}| |{{!}}{{!}}style="background-color:{{#switch:{{{dut2}}} |1 = lightgreen |2 = lightblue |3 = orange }}"|<!-- -->}} | style="text-align:left;" |{{{d}}} | style="text-align:left;" |{{{v}}}<!-- --><noinclude> {{documentation|content= ==See also== *{{tl|Para}} [[Category:Templates used by fb templates]] }}</noinclude> 2jsp0djkydj7z1y769vzk7uwrrhvboi 2813077 2156801 2020-05-08T04:44:13Z Rehman 255263 Nominated for deletion; see [[:Wikipedia:Templates for discussion/Log/2020 May 8#Template:Parameter]]. ([[WP:TW|TW]]) 2813077 wikitext text/x-wiki {{Template for discussion/dated|action=|page=Parameter|link=Wikipedia:Templates for discussion/Log/2020 May 8#Template:Parameter|help=off|bigbox={{#invoke:Noinclude|noinclude|text=yes}}}} <noinclude>{{update}}</noinclude> <!-- Template:Parameter Parameters p : parameter r : required ut : use together dut : don't use together d : description v : values -->|- |<code>{{{p}}}</code> |{{{{{r}}}}} |style="background-color:{{#switch:{{{ut}}} |1 = lightgreen |2 = lightblue |3 = orange }}"| |style="background-color:{{#switch:{{{dut}}} |1 = lightgreen |2 = lightblue |3 = orange }}"|<!-- -->{{#ifeq:{{{dut2}}}|{{non existing|dut2}}| |{{!}}{{!}}style="background-color:{{#switch:{{{dut2}}} |1 = lightgreen |2 = lightblue |3 = orange }}"|<!-- -->}} | style="text-align:left;" |{{{d}}} | style="text-align:left;" |{{{v}}}<!-- --><noinclude> {{documentation|content= ==See also== *{{tl|Para}} [[Category:Templates used by fb templates]] }}</noinclude> 1om4s06s0d815yf7pniejvz1hjhlohe 2813078 2813077 2020-05-15T20:24:47Z TheTVExpert 2902892 [[:Wikipedia:Templates for discussion/Log/2020 May 8#Template:Parameter]] closed as redirect ([[WP:XFDC|XFDcloser]]) 2813078 wikitext text/x-wiki #REDIRECT [[Template:Para]] etxz8ysfpb4d8rw7okp3bttiggfws72 2813079 2415162 2026-04-20T13:24:38Z Paine Ellsworth 184305 Paine Ellsworth moved page [[Template:Parameter]] to [[Draft:Move/Parameter]] without leaving a redirect: per move request on talk page - see [[WP:PM/C#4]] - round robin swap 2813079 wikitext text/x-wiki #REDIRECT [[Template:Para]] etxz8ysfpb4d8rw7okp3bttiggfws72 2813080 2813079 2026-04-20T13:28:20Z Paine Ellsworth 184305 Paine Ellsworth moved page [[Draft:Move/Parameter]] to [[Template:Para]] without leaving a redirect: per move request on talk page - see [[WP:PM/C#4]] - round robin swap 2813079 wikitext text/x-wiki #REDIRECT [[Template:Para]] etxz8ysfpb4d8rw7okp3bttiggfws72 2813081 2813080 2026-04-20T13:30:33Z Anomie 27260 Protected "[[Template:Para]]": Restore protection for this title after a move; the redirect is very widely used, and likely will continue to be used much more than the longer name it was moved to. ([Edit=Require template editor access] (indefinite) [Move=Require template editor access] (indefinite)) 2813079 wikitext text/x-wiki #REDIRECT [[Template:Para]] etxz8ysfpb4d8rw7okp3bttiggfws72 2813082 2813081 2026-04-20T13:31:32Z Paine Ellsworth 184305 follow up to page move - retarget and add [[WP:RCAT|rcat template]]s 2813082 wikitext text/x-wiki #REDIRECT [[Template:Parameter]] {{Rcat shell| {{R from move}} {{R from template shortcut}} }} 7s8mokgsoussxh3gkyodvsagezu2kca 2813083 2813082 2026-06-06T08:30:12Z Jtneill 10242 39 revisions imported from [[:w:Template:Para]] 2813082 wikitext text/x-wiki #REDIRECT [[Template:Parameter]] {{Rcat shell| {{R from move}} {{R from template shortcut}} }} 7s8mokgsoussxh3gkyodvsagezu2kca Learn how to "jam" 0 92141 2812998 2812722 2026-06-06T00:20:30Z Kirby - Electrotechnics 3074947 Removed from the main "Category:Music" 2812998 wikitext text/x-wiki <!--Note to editors: I have wondered whether it would be better to convert this specifically for guitar, and make another specifically for piano since it lacks interesting specifics (i.e.: Perhaps knowing how to transpose notes from string to string on a guitar, etc...). My references are basically my life experience with music (25 year old, guitar/piano teacher for 2 years, 15 year experience) and learning through wikipedia. --> {{TOCright}} ''[for beginners]'' ---- '''to jam:''' to play improvised or unrehearsed music with others ==Intro & the basics== This post is most applicable for '''''guitar, bass, and piano'''''. This also applies to the music of western countries which has a total of 12 unique notes [[w:Equal_temperament|(12-TET)]]. To be differentiated from e.g.: Arabian music (24-TET), or others (19-TET, 31-TET). Assuming that you're a "beginner", start by researching the proper technique to handle your instrument (e.g.: guitar + proper technique) before continuing. You want to play efficiently, not sloppy. Try to understand the logical reasons behind the technique and don't practice without it (it will eventually become natural to you after months of practice). ==Technical terms & study== Research the following terms and information. Also take notes of any synonyms you may encounter. * [[w:Semitone|half-step]] —— the distance(interval) from one note to the next possible note higher or lower in pitch * [[w:Whole_step|whole step]] —— two half-steps * [[w:Sharp_%28music%29|sharp ( # )]] —— when you see this next to a note it means you play that note a semitone (half-step) higher from where it's usually played * [[w:Flat_%28music%29|flat ( ♭ )]] —— the opposite of #; a semitone lower * [[w:Musical_scale|scale]] —— a series of notes used together to produce a specific style of music. The use of scales easily allows you to create melodies, chords, and solos since it is a fabricated model of all the notes that work together. Almost every song uses a scale (mainly the major scale and the minor scale). * [[w:Chromatic_scale|chromatic scale]] —— Is all the types of notes in ''our'' music C C# D D# E F F# G G# A A# B C (keep in mind that C# is the same as D♭, and G# is the same as A♭, etc...) * [[w:Interval_%28music%29|interval]] —— the difference in semitones between two notes * '''scale pattern''' —— a pattern of intervals used to remember scales and their type (e.g.: 2,2,1,2,2,2,1 major scale pattern) * '''scale patterns:''' ** 2,2,1,2,2,2,1: (Natural) major scale (frequently used as a reference for chords) ** 2,1,2,2,1,2,2: (Natural) minor scale '''You may proceed to the next section. Explore these other scales later.''' ** 2,1,2,2,1,3,1: Harmonic minor scale ** 2,1,2,2,2,2,1 ascending and (natural) minor scale descending: Melodic minor scale ** 2,1,2,2,2,1,2: Dorian scale ** 1,2,2,2,1,2,2: Phrygian scale ** 2,2,2,1,2,2,1: Lydian scale ** 2,2,1,2,2,1,2: Mixolydian scale ** 2,1,2,2,1,2,2: Aeolian scale (same as (natural minor scale) ** 1,2,2,1,2,2,2: Locrian scale ** 3,2,1,1,3,2: Minor blues scale ** 3,1,3,1,3,1: Augmented scale ** 2,2,2,2,2,2: Whole tone scale ** 2,2,3,2,3: Major pentatonic scale ** 2,3,2,2,3: Yo scale ** 3,2,2,3,2: Minyō scale ** 1,4,2,1,4: In scale ** 2,1,2,1,2,1,2,1: (Whole-half) diminished scale ** 1,2,1,2,1,2,1,2: Half-whole diminished scale ** 2,1,2,1,2,2,2: Half-diminished scale ** 1,3,1,2,1,3,1: Arabic scale ** 1,3,1,2,1,2,2: Phrygian dominant/Spanish Gypsy/Jewish scale ** 2,1,3,1,1,3,1: Hungarian gypsy scale ** 2,2,1,2,2,1,1,1: Bebop dominant scale ** 2,1,1,1,2,2,1,2: Bebop dorian scale ** 2,2,1,2,1,1,2,1: Bebop major scale ** 2,1,2,2,1,1,2,1: Bebop melodic minor scale ===Finding the notes on your instrument=== Before you continue, do a research online or anywhere to find the name of all or at least one single note on your instrument (e.g.: C). Let's look at all the types of notes again: C C# D D# E F F# G G# A A# B C. Notice there isn't a note between B and C, and, E and F. You can use that fact in order to unveil all the notes of your instrument. Let's say you start from a familiar note (e.g.: C ). Then all you need to do is play the next possible higher pitch (one semitone higher) to find C# and so on until you have your entire instrument mapped out. Keep in mind there's more than just 1 of the same kind of note ; look (i.e.: '''<big>C</big>''', C#, D, D#, E, F, F#, G, G#, A, A#, B, '''<big>C</big>''', C#, D, D#, E, F, F#, G, G#, A, A#, B, '''<big>C</big>''', C#, D, etc...) once you reach the end (G#), it loops back around to A. In this example there are 3 C notes yet they are all at different pitches. ==Using this knowledge== ===Jamming=== Jamming is all about experimentation, although it helps to have a basic understanding of musical concepts. You'll need to practice your hearing in identifying different types of scales. Just familiarize yourself with the minor and major scales to begin with. Basically the minor scale will sound more dramatic than the "happy" major scale. In order to jam with someone, you need to identify what type of scale (minor, major, etc...) the song is in. You do this by trial and error while keeping in mind the patterns for those scales; 2,2,1,2,2,2,1 (major), and 2,1,2,2,1,2,2 (minor). So start on any note, sound it out, and determine whether it sounds right or not. Once you have one note, keep guessing from thereon by trying to move up a half-step or a whole step, it's one or the other. Say it was a half-step '1', then you know it continues either as '2,2' or '2,2,2' if we look at the patterns. You'll be able to identify the scale once you determine that it's either 2,2,1 or 2,2,2. Memorize the notes you found and play them in a random order that suits you, that's called "soloing" or "creating a melody". If you make a mistake you can easily quickly correct that mistake by moving up or down a half-step or by ascending or descending chromatically from that point. ===Conclusion=== So there you have it. Once you know scales and how to identify the scale of a song, you have the ability to "solo", to "jam" with it by playing random notes (whatever sounds good to you) of that scale at once (harmonically/chords), or one after the other (melodically/melody). You improve your jamming skills by improving your ability to guess the distance (interval(s)) of the next note or series of notes as how you sing it in your mind. Practice yourself to imagine the next note or melody that would sound good and guessing where it is or how far it is from your current note. Once you're satisfied with your guessing skills with melody, move on to [[w:Chord_%28music%29|chords]]. == See also == *[[Blues basics]] (for guitar) [[Category:Howtos]] [[Category:Music performance]] [[Category:Basics]] 2j36fc62jjuka7ioefqy3bd3t9b81s7 Martin Lohse 0 96427 2812970 1823042 2026-06-05T22:01:09Z Kirby - Electrotechnics 3074947 changed to 'category:Musicians' instead of category:Music 2812970 wikitext text/x-wiki {{MLTOC}} {{Wikipedia|Martin Lohse}} {{Template:25%done}} <!--- The following line adds a break so that the green LDTOC box and the grey table of contents do not touch. ---> <br> ==Synopsis== This resource covers the [[w:music styles|music style]], [[w:composition techniques|composition techniques]] and methods of '''Martin Lohse''', a [[w:Denmark|Danish]] 21st century [[w:classical composers|classical composer]] and visual artist born May 29, 1971 in Copenhagen, [[w:Denmark|Denmark]] <ref>[http://www.naxos.com/composerinfo/Martin_Lohse/25411.htm Naxos.com. Biography; Lohse, Martin]</ref> ==[[Martin Lohse/Short biography|Short biography]]== {{Template:50%done}} # {{50Percent}} Martin Lohse ==[[Martin Lohse/Music style|Music style]]== {{Template:100%done}} # {{100Percent}} General description ## {{100Percent}} Mobile # {{100Percent}} Neoclassic ## {{100Percent}} Modern ## {{100Percent}} Romantic ## {{100Percent}} Baroque ## {{100Percent}} Renaissance/mediaeval # {{100Percent}} Transcendental/minimalism # {{100Percent}} Electronic ## {{100Percent}} Chambermusic with electronic ## {{75Percent}} Pure electronic ==[[Martin Lohse/General techniques|General techniques]]== {{Template:25%done}} # {{100Percent}} Mobile ## {{100Percent}} Description # {{0Percent}} Poly-music ## {{0Percent}} Description ## {{0Percent}} Poly-layers of music ## {{0Percent}} Rhythm across harmony # {{0Percent}} Infiniti rows ## {{0Percent}} Description ## {{0Percent}} Golden sections ## {{0Percent}} Primes # {{0Percent}} 5-tone lydian modus # {{0Percent}} Canon # {{0Percent}} Delay ==[[Martin Lohse/Harmony|Harmony]]== {{Template:25%done}} # {{50Percent}} All-interval chords ## {{100Percent}} Description ## {{50Percent}} Chord successions ### {{75Percent}} Two plus two voices ### {{0Percent}} Sequences ## {{0Percent}} Minor third + tritonus # {{0Percent}} Major third ## {{0Percent}} Choral ## {{0Percent}} Patterns # {{0Percent}} Sequences ## {{0Percent}} All-interval chords ## {{0Percent}} Major third # {{75Percent}} Shivering chords # {{0Percent}} Golden section ## {{0Percent}} Interval-music ==[[Martin Lohse/Rhythm|Rhythm]]== {{Template:0%done}} # {{0Percent}} Tempo ## {{0Percent}} Accelerando/decelerando ## {{0Percent}} Polytempoes # {{0Percent}} Golden section ## {{0Percent}} Description ## {{0Percent}} Rhythm # {{0Percent}} Free notation ==[[Martin Lohse/Melody|Melody]]== {{Template:0%done}} # {{0Percent}} Major third # {{0Percent}} All-interval chords # {{0Percent}} Golden secton ## {{0Percent}} Interval-music ==[[Martin Lohse/Chronology of Techniques|Chronology of Techniques]]== {{Template:0%done}} # {{0Percent}} 1996-1999 # {{0Percent}} 1999-2004 # {{0Percent}} 2004-2009 # {{0Percent}} 2009- ==[[Martin Lohse/Analyses of works|Analyses of works]]== {{Template:0%done}} # {{0Percent}} Smoke ## {{0Percent}} Short analysis ## {{0Percent}} Full analysis # {{0Percent}} Entity ## {{0Percent}} Short analysis ## {{0Percent}} Full analysis ==[[Martin Lohse/Classifications of works|Classifications of works]]== {{Template:0%done}} # {{0Percent}} Chronology of works # {{0Percent}} Works classified by techniques ==References== <references /> [[Category:Music composers]] [[Category:Pages moved from Wikibooks]] [[Category:Musicians]] trbw96ulml9pzh7tu3y4ct7rokfj1r2 Ten stupid things beginners do to mess up their contradance experience 0 97046 2813007 1537860 2026-06-06T00:30:32Z Kirby - Electrotechnics 3074947 Removed from the main "Category:Music" 2813007 wikitext text/x-wiki {{TOCright}} ''by Blain Nelson and other contributors'' === Preface === The purpose of this document is to help beginning contradancers have a positive experience so that, hopefully, more of them will choose to stick with the dance form and build the [[Wikipedia:contradance|contradance]] community. It seeks to do so by pointing out some of the common patterns beginners fall into that lead to less-than-satisfactory experiences and the decision to leave the dance form too quickly. While the title labels these as ''Stupid Things'', that's as much to create a snappy title as it is a fair description. They don't require much stupidity in practice, and many intelligent people have done them. The reasons they are chosen are quite understandable, but they remain counterproductive and rather easily avoidable if given the information this document intends to present. Some of them are listed to create a list of ten things, rather than because they're stupid, and those sections still have information useful to the beginner. Oh, and there are more than 10 things listed. This really doesn't break anything. You'll see. The ''Stupid Things'' are presented in the approximate order they will occur in the dance experience, with the first two taking place before the dance starts, and the rest happening during the dance. === Table of contents === #[[/Dressing Impractically/]] #[[/Skipping the Beginners Workshop/]] #[[/Waiting to Be Asked/]] #[[/Sticking to Beginner Friends/]] #[[/Sitting out Dances/]] #[[/Getting Too Fancy Too Quickly/]] #[[/Tolerating Bad Behavior from Other Dancers/]] #[[/Not Asking Questions/]] #[[/Not Counting/]] #[[/Leaving Too Early/]] #[[/Expecting Perfection/]] == Printable versions == [[:File:10 stupid pamphlet.pdf|A pamphlet version of this advice]] [[Category:Dance]] kyuzibi9t1lich0srj078u8n95gbx7n Psycholinguistics/Language and Music 0 105123 2812975 1994329 2026-06-05T22:10:35Z Kirby - Electrotechnics 3074947 changed to 'category:Musicology' instead of category:Music 2812975 wikitext text/x-wiki [[File:Ties slurs and ligature.png|right]] === Introduction === '''''"Music is the universal language of mankind"'''''. ~Henry Wadsworth Longfellow. <ref name="Quote Garden (2006).">Quote Garden (2006). Quotations About Music and Language.[http://www.quotegarden.com/music.html]</ref> At first glance, there are some clear connections between language and music. When we begin to dig even deeper though, adopting the psycholinguistic perspective, we find the two are even more closely-related than we may have initially thought! From the basic components, uses and generativity, to the brain systems and research being conducted, there is a long-standing tie between the psychology of linguistics and music. This page will serve as an overview into the fascinating world of both language and music, and begin to examine why the two are often linked together, as well as where their differences become evident. Also, past and current research into the possible benefits of using music to help with language and memory will be discussed. We will take a look at some of the problems found with neuropsychological processing, and finally present you with an opportunity to perform your own research! Regardless of whether we consider ourselves aspiring musicians, or simply love to listen, music is a huge part of everyone's lives, and is quickly becoming one of the most fascinating aspects of modern psycholinguistics. <br> <br> === Breaking It Down: Comparing the Basics of Language and Music === Similarities and differences between music and language can be seen even when each is broken down into their simplest components. From the basic sounds and meanings, to the overall sentence, story or song, music and language are closely related. {| class="wikitable" |- ! <big>Comparison Level</big> !! <big>Language</big> !! <big>Music</big> |- | <big>Basic Sound Units</big> || Language, in its most basic form, can be broken down into ''phonemes''.<ref name="Newman, A. (2011).">Newman, A. (2011). Psycholinguistics 3190 Lecture Slides, Dalhousie University.</ref>|| Music is made up of different ''notes''. <ref name="Newman, A. (2011)." /> |- | <big>Vocabulary</big> || Languages often use ''letters'' or ''symbols'' (English uses a 26-letter alphabet) to form words. || Music has twelve different notes in an ''octave'', which then repeats in a higher or lower pitch <ref name="Newman, A. (2010).">Newman, A. (2010). Psycholinguistics 3190 Lecture Slides, Dalhousie University.</ref> |- | <big>Logic</big> || Both language and music use a succession of sounds that can be seen as either "right" or "wrong". There are certain words or sentences in languages that make sense, and others that don't. || The same can be true for music <ref name="Adorno, (1956)">Adorno, T. W. (1956). Music and Language: A Fragment. ''Quasi una Fantasia, Essays on Modern Music.'' Accessed online: February 2, 2011. [https://www.msu.edu/~sullivan/AdornoMusLangFrag.html]</ref>; some note sequences sound good together, while others do not (though this can also vary across cultures)<ref name="Radice, (2011)">Radice, M. (2011). Anthropology 1100 Lecture Slides, Dalhousie University.</ref> |- | <big>Production</big> || There is a range of production abilities in language. We all fluently speak a language (whether it be verbal, sign language, Braille, etc.), we are able to communicate using some form of a language. Sometimes our communication abilities may be more limited, or sometimes we may be very adept in communicating. || With music (as with language) there is a continuum of abilities. Not everyone is so fluent in music. While we may all be able to enjoy listening to music, not everyone can play, sing or write music the way we can with languages, though some people are masters at music production. <ref name="Newman, A. (2010)." /> |- | <big>Interpretation</big> || Interpretation is an essential part of both language and music, though differs slightly in its definition for each. Interpreting a language means to understand it, so that a spoken word or sentence means the same thing to many people. || In music however, interpretation doesn't have to mean understanding, but can mean anything as simple as production or performance of music. Everyone may not have the same interpretation of a piece of music, but would still be playing the same notes when performing it <ref name="Adorno, (1956)" />. |- | <big>Function</big> || Language is used as a means of communication and is essential for creating social bonds. || Music can be used for communication as well, but is primarily a source of entertainment, or a means of personal expression <ref name="Newman, A. (2011)." /> |- | <big>Building Blocks</big> || Language is composed of phonemes, making morphemes, and syntax, which creates sentences that are built into stories and language itself. || Music begins with notes which can make chords or sequences that are combined into phrases of music that make a complete song, including affect (which gives feeling and emotion or meaning to the piece) <ref name="Newman, A. (2011)." /> |} <br> === Building on the Basics: Generativity === The term ''generativity'' is often used when talking about aspects of language composition. Generativity (similar to the concept of ''creativity'') is defined as an ability to generate, or power to produce something <ref name="Dictionary.com">Dictionary.com (2011). Generativity: define generativity. ''Ask.com''. Accessed online: February 7, 2011. [http://dictionary.reference.com/browse/generativity]</ref>. Using knowledge about the basic units and components that make up language and speech, or music, it is possible to create a vast amount and variety of combinations from these basic building blocks. ==== Language ==== [[File:Basic english syntax tree.svg|thumb|right|A basic English syntax tree showing the generative abilities of grammar]]Language is made up of phonemes (sounds) and morphemes (smallest units of meaning), which create words that combine using syntax and semantics, giving words order, meaning and grammar <ref name="Nevid, J.S. (2009).">Nevid, J.S. (2009). ''Psychology: Concepts and Applications''. Cengage Learning, Inc. Boston, MA. 592 pages</ref>. By using these words, the generativity of language is infinite <ref name="Newman, A. (2010)." />. This means that it is possible to create an endless number of different sentences by simply creating different combinations of these phonemes and morphemes. Some of these sentences are more common, and make sense to us, while others seem absurd, but are still "sentences" as far as language is concerned. This is where knowledge of grammar comes into play. Grammatical rules can be applied across situations, allowing us to identify sentences as correct or incorrect <ref name="Newman, A. (2010)." />. Using these rules, we are also able to interpret the meanings of sentences that we have never heard before, which is a necessary skill when learning to converse with others. One of the most common examples of generativity in language was created by linguist Noam Chomsky. He put forward the sentence "colourless green ideas sleep furiously" as an example of a grammatically ''correct'' sentence, but one that has no understandable meaning <ref name="Wikipedia, 2011">Wikipedia. (2011). "Colourless Green Ideas Sleep Furiously". Accessed online: February 2, 2011. [[Wikipedia:Colorless green ideas sleep furiously|Colorless green ideas sleep furiously]]</ref>. This is a perfect example of a sentence that we have likely never heard before, and know to be semantically incorrect, but are also able to identify as a legitimate sentence, having learned the rules of grammar and language. It is possible to create an infinite number of sentences using our knowledge of how language works. ==== Music ==== Music is similar to language in its hierarchical structure. Notes can be combined into chords or sequences, which can create a melody or harmony line of a song. Songs then have affect, which adds emotion and meaning to the music being played. Again, using these notes and musical "rules", the generativity of music is infinite <ref name="Newman, A. (2010)." />. Endless numbers of melodies can be created using various combinations of these musical notes, and it is possible for everyone to understand and perceive them as music. Like language, some combinations are understood to be correct, while others sound incorrect. Often, this perception of "right" or "wrong" combinations of notes is based on culture and musical upbringing and influence <ref name="Radice, (2011)" />. It has been shown that music and musicality are products of our cultural and social interactions. Though the ability to appreciate and understand music is universal, music IS and DOES different things in different cultures, and the boundaries of "music" may have a very minimal overlap between two different groups <ref name="Cross, I. (2001).">Cross, I. (2001). Music, Cognition, Culture, and Evolution. ''Annals of the New York Academy of Sciences''. 930 (28-42). DOI: 10.1111/j.1749-6632.2001.tb05723.x</ref>. That said, each culture is able to create a diverse collection of melodies, and everyone is able to perceive these as musical or "un-musical" depending on their cultural understandings. Overall, language seems more universal, whereas music is more culturally-based. Both music and language show evidence of a hierarchical syntactic structure, whether it be the structure of the phrasing of sentences, or the emphasis of the melody lines. As well, both language and music have vast generative powers, as we could create an infinite number of sentences in language, or melodies in music. We will always be coming across new sentences or sound sequences that we have never heard before. <br> === Brain Systems: Beyond the Surface === ==== Language ==== [[File:Brain Surface Gyri.SVG|thumb|left|Graphical depiction of the brain, highlighting Broca's Area (blue), Wernicke's Area (green) and the Primary Auditory Cortex (pink)]]The ability to speak human languages, and process and understand complex sentences has long been considered the distinguishing feature that sets humans above all other species intellectually <ref name="Patel, A.D. (2007).">Patel, A.D. (2007). "Language Music and the Brain". Oxford University Press. New York, NY. 528 pages</ref>. What evolutionary advances have allowed for this process in the human brain? <br> ♦ Several specific areas in the brain have been identified as being crucial for language processing. '''Broca's Area''' (first identified by Paul Broca in 1861) is located in the left inferior frontal cortex, and was the first "language center" of the brain to be singled out <ref name="Debuc, B. (2002).">Debuc, B. (2002). "The Brain From Top to Bottom". Canadian Institutes of Health Research: Institute of Neurosciences, Mental Health and Addiction. Accessed online: February 7, 2011. [http://thebrain.mcgill.ca/flash/d/d_10/d_10_cr/d_10_cr_lan/d_10_cr_lan.html]</ref>. Broca identified that damage to this area of the brain affected an individual's ability to express verbal or written language, despite any deficits to the person's ''understanding'' of language, or any other physical deficits that could have been responsible for the impairment. <br> ♦ Shortly after, a second language center was identified. Carl Wernicke discovered an area in the posterior left temporal lobe, which he named '''Wernicke's Area'''. He realized that damage to this area caused incoherent or nonsense speech production from individuals, implying that they had deficits in understanding language as opposed to actual language production <ref name="Debuc, B. (2002)." />. <br> ♦ Viewed ''most'' simplistically, the left hemisphere of the brain (in respect to language) can be viewed as a neural loop, with Broca's Area at the frontal end controlling language output, and Wernicke's Area at the posterior end controlling the processing of language input <ref name="Debuc, B. (2002)." />. A large bundle of nerves complete the "loop", and join these two areas. This bundle of nerve fibres is known as the '''arcuate fasciculus''' <ref name="Newman, A. (2011)." />. This configuration is not exclusive to people who rely on verbal communication, as the same set-up and functional distribution is seen in individuals that use sign language <ref name="Debuc, B. (2002)." />. <br> ♦ One other area of the brain that is crucial for language was discovered much more recently. Introduced by Norman Geschwind, the '''inferior parietal lobule''' (also known as Geschwind’s territory) is connected to both Broca's and Wernicke's areas by nerve bundles, and is a key addition to understanding the neuroscience of language in the brain <ref name="Debuc, B. (2002)." />. So, not only can information travel between these areas by way of the arcuate fasciculus, but could also be passed through this inferior parietal lobule. It is necessary to note the important location of this lobule, lying at the connection between the auditory, visual and somatosensory cortexes, all of which the lobule itself is also connected to <ref name="Debuc, B. (2002)." />. Hence, the neurons in this region are capable of processing auditory, visual and functional aspects of words; a necessity for understanding language. Though very simple versions of the inferior parietal lobule exist in other species, it seems to have been one of the last evolutionary steps in the human brain, assisting in a possible explanation for why our language abilities are superior to our relative species <ref name="Debuc, B. (2002)." />. <br>Though it seems that the majority of language processing occurs in the left hemisphere for most people, the right hemisphere plays an important role as well. Without the right hemisphere, it is almost impossible to distinguish between the literal (''what'' words are being used) and figurative (''how'' the words are being used) meaning of language. People who have damage to the right hemisphere of their brain are not able to distinguish between denoting and connoting; they can not understand parts of language such as emotional connotation or sarcasm <ref name="Debuc, B. (2002)." />. ==== Music ==== There are many different ways of thinking about music, and it can be argued that different methods of music processing require different areas of the brain. <br> ♦ If you were to simply ''hear'' music, you would be relying on the '''auditory cortex''' of the brain, in which different cells correspond and respond to different frequencies of sound. The core auditory cortex regions are responsible for analysis of pitch and volume, whereas outer surrounding regions analyze timbre, rhythm and melody <ref name="Fick, S. & Shilts E. (2006).">Fick, S. & Shilts E. (2006). This is Your Brain on Music. ''Canadian Geographic Magazine''. Accessed online: February 3, 2011. [http://www.canadiangeographic.ca/magazine/jf06/alacarte.asp]</ref> <br> ♦ If you were to imagine, or think about a song in your head, the areas of the brain used would differ slightly. Again, the auditory cortex would be used, but only in a few isolated areas and to a lesser degree than when you listen to music. As well, imaging a song in your head requires you to first recall the lyrics and melody from long-term memory stores, and then hold the song in your working short-term memory. This means also using the '''inferior frontal gyrus''' (memory recall), and '''dorsolateral frontal cortex''' (working memory) <ref name="Fick, S. & Shilts E. (2006)." />. <br> ♦ One of the most widespread activities, in terms of brain areas used, is playing music. The auditory cortex is needed to hear the music, and different feedback processes occur here to let us know whether we have achieved the correct pitch and melody. The '''visual cortex''' is also active, as we read (or even simply imagine) a piece of sheet music. The '''parietal lobe''' is needed for several of the complex tasks such as estimating the positioning of our fingers to play a certain note. The '''motor cortex''' helps us perfect the fine motor skills needed to play. Our '''sensory cortex''' is activated with each touch of the instrument as we play. The '''frontal lobe''' is used in the planning and overall coordination of our movements and activities, and the '''cerebellum''' works with the motor cortex to allow the production of the smooth, flowing movements often required in playing an instrument. The '''premotor area''' is also involved in the timing and sequence of our movements, but the ''exact'' role of this area of the brain in processing has still not been concretely defined <ref name="Fick, S. & Shilts E. (2006)." />. <br> ♦ Have you ever had the sensation of getting goosebumps when listening to a piece of music? This emotional reaction to music actually uses other areas of the brain as well! When you get chills or react to a piece of music that you like, this reaction is caused by the structures in the inner brain, such as the '''ventral tegmental area'''. These inner structures are responsible for the pleasure of "reward", such as receiving food when we are really hungry. If you hear a song you really like, the amygdala is also inhibited (as it is responsible for ''negative'' emotions, like fear). This emotional reaction can vary depending on our feelings about the song <ref name="Fick, S. & Shilts E. (2006)." />. Clearly, all of the various aspects of music production and processing use a large variety of skills and abilities, and therefore, a large number of brain areas and structures. When all of these areas of our brain start working together we are able to develop a full appreciation of all of the aspects of music. ==== Similar Systems ==== Music and language are processed using very complex brain systems, but it turns out that the processing of both music and language require many ''similar'' areas of the brain. <br> One very interesting similarity is seen in the brain activity of processing language, and that of musicians specifically. Many scientists understand that much of language processing occurs in the left hemisphere of the brain (in the majority of people). Recently, it has been found that musicians use these areas in the left hemisphere for music processing as well, whereas non-musicians typically rely on the right hemisphere. This has led to the belief that musicians may actually process music more analytically (as they engage more of the left hemisphere, similar to the areas used for language processing) than non-musicians <ref name="Gordon, H. (2006).">Gordon, H. (2006). Brain Waves. ''Canadian Geographic Magazine''. Accessed online: February 3, 2011. [http://www.canadiangeographic.ca/magazine/jf06/alacarte.asp]</ref>. Many areas of the brain that are used in processing language and music are almost identical. Now, with the help of positron emission tomography (PET), it is easy to visualize the active areas of the brain when attempting to process language or music. Some of the areas where we see the most overlap include the motor cortex (primary motor cortex ''and'' surrounding areas), Broca's area, both the primary and secondary auditory cortex, the cerebellum, the basal ganglia and thalamus, and the temporal pole <ref name="Brown, S., Martinez, M.J., & Parsons, L.M. (2006)">Brown, S., Martinez, M.J., & Parsons, L.M. (2006). Music and language side by side in the brain: a PET study of the generation of melodies and sentences. European Journal of Neuroscience, 23, 2791-2803.</ref>. Also, though there is a general belief that language primarily uses the left hemisphere, while music primarily uses the right hemisphere, there were several structures in ''both'' hemispheres that were used for both language and music. Again, PET scans were used to record the electrical activity in the brain. When music was played, the front of the brain and the right hemisphere were primarily engaged, though the majority of visible areas showed ''some'' activity. When processing language, the front of the brain and the left hemisphere were primarily used, though again there was activation of structures in many different areas. Then, of course, when ''both'' language and music were being processed at the same time, there was a concentration of brain activity in the frontal structures and in ''both'' hemispheres <ref name="Chudler, E. (2011).">Chudler, E. (2011). "The Brain and Nervous System". Human Diseases and Conditions. Accessed online: February 18, 2011. [http://www.humanillnesses.com/Behavioral-Health-A-Br/The-Brain-and-Nervous-System.html]</ref>. <br> <br> === Problems: What Happens When Things Go Wrong? === As with many areas of neuroscience, there is always a question of the possible effects that may be seen when imperfections or abnormalities occur in the brain. Brain damage can occur as a result of injury (damage to the brain, causing alterations), or as a result of congenital effects (abnormalities from birth). <ref name="Wikipedia Amusia">Wikipedia (2011). "Amusia" [[Wikipedia:Amusia|Amusia]] Accessed online: March 1, 2011. </ref>[[File:A_C_D_notes.svg‎|thumb|right|Amusia (or "musical deafness") can affect an individual's ability to correctly detect pitch]] It is believed that humans are born musically-inclined, and the damage to related brain areas causes problems in the ability to process music properly. One of the common results of brain damage (as it pertains to music processing) is referred to as '''Amusia'''. One of the common names for amusia is "musical deafness", because it affects an individual's ability to properly detect pitch (recognize out-of-tune notes, for example), as well as to recognize familiar songs. <ref name="Wikipedia Amusia" /> In terms of music production, amusia affects an individual's ability to sing, whistle or hum, write music (also known as "agraphia"), or play an instrument (musical apraxia). <ref name="Wikipedia Amusia" /> It is difficult to discriminate between melodies, and songs that may have been well-known before the brain damage occurred may become unfamiliar. It has been thought that individuals with congenital amusia may also experience difficulties discriminating and identifying changes in intonations as well (raising the ''tone'' of your voice at the end of a question, for example), which additionally affects their language processing abilities. <ref name="Wikipedia Amusia" /> <br> <br> === Music and Memory: What Have We Learned? === The idea that music may help improve memory has been very well-researched over the years. With new information about the brain, and new techniques for neuroimaging, it has become possible to extend research further and further each year, targeting more specific areas of the brain, and observing different groups of people. For example, we are now able to observe the role of music in the brains of infants through early childhood. Research has shown that if babies actively engage in music education or musical participation, they will show increased brain development<ref name="Hodges, D.A. (2000).">Hodges, D.A. (2000). Implications of Music and Brain Research. ''Music Educators Journal''. 87(2): 17-22</ref>. As well, many of these musical abilities (though different across cultures) are essentially innate and universal. Musical training early in life may also help improve one's memory. MRI scans allow researchers to view specific areas of the brain, and they have shown that, on average, the left temporal regions of the brain are actually larger in musicians than non-musicians <ref name="Chan, A.S., Ho, Y.C. & Cheung, M.C. (1998).">Chan, A.S., Ho, Y.C. & Cheung, M.C. (1998). Music training improves verbal memory. ''Nature''. 396(128) doi:10.1038/24075.</ref>. This area of the brain assists in verbal memory tasks, and therefore should allow musicians to perform more successfully on tests of verbal memory. Chan, Ho and Cheung collected evidence showing that individuals who had been musically trained before age 12 actually did perform better on tests of verbal memory than individuals who had not received this early training. It is understood that environmental factors will come into play, but still supports the idea that music could be extremely beneficial to memory and brain development, especially when it occurs early in life. <br> <br> === Try it yourself! === As we have seen, music and language are very closely-related. They use similar areas of the brain in processing, can be paired together to assist in memory formation, and even the structure and generativity of the two are very similar. Clearly, music is an important addition to the study of linguistics and psychology, and enhances the understanding of how these topics have developed. Now, using everything we have learned about language and music, test how much you now know about these topics below! <br> ==== Interactive Learning Exercises ==== Apply your knowledge of psycholinguistics, language and music '''[[Learning and Music - Interactive Learning Exercises|here]]''' <br> <br> === References === <references /> '''Images''' #http://en.wikiversity.org/wiki/File:Alternative.jpg #http://en.wikiversity.org/wiki/File:Brain_image1.jpg #http://en.wikiversity.org/wiki/File:Hdc_0000_0001_0_img0004.jpg #http://en.wikiversity.org/wiki/File:A_C_D_notes.svg [[Category:Psycholinguistics|{{SUBPAGENAME}}]] [[Category:Musicology]] o0fsrwtnhuru7pzfkj82ulft4kr7aml Learning and Music - Interactive Learning Exercises 0 109494 2812983 1822916 2026-06-05T22:23:52Z Kirby - Electrotechnics 3074947 changed to 'category:Music education' instead of category:Music 2812983 wikitext text/x-wiki == '''Try it Yourself! Apply Your Knowledge''' == Ok, it's time to put your knowledge to the test! Let's see what you've learned about language, music, and what happens when we combine the two. Try Part 1 to remind yourself about the basic psychology of language and music. Attempt to complete the whole section without looking at the main page (unless you get really stumped!), then move on to Part 2 for a fun exercise on advertising. See how adding a catchy melody to a slogan can make the product stick in peoples' minds! In Part 3, you can test the effects of music, language and memory on your own friends and family. See how your results compare with the current research. === '''Part 1 - What Have You Learned? (The Basics)''' === Try to answer these questions to see how much you can remember, and how much you've learned about language and music. Try to complete the section without referring back to the main page, but if you get stuck you may need to refresh your memory [[Psycholinguistics/Language and Music|here]] *For both language ''and'' music, list: :'''a)''' Basic sound units :'''b)''' "Vocabulary" used :'''c)''' Function and/or purpose *Define ''"generativity"'', and: :'''a)''' Justify why the generativity of both language and music has been deemed infinite. :'''b)''' Explain how culture plays a role in each one *Describe the similarities and differences between Broca's Area and Wernicke's Area, and explain why each is important for language and speech *Explain how early exposure to music training may affect an individual's memory. Does this earlier exposure seem in aid or inhibit memory development? *List one important brain area used in the process of: :'''a)''' Listening to music :'''b)''' Playing music :'''c)''' Processing '''''both''''' language and music === '''Part 2 - See For Yourself''' === Music intensifies every aspect of an advertisement, and gives a sense of stimulation to the viewer or listener that helps cement the message of the ad itself (Hecker, 1984). Many times, you likely find yourself humming a song or jingle from a well-known advertisement, realizing that the tune is stuck in your head. Naturally, this is what the advertisers want: an easily identifiable way of helping you remember their product. This is just one more example of how powerful the combination of music and language can be. In order to test out this concept, make a list of several well-known radio and television commercials with familiar "theme songs". Try humming the tune for the product slogan to your friends and family, and see if they can recite the line back to you. Chances are, they will know the tune, phrase product, and even the company responsible for the ad. Use the table below as a starting point, and try to fill in a selection of ads you come across. The first line shows an example of how to use the table: {| class="wikitable" |- ! Product Name !! Company !! Tv/Radio? !! Slogan !! # Correctly identified? !! # incorrectly identified? |- | Product 'A' || cb inc. || Radio || No one beats the taste of cb! || 10 || 2 |- | B || || || || || |- | C || || || || || |} '''Follow-up questions:'''<br> a) Did your results support the concept relating music and language?<br> b) Were people able to correctly identify the slogans? <br> c) If you were an advertiser, would this influence your decision to ad music to your slogan? <br> <br> '''Essay Topic'''<br> d) Research the different cognitive or neuroscientific theories surrounding the link between language, memory and music (ie: ''why'' does music help people remember?): <br> :i) List the theories<br> :ii) List the researchers involved<br> :iii) List supporting evidence for each theory<br> :iv) Has any contradictory evidence been discovered? What was it, and how did it affect the theory?<br> :v) Which theory do you feel is the most likely? Explain. Try performing the experiment the other way around...if you tell people a product slogan, would they be able to hum the jingle that goes along? Another twist on this experiment that you can try is to write a sentence out on paper. Try to make the sentence long enough that it is not ''easy'' to remember the first time you hear it. Now show a few people this sentence, read it to them, and see if they can remember it 15 minutes later. Using the same sentence, add a tune to the words, and sing it to a different people. Again, see if they can sing it back to you 15 minutes later. Which group was more successful? Hecker, S. (1984), Music for advertising effect. ''Psychology and Marketing'', 1: 3–8. doi: 10.1002/mar.4220010303 === '''Part 3 - Perform Your Own Experiment''' === If you would like to try an experiment (somewhat similar to the one presented by Steven Smith, 1985), you will need three sets of test subjects (or you may perform the experiment as a ''within-subjects'' design, meaning you will simply need extra lists of words). Create a list of words (30 words), and present each individual in each set with the same list. One group will be in a room with music playing while they study the list of words for 3 minutes, the second group in a silent room, and the third group in a room with ambient noise (ex: background television, etc.). After the list of words has been presented for 3 minutes, give the individuals a 10 minute break, then return them to the same room with the same conditions as they had been in previously. Ask them to write out as many of the words as they can remember, and see which group (under which conditions) is able to remember the most words. To alter the experiment slightly, try keeping some groups the same for testing (i.e: learning ''and'' testing both occur in a silent room), while other groups change (learning in a room with music playing, but testing in a silent room). <br> Which combination seems to produce the optimal conditions for learning and recall? <br> Why do you think this might be? <br> Is there any possibility of a deleterious effect?. Smith, S. (1985). Background Music and Context-Dependent Memory. ''The American Journal of Psychology''. 98(4): 591-603. ==='''Part 4 - Essay Questions'''=== The following questions are to be answered in essay format. It will be helpful to use the other chapters from the *[http://en.wikiversity.org/wiki/Psycholinguistics Psycholinguistics homepage] in formulating your answers<br> '''1)''' Explain the basic symptoms and problems associated with amusia. How is amusia similar/different from aphasia, acalculia, and alexia?<br> '''2)''' For each item in the following list, explain how it compares with music in terms of innate (inborn) abilities, and level of generativity: :i) Mathematics :ii) Prosody/Emotion and expression :iii) Gesture/Body Language<br> '''3)''' How do the neural components of ''speech'' production and writing compare with those of playing, writing and singing in music? :i) Which areas of the brain are used in both? :ii) Which key areas of the brain are different, and what different skills require the addition of this different brain structure? :iii) Which brain areas are affected when we see deficits in the production of each? :iv) How is speech ''perception'' similar to musical perception? How is it different? :v) Is there a musical deficit equivalent to dyslexia? What is it, and how does it affect an individual (internal/cognitive AND external/behavioural processes)<br> '''4)''' If music is culturally-based, at what age can children discriminate between music that is familiar or unfamiliar to their own culture? :i) What processes cause this? :ii) How is this similar to second language acquisition (ie: how does age affect a child's ability to quickly learn the music (or language) of a foreign country)?<br> '''5)''' Try to imagine you are teaching someone to read and play music: :i) Can you identify the music equivalent for each of the synthetic methods of reading (alphabet method, synthetic phonic method, syllable method)? :ii) Try to do the same for the analytic methods of reading (word method, sentence method, phrase method). :iii) For each method that you found a musical comparison, list why (or why not) you think it would be effective in teaching music. Which method would you deem ''most'' effective? :iv) For the method you consider most effective in teaching music, compare this method to its uses in teaching reading. Would you still consider it the best choice? Why or why not? [[Category:Music education]] 67iwe6w04ho6d92f5zvncpi2plz3f63 Fiddlers Green Beyond Violin 0 115962 2812868 2297560 2026-06-05T15:46:21Z Kirby - Electrotechnics 3074947 Removed from the main "Category:Music" 2812868 wikitext text/x-wiki __NOTOC__ __NOEDITSECTION__ <!--------------------------------> <!-- HEADER SECTION --> <!--------------------------------> {| cellspacing="0" cellpadding="0" style="margin:0em 0em 1em 0em; width:100%" | style="width:100%; vertical-align:top; border:1px solid Sienna; background-color:Cornsilk;" | <div style="border-bottom:1px solid Sienna; background-color:Wheat; padding:0.2em 0.5em 0.2em 0.5em; font-size:110%; font-weight:bold;">Introduction to Lesson One<nowiki> </nowiki> <!---[[Topic:Fiddle|Fiddle]]---> </div> <div style="padding:2em 5em 0em 3em;"> <!-- Top, Right, Bottom, Left --> {{testing}} {{nonformal}} {{workshop}} ==Prerequisites== Interested in playing fiddle and: * You are taking advantage of [[Violin 101]] and want to learn more, or * You already know how to play [[violin]] at an advanced novice or intermediate level. <!----(Contributors: Add [[Help:Icons]]. For a list of available colors, --> <!---go to [[w:Web colors|Wikipedia's HTML Web Color Tables]].))---> ==Welcome== :Hello and welcome to {{font|color=green|'''Fiddler's Green'''}} at Wikiversity! This course is designed to help you break out of violin indoctrination and free your mind and body so you can soar into the ethereal realm of the eternal emerald paradise, Fiddler's Green. First things first: we need to un-learn lots of things we were taught, such as how the bow is held, [http://www.youtube.com/watch?v=4VPuWIgmDmE&NR=1 how the fiddle is held], and incessant use of vibrato. Join Kevin Burke for Morning Dew at a[[http://www.youtube.com/watch?v=fjyG3rW_yLA house party]] - he handles the bow like Hemingway handles a typewriter: no superfluous nothin'! </div> |} <!--------------------------------> <!-- ONE BOX SECTION --> <!--------------------------------> {| cellspacing="0" cellpadding="0" style="margin:0em 0em 1em 0em; width:100%" | style="width:100%; vertical-align:top; border:1px solid Gold; background-color:LightYellow;" | <div style="border-bottom:1px solid Gold; background-color:#ffffaa; padding:0.2em 0.5em 0.2em 0.5em; font-size:110%; font-weight:bold;">Learn fiddle (despite violin training!</div> <div style="padding:1em 5em 1em 3em;"> <!--Top, right, bottom, left --> <!-- PICTURE ON ONE BOX SECTION - optional --> <!-- [[Image:Nuvola apps package edutainment.png|right|64px]] --> ;Basic differences from violin playin' :The best and most fiddle sounding is not "violinistic". Although all rules are made to be broken, to start out you should eliminate the over use of vibrato, which, in the beginning, includes almost all vibrato. (Once you get in a fiddle playin' mode you can break that rule all you want. But first things first; it is necessary to break the old bad habits that you learned from your violin teacher). Learn to play with punch like a cowpoke at a Saturday night hoe down, or an Irish rogue in a crowded pub. This is [http://www.youtube.com/watch?v=7I6nC-RDwAc&NR=1 party music]. Forget all of those Italian tempo words like ''ritartando'' - fiddle music is usually just played at one speed (mostly just fast). To help you break out from all that classical training, take a look at some right hand stuff, the "chop" technique; you probably won't be ready to use it right off the bat, but it gives you a good idea of the direction you will be moving in and how radical a departure you will be taking from all that orchestra stuff you learned as a kid. [http://www.youtube.com/watch?v=RAmvloansmU&feature=related Richard Greene's "Chop"] technique is demonstrated in a shorter video [http://www.youtube.com/watch?v=b0nvpytrdVQ&feature=related here] with the expectation that you have previously viewed [http://www.youtube.com/watch?v=b0nvpytrdVQ&feature=related]. Watch all of these videos several times in various orders and break out your fiddle and bow and give it a shot. [http://www.youtube.com/watch?v=mNYg3JGRxzk Here] is how the technique gets blended into a somewhat violinistic rock performance. </div> |} <!--------------------------------> <!-- TWO BOX SECTION --> <!--------------------------------> {| cellspacing="0" cellpadding="0" style="margin:0em 0em 1em 0em; width:100%" | style="width:60%; vertical-align:top; border:1px solid Gold; background-color: LightYellow;" rowspan="1"| <div style="border-bottom:1px solid Gold; background-color:#ffffaa; padding:0.2em 0.5em 0.2em 0.5em; font-size:110%; font-weight:bold;">Book larnin'</div> <div style="padding:0.4em 1em 0.3em 1em;"> <!-- PICTURE ON LEFT OF TWO BOX SECTION --> ===Survey the styles and find your groove!=== * Download sheet music from http://www.thesession.org * [[[Wikibooks:Compendium of Fiddle Styles|Text: Compendium of Fiddle Styles on WikiBooks]]] *[[[Wikibooks:Fiddlers Companion|Fiddlers Companion]]] Fiddlers Companion Tunes A-Z * [[http://www.cranfordpub.com/recordings/Coleman.htm Paul Cranford]] offers Michael Coleman recordings and transcriptions with some generous on line free shots. Take advantage, and, by the way, any of his books contain excellent material exemplifying the unique blend of Cape Breton style fiddle music, much of it composed in the [[w:Monastic silence]] of a lighthouse. : [http://www.cranfordpub.com/reviews/CBFiddlersReview.htm The Cape Breton Fiddlers Collection] *''Fiddling Across America'' by David Reiner & Peter Anick ::Published by Mel Bay 1989 Pacific Missouri * ''Boston Fiddle:The Dudley Street Tradition'' by Frank Ferrell [http://www.amazon.com/gp/product/B004W6GHBY/ref=pd_lpo_k2_dp_sr_1?pf_rd_p=486539851&pf_rd_s=lpo-top-stripe-1&pf_rd_t=201&pf_rd_i=B0000003EJ&pf_rd_m=ATVPDKIKX0DER&pf_rd_r=1QEFB6FNRC7X3VGW8FAD Listen] online. * ''English, Welsh,Scottish & Irish FIDDLE TUNES'' by Robin Williamson. Oak Publications. 1976. New York|London|Sydney This book has a [[Fiddlers Green Beyond Violin/Notes|study guide]] *'' Appalachian Fiddle'' by Miles Krassen Oak Publications 1995 * ''Cajun Fiddle'' by Craig Duncan Mel Bay 1995 * ''Twenty Irish Fiddle Tunes'' Kevin Burke. CD-BUR-TW01 Level 3 Includes music Kevin plays once slow, once at tempo. Grace notes, rolls, double-stops and other ornaments. </div> | style="padding:0em 0.5em 0em 0.5em;" | | style="width:40%; vertical-align:top; border:1px solid #abd5f5; background-color:#f1f5fc;" | <div style="border-bottom:1px solid #abd5f5; background-color:#d0e5f5; padding:0.2em 0.5em 0.2em 0.5em; font-size:110%; font-weight:bold;">Get a guitar accompanist and practice playing in ensemble!</div> <div style="padding:0.4em 1em 0.3em 1em;"> <!-- PICTURE ON RIGHT OF TWO BOX SECTION --> [[File:CourseInstructor.jpg|200px|Course instructor on guitar]] ;Fiddle styles Decide which style you are most interest in by reading these articles. All of them were started by this teacher so if you have questions, use the email link below. * [[w:American fiddle]] * [[w:Old time fiddle]] * [[w:Rock violin]] * [[w:Blues fiddle]] * [[w:Canadian fiddle]] * [[w:Metis fiddle]] * [[w:Cajun fiddle]] * [[w:Western swing fiddle]] * [[w:Bluegrass fiddle]] * [[w:Quebec fiddle]] * [[w:Klezmer fiddle]] </div> |} <!--------------------------------> <!-- THREE BOX SECTION --> <!--------------------------------> {| cellspacing="0" cellpadding="0" style="margin:0em 0em 1em 0em; width:100%" | style="width:40%; vertical-align:top; border:1px solid Gold; background-color: LightYellow;" rowspan="2"| <div style="border-bottom:1px solid Gold; background-color:#ffffaa; padding:0.2em 0.5em 0.2em 0.5em; font-size:110%; font-weight:bold;">Mandatory listening</div> <div style="padding:0.4em 1em 0.3em 1em;"> <!-- PICTURE ON LEFT OF THREE BOX SECTION --> {{center top}} <!---[[File:Virginia-stringband-1937.jpg|200px]]---> [[File:Violin mute.jpg|200px|Mute to allow vigorous bowing when you can't play loudly.]]<br> <small>Mutes allow vigorous bowing practice when you need to keep the volume down.</small>{{center bottom}} ===Listening is critical=== * [http://www.youtube.com/watch?v=nPAXXOhxh1M Old timer from W Virginia] *[http://www.youtube.com/watch?v=lXniM9Ml8R0&NR=1&feature=fvwp Boogie Woogie Fiddle Country Blues Rock] * [http://www.ted.com/talks/natalie_macmaster_and_donnell_leahy_play_the_cape_breton_fiddle.html Natalie MacMaster Cape Breton] * [http://www.youtube.com/watch?v=XGd21qshXDc Ahokan Farewell ] *[http://www.youtube.com/watch?v=4eEsXQt06Pc&feature=related Jay and Molly] * [http://www.lunasa.ie/ Cregg's Pipes (Link 8)] * [http://www.youtube.com/watch?v=acsNgZbvF8E&feature=related Masons Apron] played by Marty Hayes * [http://www.youtube.com/watch?v=R2jkkKYz2x0&feature=related Tamlin Canada's Fiddlers 3 with Trad Dancers] * [http://www.youtube.com/watch?v=eNigFpaBBf8 Aly Bain plays Bonaparte's Retreat] * [http://www.youtube.com/watch?v=DQEpVP9RwCg King of the Faeries] played by the legendary Scotty Fitzgerald * [http://www.youtube.com/watch?v=NGtckNUY7b8 Percussive Metis style with commentary] * [http://www.nashvilleoldtime.org/Tunes%20-%20Listen%20and%20Learn/Jukebox/index.shtml Old Time Jukebox] == Course originator Geof Bard posts as Wikidgood == [[File:Serious fiddlemusicman cropR 12pct(1).jpg|left|The primary instructor of this class is Geof Bard. Without the tireless assistance of the Wikiverity administrators and other contributors this class would not have been possible.]] ... and if you can find it, course instructor Geof Bard was recorded as here playing McPherson’s Lament, a sad bagpipe style tune of a rogue fiddler who met his demise and played one last tune “upon the gallers tree”. </div> | style="padding:0em 0.5em 0em 0.5em;" | | style="width:60%; vertical-align:top; border:1px solid #abd5f5; background-color:#f1f5fc;" | <div style="border-bottom:1px solid #abd5f5; background-color:#d0e5f5; padding:0.2em 0.5em 0.2em 0.5em; font-size:110%; font-weight:bold;">Geof and Scott, a damned good guitarist demonstrate the spirit of collaboration which is essential in makin' music.</div> <div style="padding:0.4em 1em 0.3em 1em;"> <!-- PICTURE ON TOP RIGHT OF THREE BOX SECTION --> [[File:FourHandGuitar.jpg|200px|Collaboration is the key!]] ;This course will use humor, graphics and collaboration at all times. * Listen more! Note the percussion on [http://www.youtube.com/watch?v=NGtckNUY7b8 this] clip. It is from French-Canadian-First Nations fiddle. * Keep listening. For Irish style you can't beat [http://www.youtube.com/watch?v=6oKyL0VAkDw Rollin' In the Barrel, Youngest Daughter Reel]. <!----> </div> |- | style="padding:0em 0.5em 0em 0.5em;" | | style="width:60%; vertical-align:top; border:1px solid #abd5f5; background-color:#f1f5fc;" | <div style="border-bottom:1px solid #abd5f5; background-color:#d0e5f5; padding:0.2em 0.5em 0.2em 0.5em; font-size:110%; font-weight:bold;">Lesson One Wrap Up</div> <div style="padding:0.4em 1em 0.3em 1em;"> <!-- PICTURE ON BOTTOM RIGHT OF THREE BOX SECTION --> [[Image:Crystal Clear action bookmark.svg|right]] ; What this course covers * Fiddle styles especially indigenous and Anglo-Celtic ''as developed in North America''. * Southern fiddle styles, especially from the Piedmont area of Virginia and the Appalachia, will be studied and taught only as they have influenced the above said Yankee fiddle traditions * Canadian fiddle styles, since they are influential on Down East and Northwest Fiddle, will be considered, studied, and absorbed, but this course will not teach those styles unless a Canadian fiddle instructor signs on. </div> |} <!--------------------------------> <!-- FOOTER --> <!--------------------------------> {| cellspacing="0" cellpadding="0" style="margin:0em 0em 1em 0em; width:100%" | colspan="2" style="width:100%; vertical-align:top; border:1px solid Sienna; background-color:Cornsilk;" | <div style="border-bottom:1px solid Sienna; background-color:Wheat; padding:0.2em 0.5em 0.2em 0.5em; font-size:110%; font-weight:bold;">Conclusion title</div> <div style="padding:2em 5em 0em 3em;"> <!--Top, right, bottom, left --> <!-- PICTURE ON FOOTER SECTION --> [[Image:Crystal_Clear_app_kdict.png|96px|right]] ==Lesson 2== :Continue this lesson on [[/Lesson 2/|More fiddle...]] ==Contact your instructor== :* Your instructor for this class is [[User:Geofferybard|Geof Bard]]. To email me, you simply {{email|user=Geofferybard|subject=Question on {{PAGENAME}}|text=Click Here}}. [[Image:Crystal Clear app xfmail.png|32px]] ==Resources== {{Reflist}} Lester McCumbers demonstrating [http://www.youtube.com/watch?v=4VPuWIgmDmE&NR=1 how to hold the fiddle] Appalachia style, violin teachers beware! Class [[/Notes/]]. Lesson on [[/How to Accompany/]]. </div> |} [[Category:Fiddle]] by1bguulwozmg08od0s58325jwhzx9n Fiddlers Green Beyond Violin/Lesson 2 0 115969 2812869 2297559 2026-06-05T15:46:46Z Kirby - Electrotechnics 3074947 Removed from the main "Category:Music" 2812869 wikitext text/x-wiki {{TOCright}} Please listen to [http://www.youtube.com/watch?v=SrsGpPA5TBA Old English style rock]. This will not only focus on chopping but the bow will be the main focus. == How to accompany == [[File:Virginia-stringband-1937.jpg|200px]] A great demonstration of how to accompany singers in a fiddle band is provided by the clip of Jay Ungar and Molly Mason performing [http://www.youtube.com/watch?v=BoDVEIUR4xs&NR=1&feature=fvwp Home Grown Tomatoes] with their band Swingology. It's a song written by Guy Clark. If you need convincing that fiddle is essential to performance of this kind of song, consider [http://www.youtube.com/watch?v=t5-GxuCqWAM&feature=related this] clip form the Central Ohio Folk Festival. The clip might be useful for fiddle practice as a kind of "Plus One" background track, but the real take home point is how this lineup just screams "Fiddler Needed"! === Don't improvise too much too early === [[File:Fiddler's Green (62).jpg|500px]] If you are the lead instrument, lay down the "head", as jazz musicians say, before getting too creative or you may confuse others in the band or audience. <ref>source=Banjo Ben|http://www.youtube.com/user/BanjoBen1|Red Haired Boy|Flatpicking Lesson|2010|http://www.youtube.com/watch?v=ji7w6f1mJbM&feature=relmfu</ref> If you are in improve mode, know your pentatonic scales and hit that blue note. The [[w:flat third]] and the [[w:flat five]] are your friends ! === Learning resources === Although traditional music purists prefer learning by ear rather than learning "by note", in modern times a plethora of educational material is available, often by well known composers and performers. According to ethnomusicologist Lynn Whidden, old time traditions such as Metis fiddling are transmitted directly rather than through the "passive" mode of teacher-student. "Like other things in aboriginal culture it wasn't consciously taught...[it was taught] very actively."<ref>http://www.youtube.com/watch?v=NGtckNUY7b8|Bare URL|Full citation at Metis fiddle article. Citation upgrade needed.</ref> * Mark O'Connor: Notable sources of instruction in the Old Time fiddle tradition include Mark O'Connor, who, despite a varied repertoire including jazz, bluegrass and swing, is recognized as an exemplary player of Texas style Old Time Fiddling. In 2009, O'Connor released Books 1 and 2 of his 10-book ''O'Connor Method - A New American School of String Playing''<ref>{{Cite web|title = O'Connor Violin Method |url=http://markoconnor.com/index.php?page=about&family=method |publisher = Mark O'Connor Musik International}}</ref> O'Connor pedagogy is also offered in conjunction with Berkelee College of Music and the East Tennessess State University<ref>http://markoconnor.com/index.php?page=about&family=fiddle</ref> Mark O'Connor Fiddle Camps is a 501(c)3 corporation with an arts advisory board including O'Connor's notable collaborator Yo Yo Ma. The camps include other styles, but faculty relevant to Old Time fiddle include Shane Cook (Canadian Fiddling),Daniel Carwile (Texas Fiddling)and Matt Glaser (Bluegrass Fiddling).<ref>http://markoconnor.com/?l=1n9</ref> * Jay Ungar: Jay Ungar, one of the more famous American fiddlers, while popular with many casual OT fiddling novices, is not a pure old time fiddler. He is however an educator who has elevated the social prominence of all forms of Americana and American culture through his composition and fiddle camps. He achieved fame in fiddle circles due to use of his material in the film industry, notably Ashokan Farewell and he operates fiddle camps in New York State.<ref>[http://www.jayandmolly.com/ashokanfaq.shtml The Story of Ashokan Farewell]</ref> The educational offerings by this notable fiddler and pedagogue provide an interesting secondary source suggesting a categorization scheme for various genre and styles. These include: *Western & Swing:Swing dance, two-step, waltz, square dance|squares & contras, fiddle, mandolin, guitar, piano, singing, vocal workshops, swing & country band clinics, music theory and more. *Northern style: Contras, squares, English, Quebecois, Scandinavian dances, New England, Quebecois & Scandinavian fiddle, guitar, piano, singing, band clinics and more. *Southern: Squares, contras, clogging, Cajun & Zydeco dance, fiddle, banjo, guitar, singing, band clinics, callers workshop and more. *Other:Cajun, Swing, squares, contras, fiddle workshops, song. Guitar:Blues, Rags, Jazz, Flatpicking. Summer Dances at Ashokan Summer Dance and Dinner at Ashokan. *See also:[http://www.fiddle.com/Links-Camps-Makers-.page Fiddler magazine list of fiddle camps] == References == {{Reflist}} [[Category:Fiddle]] 2742r3nsnyizvphnxt1ivknf2x7z4or Country music 0 116012 2812859 1725806 2026-06-05T15:26:40Z Kirby - Electrotechnics 3074947 changed to 'category:Music genres and styles' instead of category:Music 2812859 wikitext text/x-wiki {{Wikipedia}} Country music is a genre of American popular music that originated in the Southern United States in the 1920s. It takes its roots from the southeastern genre of American folk music and Western music.<ref>[[Wikipedia: Country music]]</ref> == Readings == ==Resources== * [[Fiddle]] * [[Fiddlers Green Beyond Violin]] * [[Ethnography of Fiddle]] * [[Violin 101]] * [[Ear training]] == See Also == * [[Guitar]] * [[Building A Cigar Box Guitar]] * [[The music industry]] == References == {{Reflist}} [[Category:Music genres and styles]] ne7wv6iujnif23b20xo18g3837jjpwy Category:Fiddle 14 116069 2812866 778115 2026-06-05T15:44:32Z Kirby - Electrotechnics 3074947 changed to 'category:Music instruments' instead of category:Musical instruments 2812866 wikitext text/x-wiki This category is for pages related to fiddles and violins. [[Category:Music instruments]] rvye1b9lwfvxeiqu6d00f18zmil85ov Portal:Fiddle 102 116252 2812867 1580725 2026-06-05T15:45:17Z Kirby - Electrotechnics 3074947 Removed from the main "Category:Music" 2812867 wikitext text/x-wiki {{Disambiguation portal|[[Portal:Fiddle|Fiddle]] ** [[Violin 101]], for beginners (co-requisite for more advanced courses) ** [[Fiddlers Green Beyond Violin]] Down home nitty gritty workshop in practice ** [[Ethnography of Fiddle]] Academic orientation to using the web to research and study ethnomusicology of fiddle. * [[Fiddlers Green Beyond Violin/How to Accompany|How to Accompany with Fiddle]] * [[School:Music and Dance|School:Music]] * [[Portal:Music|Department of Folk, Traditional and Country Music]] <!---* [[Portal:Writing Center|The Writing Center]] (for help---< <!---* [[English Language Reference Desk]]---> * [[:Category:Fiddle]] *[[Category:Music]]|[[Fiddlers Green Beyond Violin]] [[Ethnography of Fiddle]] [[Violin 101]] [[Ear training]] ==Related topics== [[Guitar]] [[Building A Cigar Box Guitar]] [[The music industry]]}} crrp5csjv65qx18i6q5qsefgqv1hxrl Engineering Projects/Music projects 0 118594 2812883 2111052 2026-06-05T15:56:22Z Kirby - Electrotechnics 3074947 changed to 'category:Music related projects' instead of category:Musical instruments 2812883 wikitext text/x-wiki ==Idea== Music projects involve creating new [[wikipedia:Hydraulophone|instruments]], sampling in a new way and creating new sounds. For example, [[Wikipedia:Hall effect|hall effect]] probes, magnents and a bean can can be combined to [http://www.youtube.com/watch?v=47ijWfiIMT8&feature=fvsr create] a combination player piano drum and mixer. Ball bearings and metal washers can be combined into a [http://www.youtube.com/watch?feature=player_embedded&v=wreP8FMupyM drum box]. ==SubSystems== :Synthesizers :MIDI ==Tutorials== ==College/Universities== [[/Howard Community College/]] [[Category:General_Engineering_Projects_2010-2012]] [[Category:Music related projects]] q41p2rh02k79izbaq68v8hz3oz8cj9f Hymnology 0 120136 2812984 2477403 2026-06-05T22:24:30Z Kirby - Electrotechnics 3074947 changed to 'category:Musicology' instead of category:Music 2812984 wikitext text/x-wiki [[Hymnology|Sacred Music]] '''The 1000 Songs Project''' This project is an attempt to create a taxonomy of songs intended for congregational singing in a Christian church. It will be built and maintained by students, and then used as a resource and guide by more students. It is a hymnal with 1000 songs, designed to represent a balance of historical traditions as well as contemporary culture. It is also a hymn companion, with background information regarding the source, context and material for each song. It is also an anthology of music and texts, providing written scores for study and arrangement, as well as a creative outlet for composition and arranging students. It is also a resource for worship leaders, who might find a new text or tune for a song, or perhaps some background information that will help. Let's get started! THE LIST OF [[1000 Songs]] 200 Bible songs 150 psalms 6 canticles, 12 hymn fragments 32 prophetic songs Use a minimum of three sources for every assertion you make (our text and Wikipedia are two options), and LINK TO YOUR SOURCE every detail that is not just common knowledge! No need for you to write something that someone else has already written—link to the thing that says what you would say. Use Wikipedia as one of your sources, and EDIT the article if you have something to add to it. (Extra credit for improving a Wikipedia article!) What to include: Why might this passage be included as an example of biblical songs? Was it intended to be sung by a congregation? If so, how might that work in your church today? Find out some information about the writer, about the historical and cultural setting. (Beware of higher criticism!) Explain the content of the material itself. Pick or create your own “Editor’s Choice” version of this text, suitable for singing. Choose your translation or possibly metricized version, or make your own. Include other versions, such as Tate& Brady or Sternhold&Hopkins, other metricized versions, various translations that might work for singing, etc. Find musical settings out there—someone’s been singing this passage for thousands of years, including today. What’s that text and music look and sound like? Extra credit for each if you include music, a chart, and/or a recording. 100 Pre-Reformation 50 Early Christian hymns (AD 100-400) 20 Greek hymns 10 Odes of Solomon 20 Syrian hymns 15 Byzantine hymns (AD 400-1000) 15 Latin hymns (AD 400-1000) 10 Late Middle Ages (AD 1000-1500) 10 Pre-Reformation hymns (AD 1200-1500) 100 Early Reformation (AD 1500-1600) 40 Lutheran chorales (Luther’s psalm settings included above in “psalms”) 10 Reformed (metricized psalter included above in “psalms”) 30 Anglican 20 Catholic Counter-Reformation 200 Second-Generation Reformation (AD 1600-1700) 50 Lutheran pietist 50 Anabaptist (Radical Reformed) 50 Reformed (new generation metricized psalms, included above in “psalms”) 50 Anglican chorales and Moravian hymns 200 British hymns (AD 1700-1800) 50 Isaac Watts (150 psalms included above in “psalms”) 50 Charles Wesley hymns 50 Evangelical hymns 50 Olney hymns 200 American/English songs (AD 1800-1900) 50 singing school, spirituals and campmeeting songs 50 transitional, Oxford, Victorian hymns (50 early American psalmody and shaped-note tunes) (50 Lowell Mason tunes) 50 Sunday School songs 50 Gospel Songs (Sankey and Bliss) 200 American/English songs (AD 1900-2000) 20 hymns 20 Black gospel 20 Southern gospel 20 Sentimental (1900-1950) 20 Youth-oriented (1950-1970) 20 Evangelism-oriented (1960-1980) 20 Praise songs (1960-1990) 20 presentational-to-congregational (1990-2010) 20 postmodern self-orientation (2000-present) [[Category:Religion]] [[Category:Musicology]] ts58pfkgkc7ae6yb6iin0tme7ozy6m1 1000 Songs 0 120138 2813011 2621730 2026-06-06T00:34:30Z MathXplore 2888076 added [[Category:Songs]] using [[Help:Gadget-HotCat|HotCat]] 2813011 wikitext text/x-wiki {{TOCright}} == Presentation == ''From https://en.wikiversity.org/w/index.php?title=User_talk:TheProf&oldid=791805'' ''06 X 11'' The purpose and contents of this learning resource is to be hymnology project. The plan is for it to be open content, contributed by students and others. Much of the content will be short explanations with wikipedia links. Other content will be student compositions in pdf and/or audio files. The "songs" are mostly public domain hymns, with student editing and tunes supplied. (...) This project is specifically sacred/congregational music. It should not be the only learning project in the Sacred Music division, and it seems there should be another layer between Sacred Music, and even Hymnology, and the 1000 Songs Project. == 200 Bible songs == === 150 psalms === [[1000 Songs/How blessed is the man who walks not Ps 001|How blessed in the man who walks not Ps 001]] [[1000 Songs/Why do the nations rage Ps 002|Why do the nations rage Ps 002]] [[1000 Songs/My God my God why have You forsaken me Ps 022|My God my God why have You forsaken me Ps 022]] [[1000 Songs/The Lord is my shepherd I shall not want Ps 023|The Lord is my shepherd I shall not want Ps 023]] [[1000 Songs/The earth is the Lord's and the fulness thereof Ps 024|The earth is the Lord's and the fulness thereof Ps 024]] [[1000 Songs/How blessed is the man whose transgression is forgiven Ps 032|How blessed is the man whose transgression is forgiven Ps 032]] [[1000 Songs/Sing joyfully to the Lord you righteous Ps 033|Sing joyfully to the Lord you righteous Ps 033]] [[1000 Songs/Why are you downcast O my soul Ps 042|Why are you downcast O my soul Ps 042]] [[1000 Songs/God is our refuge and strength Ps 046|God is our refuge and strength Ps 046]] [[1000 Songs/Have mercy on me O God Ps 051|Have mercy on me O God Ps 051]] [[1000 Songs/May God arise may his enemies be scattered Ps 068|May God arise may his enemies be scattered Ps 068]] [[1000 Songs/Sing to the Lord a new song Ps 098|Sing to the Lord a new song Ps 098]] [[1000 Songs/Praise the Lord Praise O servants of the Lord Ps 113|Praise the Lord Praise O servants of the Lord Ps 113]] [[1000 Songs/Give thanks to the Lord for He is good Ps 118|Give thanks to the Lord for He is good Ps 118]] [[1000 Songs/Blessed are those whose way is blameless Ps 119|Blessed are those whose way is blameless Ps 119]] [[1000 Songs/Out of the depths I cry to You O Lord Ps 130|Out of the depths I cry to You O Lord Ps 130]] [[1000 Songs/Praise the Lord How good it is to sing praises to our God Ps 147|Praise the Lord How good it is to sing praises to our God Ps 147]] [[1000 Songs/Praise the Lord Praise the Lord from the heavens Ps 148|Praise the Lord Praise the Lord from the heavens Ps 148]] [[1000 Songs/Praise the Lord Sing to the Lord a new song Ps 149|Praise the Lord Sing to the Lord a new song Ps 149]] [[1000 Songs/Praise the Lord Praise God in his sanctuary Ps 150|Praise the Lord Praise God in his sanctuary Ps 150]] === 6 canticles === [[1000 Songs/My soul magnifies the Lord Luke 1:46-55|My soul magnifies the Lord Luke 1:46-55]] [[1000 Songs/Praise be to the Lord the God of Israel Luke 1:67-79|Praise be to the Lord the God of Israel Luke 1:67-79]] [[1000 Songs/Glory to God in the highest Luke 2:14|Glory to God in the highest Luke 2:14]] [[1000 Songs/Now dismiss your servant in peace Luke 2:29-32|Now dismiss your servant in peace Luke 2:29-32]] === 12 hymn fragments === [[1000 Songs/Have this mind among yourselves which is also in Christ Jesus Phil 2:5-11|Have this mind among yourselves which is also in Christ Jesus Phil 2:5-11]] [[1000 Songs/Awake O sleeper and rise from the dead Eph 5:14|Awake O sleeper and rise from the dead Eph 5:14]] [[1000 Songs/He appeared in a body 1 Tim 3:16|He appeared in a body 1 Tim 3:16]] [[1000 Songs/If we died with Him we will also live with Him 2 Tim 2:11-13|If we died with Him we will also live with Him 2 Tim 2:11-13]] [[1000 Songs/Worthy are You to take the scroll Rev 5:9-10 (8-14)|Worthy are You to take the scroll Rev 5:9-10 (8-14)]] === 32 prophetic songs === [[1000 Songs/I will sing unto the Lord, for He has triumphed gloriously Exod 15:1-18|I will sing unto the Lord, for He has triumphed gloriously Exod 15:1-18]] [[1000 Songs/Give ear, O heavens and I will speak Deut 32:1-43|Give ear, O heavens and I will speak Deut 32:1-43]] [[1000 Songs/For unto us a child is born Is 9:6|For unto us a child is born Is 9:6]] [[1000 Songs/The desert and the parched land will be glad Is 35|The desert and the parched land will be glad Is 35]] [[1000 Songs/Who has believed our message Is 53|Who has believed our message Is 53]] [[1000 Songs/The Lord is a jealous and avenging God Nahum 1:2-8|The Lord is a jealous and avenging God Nahum 1:2-8]] [[1000 Songs/O Lord I have heard the report of You Hab 3:2-19|O Lord I have heard the report of You Hab 3:2-19]] == 100 Pre-Reformation == === 50 Early Christian hymns (AD 100-400) === === 20 Greek hymns === [[1000 Songs/Shepherd of Eager Youth (Clement of Alexandria)|Shepherd of Eager Youth (Clement of Alexandria)]] [[1000 Songs/Sunset to Sunrise Changes Now (Clement of Alexandria)|Sunset to Sunrise Changes Now (Clement of Alexandria)]] [[1000 Songs/Hymn to Christ the Savior "Bridle of colts untamed" (Clement of Alexandria)|Hymns to Christ the Savior "Bridle of colts untamed" (Clement of Alexandria)]] [[1000 Songs/O Gladsome Light of the Father Immortal (anon)|O Gladsome Light of the Father Immortal (anon)]] [[1000 Songs/Oxyrhynchus hymn fragment|Oxyrhynchus hymn fragment]] [[1000 Songs/What new mystery then is this (Melito of Sardis)|What new mystery then is this (Melito of Sardis)]] === 10 Odes of Solomon === [[1000 Songs/The Lord abides upon my head like a crown (ode 01)|The Lord abides upon my head like a crown (ode 01)]] [[1000 Songs/As the hand moves over the harp and the strings speak ode 06|As the hand moves over the harp and the strings speak ode 06]] [[1000 Songs/As the impulse of anger against evil (ode 07)|As the impulse of anger against evil (ode 07)]] [[1000 Songs/I stretched out my hands and approached my Lord (ode 42)|I stretched out my hands and approached my Lord (ode 42)]] === 20 Syrian hymns === [[1000 Songs/Lord Jesus Think on Me (Synesius of Cyrene)|Lord Jesus Think on Me (Synesius of Cyrene)]] [[1000 Songs/Light is arisen upon the just (Ephriam of Syria)|Light is arisen upon the just (Ephriam of Syria)]] [[1000 Songs/The Pearl (Ephriam of Syria)|The Pearl (Ephriam of Syria)]] === 15 Byzantine hymns (AD 400-1000) === [[1000 Songs/Only-begotten Son and Word of God|Only-begotten Son and Word of God]] [[1000 Songs/Save the world, O Savior (Romanus)|Save the world, O Savior (Romanus)]] [[1000 Songs/The Day of Resurrection (John of Damascus)|The Day of Resurrection (John of Damascus)]] [[1000 Songs/Come ye faithful raise the strain (John of Damascus)|Come ye faithful raise the strain (John of Damascus)]] [[1000 Songs/Trisagion Hymn (John Chrysostem)|Trisagion Hymn (John Chrysostem)]] [[1000 Songs/Let all mortal flesh keep silence (St. James)|Let all mortal flesh keep silence (St. James)]] [[1000 Songs/When we were buried with you in baptism (Vespers of Palm Sunday, apolytikion)|When we were buried with you in baptism (Vespers of Palm Sunday)]] === 15 Latin hymns (AD 400-1000) === [[1000 Songs/Lorica--I bind unto myself today|Lorica--I bund unto myself today (Patrick)]] [[1000 Songs/Aeterna rerum Conditer--Framer of the earth and sky (Ambrose)|Framer of the earth and sky (Ambrose)]] [[1000 Songs/Deus Creator omnium--Maker of all things God most high (Ambrose)|Deus Creator omnium--Maker of all things God most high (Ambrose)]] [[1000 Songs/Iam surgit hora tertia--Come thou savior of our race (Ambrose)|Iam surgit hora tertia--Come thou savior of our race (Ambrose)]] [[1000 Songs/Splendor paternae gloriae--O splendor of God's glory bright (Ambrose)|Splendor paternae gloriae--O splendor of God's glory bright (Ambrose)]] [[1000 Songs/Claro paschali gaudio--That Easter day with joy was bright|Claro paschali gaudio--That Easter day with joy was bright]] [[1000 Songs/Veni veni Emmanuel--O come O come Emmanuel|Veni veni Emmanuel--O come O come Emmanuel]] [[1000 Songs/Te Deum laudamus--We praise Thee O God|Te Deum laudamus--We praise Thee O God]] [[1000 Songs/Kyrie eleison--Lord have mercy|Kyrie eleison--Lord have mercy]] [[1000 Songs/Gloria in excelsis Deo--Glory to God in the highest|Gloria in excelsis Deo--Glory to God in the highest]] [[1000 Songs/Sanctus Sanctus Sanctus--Holy holy holy|Sanctus Sanctus Sanctus--Holy holy holy]] [[1000 Songs/Of the Father's love begotten (Prudentius)|Of the Father's love begotten (Prudentius)]] [[1000 Songs/All glory laud and honor (Theodulph of Orleans)|All glory laud and honor (Theodulph of Orleans)]] [[1000 Songs/Veni Creator Spiritus--Come Holy Ghost our souls inspire|Veni Creator Spiritus--Come Holy Ghost our souls inspire]] [[1000 Songs/jubilus|jubilus]] === 10 Late Middle Ages (AD 1000-1500) === [[1000 Songs/Victimae paschali laudes--Christians to the Paschal victim (Wipo of Burgundy)|Victimae paschali laudes--Christians to the Paschal victim (Wipo of Burgundy)]] [[1000 Songs/Dies irae--Day of wrath O day of mourning (Thomas of Celano)|Dies irae--Day of wrath O day of mourning (Thomas of Celano)]] [[1000 Songs/Stabat mater--At the cross her station keeping|Stabat mater--At the cross her station keeping]] [[1000 Songs/Alone thou goest forth O Lord (Peter Abelard)|Alone thou goest forth O Lord (Peter Abelard)]] [[1000 Songs/Jesus the very thought of Thee (Bernard of Clairvaux)|Jesus the very thought of Thee (Bernard of Clairvaux)]] [[1000 Songs/Jesus thou joy of loving hearts (Bernard of Clairvaux)|Jesus thou joy of loving hearts (Bernard of Clairvaux)]] [[1000 Songs/Jerusalem the golden (Bernard of Cluny)|Jerusalem the golden (Bernard of Cluny)]] [[1000 Songs/Symphony of the harmony of celestial revelations (Hildegard of Bingen)|Symphony of the harmony of celestial revelations (Hildegard of Bingen)]] [[1000 Songs/Adoro te devote--Humbly I adore Thee (Thomas Aquinas)|Adoro te devote--Humbly I adore Thee (Thomas Aquinas)]] === 10 Pre-Reformation hymns (AD 1200-1500) === [[1000 Songs/Canticle of the Sun (Francis of Assisi)|Canticle of the Sun (Francis of Assisi)]] [[1000 Songs/All creatures of our God and King (Francis of Assisi)|All creatures of our God and King (Francis of Assisi)]] [[1000 Songs/Discendi amor santo--Come down O love divine|Discendi amor santo--Come down O love divine]] == 100 Early Reformation (AD 1500-1600) == === 30 Lutheran chorales (Luther’s psalm settings included above in “psalms”) === [[1000 Songs/A Mighty Fortress Is Our God (Martin Luther)|A Mighty Fortress Is Our God (Martin Luther)]] [[1000 Songs/Out of the Depths I Cry (Martin Luther)|Out of the Depths I Cry (Martin Luther)]] [[1000 Songs/Christ Jesus Lay in Death's Strong Bands (Martin Luther)|Christ Jesus Lay in Death's Strong Bands (Martin Luther)]] [[1000 Songs/Come Holy Ghost God and Lord (Martin Luther)|Come Holy Ghost God and Lord (Martin Luther)]] [[1000 Songs/Christ Who Freed Our Souls From Danger (Martin Luther)|Christ Who Freed Our Souls From Danger (Martin Luther)]] (Justus Jonas) Hans Sachs Paul Speratus [[1000 Songs/All glory be to God on high (Nicolaus Decius)|All glory be to God on high (Nicolaus Decius)]] Nicolaus Hermann Paul Eber Nicolaus Selnecker Bartholomaeus Ringwaldt Martin Schalling Martin Moller [[1000 Songs/Wake awake for night is flying (Philipp Nicolai)|Wake awake for night is flying (Philipp Nicolai)]] [[1000 Songs/O morning star how fair and bright (Philipp Nicolai)|O morning star how fair and bright (Philipp Nicolai)]] === 10 Lutheran chorale tunes === [[1000 Songs/WACHET AUF (Philipp Nicolai)|WACHET AUF (Philipp Nicolai)]] [[1000 Songs/WIE SCHON LEUCHTET (Philipp Nicolai)|WIE SCHON LEUCHTET (Philipp Nicolai)]] [[1000 Songs/PASSION CHORALE (Hans Leo Hassler)|PASSION CHORALE (Hans Leo Hassler)]] [[1000 Songs/ST THEODULPH (Melchior Teschner)|ST THEODULPH (Melchior Teschner)]] [[1000 Songs/ES IST EIN ROS'ENTSPRUNGEN (Michael Praetorius)|ES IST EIN ROS'ENTSPRUNGEN (Michael Praetorius)]] === 10 Reformed (metricized psalter included above in “psalms”) === [[1000 Songs/OLD 100TH (Louis Bourgeois)|OLD 100TH (Louis Bourgeois)]] [[1000 Songs/OLD 124 (Louis Bourgeois)|OLD 124 (Louis Bourgeois)]] [[1000 Songs/NUNC DIMITTIS (Louis Bourgeois)|NUNC DIMITTIS (Louis Bourgeois)]] [[1000 Songs/RENDEZ A DIEU (Louis Bourgeois)|RENDEZ A DIEU (Louis Bourgeois)]] === 30 Anglican === Sternhold and Hopkins Sternhold and Hopkins Sternhold and Hopkins Sternhold and Hopkins Sternhold and Hopkins [[1000 Songs/While shepherds watched their flocks by night (Tate and Brady)|While shepherds watched their flocks by night (Tate and Brady)]] Tate and Brady Tate and Brady Tate and Brady [[1000 Songs/HANOVER (William Croft, Tate and Brady)|HANOVER (William Croft, Tate and Brady)]] ST ANNE (William Croft, Tate and Brady) Thomas Ravenscroft [[1000 Songs/TALLIS' ORDINAL (Thomas Tallis)|TALLIS' ORDINAL (Thomas Tallis)]] [[1000 Songs/TALLIS' CANON (Thomas Tallis)|TALLIS' CANON (Thomas Tallis)]] [[1000 Songs/CAITHNESS (Scottish Psalter, Edware Millar)|CAITHNESS (Scottish Psalter, Edware Millar)]] [[1000 Songs/DUNDEE (Scottish Psalter, Edware Millar)|DUNDEE (Scottish Psalter, Edware Millar)]] === 20 Catholic Counter-Reformation === [[1000 Songs/Silent night holy night (Joseph Mohr)|Silent night holy night (Joseph Mohr)]] [[1000 Songs/STABAT MATER (Jacopone da Todi, Pope Innocent III)|STABAT MATER (Jacopone da Todi), Pope Innocent III]] [[1000 Songs/LASST UNS ERFREUEN|LASST UNS ERFREUEN]] [[1000 Songs/STILLE NACHT (Franz Gruber)|STILLE NACHT (Franz Gruber)]] 200 Second-Generation Reformation (AD 1600-1700) === 50 Lutheran pietist/orthodox === [[1000 Songs/Sing praise to God who reigns above (Johann Jakob Schutz)|Sing praise to God who reigns above (Johann Jakob Schutz)]] [[1000 Songs/Praise to the Lord the Almighty (Joachim Neander)|Praise to the Lord the Almighty (Joachim Neander)]] Joachim Neander Joachim Neander Gerhard Tersteegen [[1000 Songs/My Jesus as thou wilt (Benjamin Schmolck)|My Jesus as thou wilt (Benjamin Schmolck)]] [[1000 Songs/Open now thy gates of beauty (Benjamin Schmolck)|Open now thy gates of beauty (Benjamin Schmolck)]] [[1000 Songs/We plow the fields and scatter (Matthias Claudius)|We plow the fields and scatter (Matthias Claudius)]] [[1000 Songs/Sinners Jesus will receive (Erdmann Neumeister)|Sinners Jesus will receive (Erdmann Neumeister)]] [[1000 Songs/Print thine image pure and holy (Thomas Hansen Kingo)|Print thine image pure and holy (Thomas Hansen Kingo)]] [[1000 Songs/Jesus thy boundless love to me (Paul Gerhardt)|Jesus thy boundless love to me (Paul Gerhardt)]] [[1000 Songs/All my heart this night rejoices (Paul Gerhardt)|All my heart this night rejoices (Paul Gerhardt)]] (Paul Gerhardt) (Paul Gerhardt) === 50 Anabaptist (Radical Reformed) === Menno Simons Michael Weisse [[1000 Songs/Jesus still lead on (Nicolaus von Zinzendorf)|Jesus still lead on (Nicolaus von Zinzendorf)]] [[1000 Songs/Christian heats in love united (Nicolaus von Zinzendorf)|Christian heats in love united (Nicolaus von Zinzendorf)]] [[1000 Songs/Jesus thy blood and righteousness (Nicolaus von Zinzendorf)|Jesus thy blood and righteousness (Nicolaus von Zinzendorf)]] [[1000 Songs/Thou great high priest (Nicolaus von Zinzendorf)|Thou great high priest (Nicolaus von Zinzendorf)]] Nicolaus von Zinzendorf Nicolaus von Zinzendorf === 50 Reformed (new generation metricized psalms, included above in “psalms”) === === 50 Anglican chorales and Moravian hymns === == 200 British hymns (AD 1700-1800) == === 50 Isaac Watts (150 psalms included above in “psalms”) === [[1000 Songs/Come ye that love the Lord (Isaac Watts)|Come ye that love the Lord (Isaac Watts)]] [[1000 Songs/O God our help in ages past (Isaac Watts)|O God our help in ages past (Isaac Watts)]] [[1000 Songs/I sing the mighty power of God (Isaac Watts)|I sing the mighty power of God (Isaac Watts)]] [[1000 Songs/Joy to the world the Lord is come (Isaac Watts)|Joy to the world the Lord is come (Isaac Watts)]] [[1000 Songs/When I survey the wondrous cross (Isaac Watts)|When I survey the wondrous cross (Isaac Watts)]] [[1000 Songs/Alas and did my Savior bleed (Isaac Watts)|Alas and did my Savior bleed (Isaac Watts)]] === 50 Charles Wesley hymns === [[1000 Songs/O for a thousand tongues (Charles Wesley)|O for a thousand tongues (Charles Wesley)]] [[1000 Songs/Come thou long-expected Jesus (Charles Wesley)|Come thou long-expected Jesus (Charles Wesley)]] [[1000 Songs/Hark the herald angels sing (Charles Wesley)|Hark the herald angels sing (Charles Wesley)]] [[1000 Songs/Arise my soul arise (Charles Wesley)|Arise my soul arise (Charles Wesley)]] [[1000 Songs/And can it be that I should gain (Charles Wesley)|And can it be (Charles Wesley)]] [[1000 Songs/Christ the Lord is risen today (Charles Wesley)|Christ the Lord is risen today (Charles Wesley)]] [[1000 Songs/Jesus lover of my soul (Charles Wesley)|Jesus lover of my soul (Charles Wesley)]] Jesus Lover of my soul === 50 Evangelical hymns === === 50 Olney hymns === [[1000 Songs/Amazing grace how sweet the sound (John Newton)|Amazing grace how sweet the sound (John Newton)]] [[1000 Songs/God moves in a mysterious way (William Cowper)|God moves in a mysterious way (William Cowper)]] [[1000 Songs/There is a fountain filled with blood (William Cowper)|There is a fountain filled with blood (William Cowper)]] == 200 American/English songs (AD 1800-1900) == === 50 singing school, spirituals and campmeeting songs === [[1000 Songs/Brethren we have met to worship (George Atkins)|Brethren we have met to worship (George Atkins)]] [[1000 Songs/WONDROUS LOVE|WONDROUS LOVE]] [[1000 Songs/Down By the Riverside|Down By the Riverside]] === 50 transitional, Oxford, Victorian hymns === [[1000 Songs/O Zion Haste (Mary Ann Thomson)|O Zion Haste (Mary Ann Thomson)]] [[1000 Songs/Jesus I Am Resting Resting (Jean Pigott)|Jesus I Am Resting Resting (Jean Pigott)]] === (50 early American psalmody and shaped-note tunes) === [[1000 Songs/LENOX (Lewis Edson)|LENOX (Lewis Edson)]] [[1000 Songs/When Jesus Wept (William Billings)|When Jesus Wept (William Billings)]] [[1000 HOLY MANNA|HOLY MANNA]] [[1000 Songs/AMHERST (William Billings)|AMHERST (William Billings)]] [[1000 Songs/BROOKFIELD (William Billings)|BROOKFIELD (William Billings)]] [[1000 Songs/CHESTER (William Billings)|CHESTER (William Billings)]] [[1000 Songs/LEBANON (William Billings)|LEBANON (William Billings)]] [[1000 Songs/GREENWICH (Daniel Read)|GREENWICH (Daniel Read)]] [[1000 Songs/WINDHAM (Daniel Read)|WINDHAM (Daniel Read)]] === (50 Lowell Mason tunes) === [[1000 Songs/BETHANY (Lowell Mason)|BETHANY (Lowell Mason)]] [[1000 Songs/OLIVET (Lowell Mason)|OLIVET (Lowell Mason)]] [[1000 Songs/DENNIS (Lowell Mason)|DENNIS (Lowell Mason)]] [[1000 Songs/BOYLSTON (Lowell Mason)|BOYLSTON (Lowell Mason)]] [[1000 Songs/HAMBURG (Lowell Mason)|HAMBURG (Lowell Mason)]] [[1000 Songs/WORK SONG (Lowell Mason)|WORK SONG (Lowell Mason)]] === 50 Sunday School songs === [[1000 Songs/CHINA (William Bradbury)|CHINA (William Bradbury)]] [[1000 Songs/BRADBURY (William Bradbury)|BRADBURY (William Bradbury)]] [[1000 Songs/SOLID ROCK (William Bradbury)|SOLID ROCK (William Bradbury)]] [[1000 Songs/Just As I Am (William Bradbury)|Just As I Am (William Bradbury)]] [[1000 Songs/Sweet hour of prayer (William Bradbury)|Sweet hour of prayer (William Bradbury)]] [[1000 Songs/What can wash away my sin (Robert Lowry)|What can wash away my sin (Robert Lowry)]] [[1000 Songs/Shall we gather at the river (Robert Lowry)|Shall we gather at the river (Robert Lowry)]] [[1000 Songs/We're marching to Zion (Robert Lowry)|We're marching to Zion (Robert Lowry)]] [[1000 Songs/Low in the grave He lay (Robert Lowry)|Low in the grave He lay (Robert Lowry)]] [[1000 Songs/To God be the glory (William H. Doane)|To God be the glory (William H. Doane)]] [[1000 Songs/Jesus keep me near the cross (William H. Doane)|Jesus keep me near the cross (William H. Doane)]] [[1000 Songs/Pass me not O gentle Savior (William H. Doane)|Pass me not O gentle Savior (William H. Doane)]] === 50 Gospel Songs (Sankey and Bliss) === [[1000 Songs/Man of sorrows what a name (P. P. Bliss)|Man of sorrows what a name (P. P. Bliss)]] [[1000 Songs/Wonderful words of life (P. P. Bliss)|Wonderful words of life (P. P. Bliss)]] [[1000 Songs/It is well with my soul (P. P. Bliss)|It is well with my soul (P. P. Bliss)]] [[1000 Songs/There were ninety and nine (Ira Sankey)|There were ninety and nine (Ira Sankey)]] == 200 American/English songs (AD 1900-2000) == === 20 hymns === How great the Father's Love for Us God is Here as We Your People We Will Glorify There is a Redeemer Great is Thy Faithfulness How Great Thou Art === 20 Black gospel === Precious Lord Take My Hand By and By O Happy Day Soon and Very Soon Through It All Tell Them Go === 20 Southern gospel === He Touched Me Because He Lives === 20 Sentimental (1900-1950) === Love Lifted Me Tell Mother I'll Be There [[1000 Songs/Tis So Sweet to Trust in Jesus (Louisa Stead)|Tis So Sweet to Trust in Jesus (Louisa Stead)]] === 20 Youth-oriented (1950-1970) === Christ For Me He's Everything to Me Pass It On Walkin on the Heaven Road Boys and Girls for Jesus === 20 Evangelism-oriented (1960-1980) === Just As I Am The Savior is Waiting Room At the Cross for You Yesterday He Died for Me === 20 Praise songs (1960-1990) === Majesty Alleluia === 20 presentational-to-congregational (1990-2010) === Your Love Oh Lord We Shall Behold Him === 20 postmodern self-orientation (2000-present) === How He Loves [[Category:Vocal music]] [[Category:Songs]] 6bngfpuj7k01stlsrow76e4jlh32w99 Template:Tsh 10 132051 2813141 2008-06-06T18:12:27Z Lenoxus 2241951 [[WP:AES|←]]Redirected page to [[Template:Template shortcut]] 2813141 wikitext text/x-wiki #redirect[[Template:Template shortcut]] {{r tsh}} 43nwevlcwbdziab5qs696z459ssvpit 2813142 2813141 2011-01-06T11:23:57Z Paine Ellsworth 184305 add Rcat 2813142 wikitext text/x-wiki #redirect[[Template:Template shortcut]]{{r tsh}}{{R template}} kbkbehrs3m2b7yz0oa6c4he1jmd99eq 2813143 2813142 2011-02-15T05:58:18Z SporkBot 784491 Remove template per [[Wikipedia:Templates for discussion/Log/2011 February 5|TFD outcome]] 2813143 wikitext text/x-wiki #redirect[[Template:Template shortcut]]{{r tsh}} 4ca3oogmjwhhanb1fqee8k2wrgt2w0r 2813144 2813143 2011-04-09T19:39:53Z Rich Farmbrough 32554 remove superflous : using [[Project:AWB|AWB]] 2813144 wikitext text/x-wiki #REDIRECT [[Template:Template shortcut]]{{r tsh}} 8nlbpinpa61jjt4zwsdhmgpzvjqtk4l 2813145 916644 2013-07-15T02:03:02Z JPaestpreornJeolhlna 718089 Fixed [[WP:RCAT|Rcats]] 2813145 wikitext text/x-wiki #REDIRECT [[Template:Template shortcut]] {{R abb}} {{R msp}} {{R tsh}} 3mpphlv2wjw52kyii645yn7ysfi2px1 2813146 2813145 2014-02-21T20:51:08Z Wbm1058 290388 remove template in the wrong namespace, removed: {{R abb}} using [[Project:AWB|AWB]] 2813146 wikitext text/x-wiki #REDIRECT [[Template:Template shortcut]] {{R msp}} {{R tsh}} 5ndekugtqxxn5oo01wr20947tdh38yr 2813147 2813146 2016-11-20T07:53:20Z Paradoctor 135346 use rcatsh 2813147 wikitext text/x-wiki #REDIRECT [[Template:Template shortcut]] {{rcatsh| {{R msp}} {{R tsh}} }} ioa2u7bu5539o0pak4nyq6j6ksdod25 2813148 2813147 2018-01-10T19:22:23Z Primefac 840239 Protected "[[Template:Tsh]]": [[WP:SEMI|semi-protecting]] highly-visible templates in response to recent template-space vandalism ([Edit=Require autoconfirmed or confirmed access] (indefinite) [Move=Require autoconfirmed or confirmed access] (indefinite)) 2813147 wikitext text/x-wiki #REDIRECT [[Template:Template shortcut]] {{rcatsh| {{R msp}} {{R tsh}} }} ioa2u7bu5539o0pak4nyq6j6ksdod25 2813149 2813148 2024-05-24T04:53:22Z Jlwoodwa 2967700 -rcat 2813149 wikitext text/x-wiki #REDIRECT [[Template:Template shortcut]] {{rcatsh| {{R tsh}} }} dbgssn12xup12fq1n1zhkcktef0d05l 2813150 2661856 2026-03-27T18:00:12Z MusikBot II 2893438 Changed protection settings for "[[Template:Tsh]]": [[Wikipedia:High-risk templates|High-risk template or module]]: 2516 transclusions ([[User:MusikBot II/TemplateProtector|more info]]) ([Edit=Require extended confirmed access] (indefinite) [Move=Require extended confirmed access] (indefinite)) 2813150 wikitext text/x-wiki #REDIRECT [[Template:Template shortcut]] {{rcatsh| {{R tsh}} }} dbgssn12xup12fq1n1zhkcktef0d05l 2813151 2813150 2026-06-06T08:30:19Z Jtneill 10242 10 revisions imported from [[:w:Template:Tsh]] 2813150 wikitext text/x-wiki #REDIRECT [[Template:Template shortcut]] {{rcatsh| {{R tsh}} }} dbgssn12xup12fq1n1zhkcktef0d05l Folk Music 0 136827 2812870 2459722 2026-06-05T15:47:16Z Kirby - Electrotechnics 3074947 changed to 'category:Music genres and styles' instead of category:Music 2812870 wikitext text/x-wiki This is the beginning of "The Department of Folk Music" in "The School of Creativity (just a stub for now). [[User:Ray Calvin Baker|Ray Calvin Baker]] ([[User talk:Ray Calvin Baker|talk]]) 00:30, 4 October 2012 (UTC) Many senior citizens remember folk songs they learned in their youth. They like to sing them, read stories about their origins and history, and some are even creative and bold enough to add verses to old folk songs, or to create entirely new songs to share with their friends. "Sing-along" activities can be greatly facilitated when song lyrics and arrangements for simple but suitable instrumental accompaniments can be made easily available. Wikimedia Commons seems like a convenient place to store such public-domain material. Projection of suitable graphical displays can be usefully employed by groups of any size, from simple equipment -- a laptop computer connected to a projector and to the internet. This Department is intended to support all of these purposes, and to encourage wider participation and use of the Wikiversity. It is my hope that this Department can be expanded to support the "sing-along" activities of the Senior Center of Caroline County (Maryland) In the town of Denton. I expect that the support and participation of my friends there can productively guide the expansion of this Department. [[User:Ray Calvin Baker|Ray Calvin Baker]] ([[User talk:Ray Calvin Baker|talk]]) 00:30, 4 October 2012 (UTC) ==She'll Be Comin' 'Round the Mountain== This is one of the songs we enjoy singing together at the Caroline County Senior Center. We have not yet performed it correctly, with the silly motions, but we have been having fun trying. ("Toot! Toot!" -- pull the cord on a riverboat's steam whistle) She'll be comin' 'round the mountain, when she comes: "toot! toot!"<br> She'll be comin' 'round the mountain, when she comes: "toot! toot!"<br> She'll be comin' 'round the mountain,<br> She'll be comin' 'round the mountain,<br> She'll be comin' 'round the mountain, when she comes: "toot! toot!" ("Whoa, back!" -- pull on the reins of the six white horses) She'll be drivin' six white horses when she comes: "Whoa, back! toot! toot!"<br> She'll be drivin' six white horses when she comes: "Whoa, back! toot! toot!"<br> She'll be drivin' six white horses,<br> She'll be drivin' six white horses,<br> She'll be drivin' six white horses when she comes: "Whoa, back! toot! toot!" ("Scratch, scratch!" -- old red flannel underwear itches!) She'll be wearin' old red flannels when she comes: "Scratch, scratch! Whoa, back! toot! toot!"<br> She'll be wearin' old red flannels when she comes: "Scratch, scratch! Whoa, back! toot! toot!"<br> She'll be wearin' old red flannels,<br> She'll be wearin' old red flannels,<br> She'll be wearin' old red flannels when she comes: "Scratch, scratch! Whoa, back! toot! toot!" ("Whack, whack!" -- chop off the head of the old red rooster.) Oh, we'll kill the old red rooster when she comes: "Whack, whack! Scratch, scratch, Whoa, back! toot! toot!"<br> Oh, we'll kill the old red rooster when she comes: "Whack, whack! Scratch, scratch, Whoa, back! toot! toot!"<br> Oh, we'll kill the old red rooster,<br> Oh, we'll kill the old red rooster,<br> Oh, we'll kill the old red rooster when she comes: "Whack, whack! Scratch, scratch, Whoa, back! toot! toot!" ("Yum! Yum!" -- rub your tummy) Oh, we'll all have chicken an' dumplin's when she comes: "Yum! Yum! Whack, whack! Scratch, scratch, Whoa, back! toot! toot!"<br> Oh, we'll all have chicken an' dumplin's when she comes: "Yum! Yum! Whack, whack! Scratch, scratch, Whoa, back! toot! toot!"<br> Oh, we'll all have chicken an' dumplin's,<br> Oh, we'll all have chicken an' dumplin's,<br> Oh, we'll all have chicken an' dumplin's when she comes: "Yum! Yum! Whack, whack! Scratch, scratch, Whoa, back! toot! toot!"<br> ("Hi, Babe!" -- wave, "hello!") Oh, we'll all come out to meet her, when she comes: "Hi, Babe!, Yum! Yum! Whack, whack! Scratch, scratch, Whoa, back, toot! toot!<br> Oh, we'll all come out to meet her, when she comes: "Hi, Babe!, Yum! Yum! Whack, whack! Scratch, scratch, Whoa, back, toot! toot!<br> Oh, we'll all come out to meet her,<br> Oh, we'll all come out to meet her,<br> Oh, we'll all come out to meet her, when she comes: "Hi, Babe!, Yum! Yum! Whack, whack! Scratch, scratch, Whoa, back, toot! toot! This has not yet been put into a form suitable for Wikimedia Commons. Content first, then nice formatting. [[User:Ray Calvin Baker|Ray Calvin Baker]] ([[User talk:Ray Calvin Baker|talk]]) 00:56, 4 October 2012 (UTC)<br> I finished the song. [[User:Ray Calvin Baker|Ray Calvin Baker]] ([[User talk:Ray Calvin Baker|talk]]) 02:01, 4 October 2012 (UTC) ==Back to the School of Creativity== Here is a link to take you directly back to the School of Creativity, in case you would like to explore other Departments there.<br> [[http://en.Wikiversity.org/wiki/School:Creativity]]<br> [[User:Ray Calvin Baker|Ray Calvin Baker]] ([[User talk:Ray Calvin Baker|talk]]) 21:15, 5 October 2012 (UTC) [[Category:Music genres and styles]] tbrwcb6azvll73of4d72ax9uy1ihs2y Understanding Arithmetic Circuits 0 139384 2812842 2812730 2026-06-05T13:47:30Z Young1lim 21186 /* Adder */ 2812842 wikitext text/x-wiki == Adder == * Binary Adder Architecture Exploration ( [[Media:Adder.20131113.pdf|pdf]] ) {| class="wikitable" |- ! Adder type !! Overview !! Analysis !! VHDL Level Design !! CMOS Level Design |- | '''1. Ripple Carry Adder''' || [[Media:VLSI.Arith.1A.RCA.20250522.pdf|A]]|| || [[Media:Adder.rca.20140313.pdf|pdf]] || [[Media:VLSI.Arith.1D.RCA.CMOS.20211108.pdf|pdf]] |- | '''2. Carry Lookahead Adder''' || [[Media:VLSI.Arith.1.A.CLA.20260109.pdf|org]], [[Media:VLSI.Arith.2A.CLA.20260605.pdf|A]], [[Media:VLSI.Arith.2B.CLA.20260605.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]] 59idgsgun7dqv3e3gggasz3umzzc21b 2813271 2812842 2026-06-06T08:42:57Z Young1lim 21186 /* Adder */ 2813271 wikitext text/x-wiki == Adder == * Binary Adder Architecture Exploration ( [[Media:Adder.20131113.pdf|pdf]] ) {| class="wikitable" |- ! Adder type !! Overview !! Analysis !! VHDL Level Design !! CMOS Level Design |- | '''1. Ripple Carry Adder''' || [[Media:VLSI.Arith.1A.RCA.20250522.pdf|A]]|| || [[Media:Adder.rca.20140313.pdf|pdf]] || [[Media:VLSI.Arith.1D.RCA.CMOS.20211108.pdf|pdf]] |- | '''2. Carry Lookahead Adder''' || [[Media:VLSI.Arith.1.A.CLA.20260109.pdf|org]], [[Media:VLSI.Arith.2A.CLA.20260606.pdf|A]], [[Media:VLSI.Arith.2B.CLA.20260606.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]] 1h0u11vp86m79n2bg40wy8lp6ir4ru6 Sound design 0 140727 2812996 2588834 2026-06-06T00:19:42Z Kirby - Electrotechnics 3074947 changed to 'category:Music technology' instead of category:Music 2812996 wikitext text/x-wiki Welcome to the '''Department of Sound Design''', part of the [[School:Music and Dance|School of Music and Dance]]. {{launch}} {{tertiary}} [[File:An-Najah University media room Victor 2011 -1-70.jpg|thumb|right|400px|A Sound Designer editing a sound file on a workstation]] ==Content summary== Sound design is the process of specifying, acquiring, manipulating or generating audio elements. It is employed in a variety of disciplines including filmmaking, television production, theatre, sound recording and reproduction, live performance, sound art, post-production and video game software development. Sound design most commonly involves the manipulation of previously composed or recorded audio, such as music and sound effects. In some instances, it may also involve the composition or manipulation of audio to create a desired effect or mood. A sound designer is one who practices the art of sound design. ==Prerequisites== * [[School:Physics and Astronomy|Physics]] * [[School:Mathematics|Mathematics]] ==Learning Objectives== # What are sound and timbre? What sounds are perceived as what emotions? # Operate a recording studio (like Audacity). Record samples and mix them into a track. # What are the different kinds of synthesis? What are VCO, VCF, VCA, LFO and ADSR? # Score or add effects to a song or a video. ==Learning Materials== ===Texts=== * [[b:Digital_Music_Composition|Digital Music Composition]] * [[b:Designing_Sound_in_SuperCollider|Designing Sound in SuperCollider]] * [[b:Sound_Synthesis_Theory|Sound Synthesis Theory]] * [[Motivation and emotion/Book/2011/Music and emotion|Motivation and Emotion: Music]] * [[Motivation_and_emotion/Textbook/Emotion/Music|Emotion Textbook:Music]] ===Sound Design Lessons=== * ''Characteristics of sound'' In music, timbre, also known as tone color or tone quality from psychoacoustics, is the quality of a musical note or sound or tone that distinguishes different types of sound production, such as voices and musical instruments, string instruments, wind instruments, and percussion instruments. The physical characteristics of sound that determine the perception of timbre include spectrum and envelope. It is often important in sound design to differentiate, characterize and control qualities such as timbre with technology. # Wikipedia article:[[w:Timbre]] # Wikipedia article:[[w:Sonic_Visualiser]] # Wikipedia article:[[w:Acousmatic sound]] # Wikipedia article:[[w:Acoustic ecology]] # Wikipedia article:[[w:Psychoacoustics]] * ''Introduction to Sound Recording'' Sound recording and reproduction is an electrical or mechanical inscription and re-creation of sound waves, such as spoken voice, singing, instrumental music, or sound effects. The two main classes of sound recording technology are analog recording and digital recording. Sound Recording essentially means capturing a piece of sound onto a storage media, so as to archive it and review it afterwards. It is not just storing sounds but also maintaining its quality, i.e., the playback of the recorded sound must be an exact imitation of the original sound that was recorded. To do this we need the proper equipment, and some essential skills and knowledge. # Wikipedia article:[[w:Sound recording and reproduction]] # Wikiversity course: [[Getting started with sound recording]] # Wikiversity course: [[Audacity]] # Wikipedia article:[[w:Foley_(filmmaking)]] * ''Sound Editing and Synthesis'' Audio signal processing sometimes referred to as audio processing, is the intentional alteration of auditory signals, or sound, often through an audio effect or effects unit. As audio signals may be electronically represented in either digital or analog format, signal processing may occur in either domain. Analog processors operate directly on the electrical signal, while digital processors operate mathematically on the digital representation of that signal. # Wikipedia article:[[w:Audio signal processing]] # Wikipedia article:[[w:Synthesizer]] # Wikipedia article:[[w:Karplus–Strong_string_synthesis]] # [http://soundation.com/studio Soundation free DAW with Synthesis] # Wikiversity course: [[Digital Audio Workstation|DAW]] * ''Production for Sound Design'' In the application of sound design, it is important to be aware of the human perception of sound and music. Moreover, it is critical that a student of sound design study and apply the principles at hand, therefore a study of Film Scoring and the lesson items there may, for the most part, be very useful to the student of sound design. # Wikipedia article:[[w:Sonification]] # Wikipedia article:[[w:Algorithmic composition]] # [http://kc.koncon.nl/downloads/ACToolbox/ ACToolbox free Algorithmic Composition tool (for Mac)] # Wikipedia article:[[w:Leitmotif]] # Wikiversity course: [[Film scoring]] ==Assignments== ===Preliminary Study guide=== # Wikipedia article:[[w:Sound_design]] ===Activities=== * Use [http://quizlet.com/1881636/sound-design-unit-1-flash-cards/ these] and [http://quizlet.com/900141/sound-design-quiz-2-flash-cards/ these] or [http://quiz.thefullwiki.org/Sound_design these] flashcards and quizzes ==References== Additional helpful readings include: * [http://www.aact.org/people/sounddesigner.html A job description of a Sound Designer] * [http://www.thefullwiki.org/Sound_design A community sound design portal] * [https://wiki.brown.edu/confluence/display/Fall2010MUSC1250S01/Syllabus A Brown U. Syllabus] ==Active participants== Please sign up!! Active participants in this [[Portal:Learning Projects#Learning Groups|Learning Group]] * [[User:gustable]] * [[user:...|... ==See also== * [[Portal:Audio engineering]] [[Category:Music technology]] [[Category:Sound|design]] rpt4rw1aysk6m9xqq3deogzvwii155v Template:No redirect 10 148600 2813084 2395258 2025-06-27T17:59:07Z Paine Ellsworth 184305 per edit request on talk page - modify to add backlinks if using an external URL 2813084 wikitext text/x-wiki {{safesubst:<noinclude/>#if: {{safesubst:<noinclude/>#invoke:Redirect|isRedirect|{{{1}}}}} | <span class="plainlinks">[{{safesubst:<noinclude/>fullurl:{{{1}}}|redirect=no}} {{{2|{{{1}}}}}}]</span><span style="display: none">[[:{{{1}}}]]</span> | {{safesubst:<noinclude/>#if:{{{2|}}}|[[:{{safesubst:<noinclude/>FULLPAGENAME:{{{1}}}}}|{{{2}}}]]|[[:{{safesubst:<noinclude/>FULLPAGENAME:{{{1}}}}}]]}} }}<noinclude> {{documentation}} </noinclude> g461hvbn2htbyts2r1a6wp14lxhkibr 2813085 2813084 2025-06-28T05:18:36Z Jonesey95 831143 This colon is causing Linter double-colon errors in a dozen or more pages, including Template:Singular and plural/doc 2813085 wikitext text/x-wiki {{safesubst:<noinclude/>#if: {{safesubst:<noinclude/>#invoke:Redirect|isRedirect|{{{1}}}}} | <span class="plainlinks">[{{safesubst:<noinclude/>fullurl:{{{1}}}|redirect=no}} {{{2|{{{1}}}}}}]</span><span style="display: none">[[{{{1}}}]]</span> | {{safesubst:<noinclude/>#if:{{{2|}}}|[[:{{safesubst:<noinclude/>FULLPAGENAME:{{{1}}}}}|{{{2}}}]]|[[:{{safesubst:<noinclude/>FULLPAGENAME:{{{1}}}}}]]}} }}<noinclude> {{documentation}} </noinclude> 3msltq2spqk7jkcqdh5ees1s93leh57 2813086 2813085 2025-07-21T01:58:16Z Anomie 27260 Removal of the colon makes uses of the template to link file pages register as a use rather than a link, which seems to have made [[User:JJMC89 bot]] misbehave. Reverting the whole thing for now. 2813086 wikitext text/x-wiki {{safesubst:<noinclude/>#if: {{safesubst:<noinclude/>#invoke:Redirect|isRedirect|{{{1}}}}} | <span class="plainlinks">[{{safesubst:<noinclude/>fullurl:{{{1}}}|redirect=no}} {{{2|{{{1}}}}}}]</span> | {{safesubst:<noinclude/>#if:{{{2|}}}|[[:{{safesubst:<noinclude/>FULLPAGENAME:{{{1}}}}}|{{{2}}}]]|[[:{{safesubst:<noinclude/>FULLPAGENAME:{{{1}}}}}]]}} }}<noinclude> {{documentation}} </noinclude> 2qaonlqtyk07btxy9fviektgkcflxxv 2813087 2813086 2025-08-08T22:09:53Z Jonesey95 831143 Back a couple versions per the talk page 2813087 wikitext text/x-wiki {{safesubst:<noinclude/>#if: {{safesubst:<noinclude/>#invoke:Redirect|isRedirect|{{{1}}}}} | <span class="plainlinks">[{{safesubst:<noinclude/>fullurl:{{{1}}}|redirect=no}} {{{2|{{{1}}}}}}]</span><span style="display: none">[[:{{{1}}}]]</span> | {{safesubst:<noinclude/>#if:{{{2|}}}|[[:{{safesubst:<noinclude/>FULLPAGENAME:{{{1}}}}}|{{{2}}}]]|[[:{{safesubst:<noinclude/>FULLPAGENAME:{{{1}}}}}]]}} }}<noinclude> {{documentation}} </noinclude> g461hvbn2htbyts2r1a6wp14lxhkibr 2813088 2813087 2025-11-25T17:21:31Z GTrang 1731925 Removed unnecessary usages of FULLPAGENAME. 2813088 wikitext text/x-wiki {{safesubst:<noinclude/>#if: {{safesubst:<noinclude/>#invoke:Redirect|isRedirect|{{{1}}}}} | <span class="plainlinks">[{{safesubst:<noinclude/>fullurl:{{{1}}}|redirect=no}} {{{2|{{{1}}}}}}]</span><span style="display: none">[[:{{{1}}}]]</span> | {{safesubst:<noinclude/>#if:{{{2|}}}|[[:{{{1}}}|{{{2}}}]]|[[:{{{1}}}]]}} }}<noinclude> {{documentation}} </noinclude> bgz65ivjfqe1bqd4s4r85cfxhifyi73 2813089 2813088 2025-11-25T21:47:15Z GTrang 1731925 Used FULLPAGENAME again to fix the "Multi colon escape" lint error. 2813089 wikitext text/x-wiki {{safesubst:<noinclude/>#if: {{safesubst:<noinclude/>#invoke:Redirect|isRedirect|{{{1}}}}} | <span class="plainlinks">[{{safesubst:<noinclude/>fullurl:{{{1}}}|redirect=no}} {{{2|{{{1}}}}}}]</span><span style="display: none">[[:{{{1}}}]]</span> | [[:{{safesubst:<noinclude/>FULLPAGENAME:{{{1}}}}}|{{{2|{{{1}}}}}}]] }}<noinclude> {{documentation}} </noinclude> 5pw8ky320dln3bgzenyzuppz9t6twll 2813090 2813089 2026-06-06T08:30:13Z Jtneill 10242 6 revisions imported from [[:w:Template:No_redirect]] 2813089 wikitext text/x-wiki {{safesubst:<noinclude/>#if: {{safesubst:<noinclude/>#invoke:Redirect|isRedirect|{{{1}}}}} | <span class="plainlinks">[{{safesubst:<noinclude/>fullurl:{{{1}}}|redirect=no}} {{{2|{{{1}}}}}}]</span><span style="display: none">[[:{{{1}}}]]</span> | [[:{{safesubst:<noinclude/>FULLPAGENAME:{{{1}}}}}|{{{2|{{{1}}}}}}]] }}<noinclude> {{documentation}} </noinclude> 5pw8ky320dln3bgzenyzuppz9t6twll Template:Yesno 10 148697 2813042 2289116 2025-12-06T20:28:33Z Pppery 2799023 Copy from sandbox per request 2813042 wikitext text/x-wiki {{<includeonly>safesubst:</includeonly>#switch: {{<includeonly>safesubst:</includeonly>lc: {{{1|¬}}} }} |no |n |f |false |off |0 = {{{no|}}} | = {{{blank|{{{no|}}}}}} |¬ = {{{¬|}}} |yes |y |t |true |on |1 = {{{yes|yes}}} |#default = {{{def|{{{yes|yes}}}}}} }}<noinclude> {{Documentation}} </noinclude> ap97tktff18ii0muqd5f4eezydumg5v 2813043 2813042 2026-06-06T08:30:11Z Jtneill 10242 1 revision imported from [[:w:Template:Yesno]] 2813042 wikitext text/x-wiki {{<includeonly>safesubst:</includeonly>#switch: {{<includeonly>safesubst:</includeonly>lc: {{{1|¬}}} }} |no |n |f |false |off |0 = {{{no|}}} | = {{{blank|{{{no|}}}}}} |¬ = {{{¬|}}} |yes |y |t |true |on |1 = {{{yes|yes}}} |#default = {{{def|{{{yes|yes}}}}}} }}<noinclude> {{Documentation}} </noinclude> ap97tktff18ii0muqd5f4eezydumg5v Applied mathematics/Music 0 157114 2812978 2207541 2026-06-05T22:13:22Z Kirby - Electrotechnics 3074947 changed to 'category:Music related projects' instead of category:Music 2812978 wikitext text/x-wiki Music and its chords can be geometrically mapped out.<ref name="beautiful math"/> Visual differences can be detected between different genres or musicians.<ref name="beautiful math"/> ==See also== * [[w:Mozart effect]] ==References== {{reflist|refs= <ref name="beautiful math">{{citation|title=Music Reduced to Beautiful Math|url=http://www.livescience.com/4929-music-reduced-beautiful-math.html|author=Moskowitz, Clara|date=6 May 2008|publisher=Live Science}}</ref> }} [[Category:Mathematics]] [[Category:Music related projects]] mebuee1vd9omopdft4bya8sy7j0hrdv Music/Software/PWGL 0 164792 2812855 1233655 2026-06-05T15:15:24Z Kirby - Electrotechnics 3074947 changed to 'category:Music technology' instead of category:Music 2812855 wikitext text/x-wiki '''PWGL''' is a freeware software application that facilitates computer-assisted composition of music. The homepage for PWGL is http://www2.siba.fi/PWGL/. The archive of the PWGL mailing list is here: http://blog.gmane.org/gmane.lisp.pwgl.user This wiki will be built upon in coming months by students and faculty of the City University of New York as part of a doctoral course on computer-assisted composition. Materials regarding essentials of operating the software as well as tutorials on realizing specific composition techniques will be posted here and in supporting pages. '''About PWGL''' PWGL is a program that gives the user a graphical interface to doing computer programming to create music. The interface has been designed for musicians, with many objects that allow one to see, hear, and manipulate musical materials. PWGL's interface is similar to other applications, including OpenMusic, Max/MSP, and Pd. It is most similar to OpenMusic, because both share lineage as successors to the 1980s-90s application Patchwork (the PW in PWGL refers to Patchwork.) For those familiar with Max/MSP or Pd, the biggest difference to know about PWGL is that generally all user patches are organized in the form of a tree, with many computations that happen in the "leaves" and "branches" that feed into one another and end at the bottom of the patch with one object that is the "root." The user activates the patch by evaluating this root object, which then calls all the other objects successively up the tree to the leaves, in a recursive fashion. The outermost leaves then evaluate and feed their results back down. This happens through all levels of the patch back to the root object. When the root object evaluates, it sends the final answer to the user. Users may evaluate the patch at locations other than the "root" object. The object called for evaluation will call up its own branches and leaves and output its result to the user. Other branches of the patch will not evaluate, nor will levels of the patch below this node. To evaluate an object, select it and hit 'v' (for "eValuate"!). Instructions for how to select objects are below. '''Running PWGL''' PWGL opens with two windows that appear initially blank. A larger window usually named "PWGL1" and a smaller one at the bottom called "PWGL output." The top one is the main window where you will do nearly all work, at least until you become an advanced user. The lower window outputs results when you evaluate your patch, and also outputs error messages. '''Basic Interface Navigation:''' A short video introduction to the PWGL interface is here: http://youtu.be/IzhOYpidMtY 1. '''To add an object to your patch''', simply right-click (on a Mac, control-click) and choose from the menu that appears. 2. '''The cursor’s various shapes:''' • Arrow: When moving around empty space in your patch. • Open hand: When near edge of object box. This signifies that you may move the location of the box. • Hand with pointing finger: This indicates that user may double-click to view an interface hidden within the selected box (such as with chord editor object.) • Left & right arrows: This allows user to move dividing line between left and right windows in an object (for objects with these). • Right arrow: allows user to adjust horizontal size of box. • Downward arrow: allows user to adjust vertical size of box. • Thin square crosshairs (plus sign): allows user to adjust vertical AND horizontal size of box. 3. '''To enter data into the arithmetic objects''', hover cursor over the input you want to adjust, click the left mouse button and slide up or down. A second method is to right-click on the input and select “edit-input box” from the pop-up menu. 4. '''To enter a value into object that features direct input of lists''' (such as the value object, etc.), double-click on it to enter the desired data object, type in value(s) desired, and hit ENTER or RETURN to exit the entry mode. 5. '''To select an object box''', draw a circle around it so that its outline turns red. 6. '''Actions possible when an object has been selected:''' • Select a box and hit ‘v’ key to evaluate the patch up to that point. • Select a box and hit ‘d’ to get documentation on that object. • Select a box and hit ‘t’ to see whether there is a tutorial patch for it. • For boxes with interface (such as chord editor), double-click to open it. Selecting box and hitting ‘m’ will let you see into it without double-clicking. • Hit spacebar to play musical contents of a box. • Enlarge boxes by hovering over lower right corner until cursor changes to a double arrow. • On boxes with an arrow pointing down in their lower left corner, typing + will add another input to the box and typing - will take one away. '''ENP Score Notation''' One of the noteworthy aspects of PWGL is its integration of music notation with algorithmic methods. PWGL's notation system is named "Expressive Notation Package," (ENP) and was created by composer/developer Mika Kuuskankare. ENP allows musicians to see musical ideas in the form of traditional or graphic music notation, to alter the notation manually, and also to algorithmically generate it and to analyze it for algorithmic alterations. The ENP score format is particular to PWGL but may be exported to other software in a variety of formats, including MIDI, Music XML, PNG, and Postscript. Three short video introductions to ENP notation is available here: http://youtu.be/wEdBimCpO_o, http://youtu.be/yAlw7SGVvSk, and http://youtu.be/t8-RZob4nIg . [[Category:Music technology]] 20evmk4meononjpkasg3jng60mr207h Anthology of Chromatic Harmony 0 168582 2812865 2532600 2026-06-05T15:42:21Z Kirby - Electrotechnics 3074947 Removed from the main "Category:Music" 2812865 wikitext text/x-wiki {{TOCright}} This page serves as an open-source anthology of musical examples containing diatonic and chromatic harmonies. Contributions for all types of music are welcome. At a minimum, each contribution must specify the precise location of occurrence. Further, every effort should be made to find a reliable source for each contribution. You may place a "vote of no confidence" in a contribution by reformatting the entry using ''italics.'' ''Italics indicate that an entry may be unreliable.'' Votes of no confidence will be periodically reviewed, and if deemed to be erroneous, will be moved to the discussion tab of this page before being deleted. == Secondary dominants == {| class="wikitable" |+ V/2 |- ! Song ! Performer/Composer ! Approximate Location ! Lyric Marker ! Chordal Context ! Citation |- |[[Wikipedia:Don%27t Stop Me Now|"Don't Stop Me Now"]] |[[Wikipedia:Queen (band)|Queen]] |Verse |"in ecstacy" |F: gm D7 gm |<ref>http://www.queensongs.info/the-book/songwriting-analyses/no-synth-era/jazz/dont-stop-me-now.html</ref> |- |[[Wikipedia:My Way|"My way"]] |[[Wikipedia:Frank sinatra|Frank Sinatra]] |Verse |"The final curtain" |D Dmaj7 D7 B7 em7 |<ref>http://www.hooktheory.com/theorytab/view/frank-sinatra/my-way</ref> |- |[[Wikipedia:Tears in Heaven|"Tears in Heaven"]] |[[Wikipedia:Eric Clapton|Eric Clapton]] |Chorus |"Carry on" |A: F# Bm E7 A | |} {| class="wikitable" |+ V/3 |- ! Song ! Performer/Composer ! Approximate Location ! Lyric Marker ! Chordal Context ! Citation |- |''"Bastard"'' |[[Wikipedia:Ben Folds|''Ben Folds'']] |''Verse'' |''"ties and suit"'' |''em: D G D G D G'' | |} {| class="wikitable" |+ V/4 |- ! Song ! Performer/Composer ! Approximate Location ! Lyric Marker ! Chordal Context ! Citation |- |[[Wikipedia:Baby, It%27s Cold Outside|''"Baby it's cold outside"'']] |[[Wikipedia:Frank Loesser|''Frank Loesser'']] |''Verse'' |''"I'll hold your hands"'' |''C: gm7 C7 F'' | |- |[[Wikipedia:Desperado (song)|"Desperado"]] |[[Wikipedia:Eagles (band)|The Eagles]] |Verse |"Why don't you..." |G: G G7 C |<ref> https://www.hooktheory.com/theorytab/common-chord-progressions/33 </ref> |- |[[Wikipedia:Don%27t Stop Me Now|"Don't Stop Me Now"]] |[[Wikipedia:Queen (band)|Queen]] |Verse |"turning inside out" |F: F F7 Bb |<ref>http://www.queensongs.info/the-book/songwriting-analyses/no-synth-era/jazz/dont-stop-me-now.html</ref> |- |[[Wikipedia:Don%27t Think Twice, It%27s All Right|"Don't Think Twice It's Alright"]] | [[Wikipedia:Bob Dylan|Bob Dylan]] |Verse |"At the break of dawn" |''E: E E7 A'' |<ref>http://www.ethanhein.com/wp/2011/secondary-dominants/</ref> |- |[[Wikipedia:Hey Jude|"Hey Jude"]] |[[Wikipedia:The Beatles|The Beatles]] |End of Verse |"Make it better..." |F: F F7 Bb | <ref>http://www.icce.rug.nl/~soundscapes/DATABASES/AWP/hj.shtml</ref> |- |[[Wikipedia:Lady Godiva (song)|"Lady Godiva"]] |[[Wikipedia:Peter and Gordon|Peter and Gordon]] |Verse |"Hangin down around" |G: G G7 A7 D7 | |- |[[Wikipedia:Lucille %28Little Richard song%29|"Lucille"]] |[[Wikipedia:Little Richard|"Little Richard"]] |Bridge |"I asked her friends" |C: C C7 | |- |[[Wikipedia:My Way|"My way"]] |[[Wikipedia:Frank sinatra|Frank Sinatra]] |Verse |"So I face" |D Dmaj7 D7 B7 em7 |<ref>http://www.hooktheory.com/theorytab/view/frank-sinatra/my-way</ref> |- |[[Wikipedia:The Ballad of John and Yoko|"The Ballad of John and Yoko"]] |[[Wikipedia:The Beatles|The Beatles]] |Verse |"The man in the Mac" |E: E E7 A | <ref>http://www.icce.rug.nl/~soundscapes/DATABASES/AWP/tbojay.shtml </ref> |- | [[Wikipedia:You Never Give Me Your Money|"You Never Give Me Your Money"]] | [[Wikipedia:The Beatles|The Beatles]] | Verse | "Pay No Rent..." | C-E-a-C7-F-G-C | <ref>http://www.icce.rug.nl/~soundscapes/DATABASES/AWP/yngmym.shtml</ref> |- |} {| class="wikitable" |+ V/5 |- ! Song ! Performer/Composer ! Approximate Location ! Lyric Marker ! Chordal Context ! Citation |- |[[Wikipedia:Lady Godiva (song)|"Lady Godiva"]] |[[Wikipedia:Peter and Gordon|Peter and Gordon]] |Verse |"Knees" |G: G G7 A7 D7 | |- |[[Wikipedia:Saturday in the Park (song)|"Saturday in the Park"]] |[[Wikipedia:Chicago (band)|Chicago]] |Intro & Verse |"In the Park" | am7 D9 G11 C |<ref>http://www.angelfire.com/fl4/moneychords/circleprogressions.html</ref> |- |[[Wikipedia:That's Life (song)|"That's Life"]] |[[Wikipedia:Frank Sinatra|Frank Sinatra]] | Verse |"When I'm back on top" |G: em A7 D G | |} {| class="wikitable" |+ V/6 |- ! Song ! Performer/Composer ! Approximate Location ! Lyric Marker ! Chordal Context ! Citation |- | [[Wikipedia:Creep (Radiohead song)|"Creep"]] | [[Wikipedia:Radiohead|Radiohead]] | Verse | "Couldn't look you in the eye..." | G-B7-C-Cm | <ref>http://forum.emusictheory.com/read.php?5,2444,2445</ref> |- |[[Wikipedia:Desperado (song)|"Desperado"]] |[[Wikipedia:Eagles (band)|The Eagles]] |Verse |"Things that are pleasin'" |G: G B7 Em7 | |- |"Honest Songs" |[[Wikipedia:Noah Gundersen|Noah Gundersen]] |Chorus |"And the sound it makes" |G: B7 em C G | |- |[[Wikipedia:That's Life (song)|"That's Life"]] |[[Wikipedia:Frank Sinatra|Frank Sinatra]] | Verse |"All the people say" |G: G B7 em | |} {| class="wikitable" |+ V/b7 |- ! Song ! Performer/Composer ! Approximate Location ! Lyric Marker ! Chordal Context ! Citation |} == Secondary leading tone chords == {| class="wikitable" |+ vii°/2 |- ! Song ! Performer/Composer ! Approximate Location ! Lyric Marker ! Chordal Context ! Citation |} {| class="wikitable" |+ vii°/3 |- ! Song ! Performer/Composer ! Approximate Location ! Lyric Marker ! Chordal Context ! Citation |} {| class="wikitable" |+ vii°/4 |- ! Song ! Performer/Composer ! Approximate Location ! Lyric Marker ! Chordal Context ! Citation |} {| class="wikitable" |+ vii°/5 |- ! Song ! Performer/Composer ! Approximate Location ! Lyric Marker ! Chordal Context ! Citation |} {| class="wikitable" |+ vii°/6 |- ! Song ! Performer/Composer ! Approximate Location ! Lyric Marker ! Chordal Context ! Citation |- |[[Wikipedia:Lucky (Jason Mraz song)|"Lucky]] |[[Wikipedia:Jason Mraz|Jason Mraz]] |End of verse |"When life gets hard..." | dm7 G G#° am | |} {| class="wikitable" |+ vii°/b7 |- ! Song ! Performer/Composer ! Approximate Location ! Lyric Marker ! Chordal Context ! Citation |} == Borrowed chords == {| class="wikitable" |+ I (Picardy third) ! Song ! Performer/Composer ! Approximate Location ! Lyric Marker ! Chordal Context ! Citation |- | [[Wikipedia:And I Love Her|"And I Love Her"]] | [[Wikipedia:The Beatles|The Beatles]] | Final Chord | | gm-F-gm-D | <ref> http://www.icce.rug.nl/~soundscapes/DATABASES/AWP/ailh.shtml </ref> |} {| class="wikitable" |+ i (borrowed minor tonic) |- ! Song ! Performer/Composer ! Approximate Location ! Lyric Marker ! Chordal Context ! Citation |} {| class="wikitable" |+ ii° |- ! Song ! Performer/Composer ! Approximate Location ! Lyric Marker ! Chordal Context ! Citation |} {| class="wikitable" |+ bIII |- ! Song ! Performer/Composer ! Approximate Location ! Lyric Marker ! Chordal Context ! Citation |} {| class="wikitable" |+ iv |- ! Song ! Performer/Composer ! Approximate Location ! Lyric Marker ! Chordal Context ! Citation |- |[[Wikipedia:My Way|"My way"]] |[[Wikipedia:Frank sinatra|Frank Sinatra]] |Verse |"Each and every highway" |D D7 G Gm |<ref>http://www.hooktheory.com/theorytab/view/frank-sinatra/my-way</ref> |} {| class="wikitable" |+ iv7 |- ! Song ! Performer/Composer ! Approximate Location ! Lyric Marker ! Chordal Context ! Citation |} {| class="wikitable" |+ v |- ! Song ! Performer/Composer ! Approximate Location ! Lyric Marker ! Chordal Context ! Citation |- |[[Wikipedia:Tears in Heaven|''"Tears in Heaven"'']] |[[Wikipedia:Eric Clapton|''Eric Clapton'']] |''Chorus'' |''"And carry on"'' |''A: Em F#'' | |} {| class="wikitable" |+ bVI |- ! Song ! Performer/Composer ! Approximate Location ! Lyric Marker ! Chordal Context ! Citation |} {| class="wikitable" |+ vi° |- ! Song ! Performer/Composer ! Approximate Location ! Lyric Marker ! Chordal Context ! Citation |} {| class="wikitable" |+ bVII |- ! Song ! Performer/Composer ! Approximate Location ! Lyric Marker ! Chordal Context ! Citation |} == Modulations == {| class="wikitable" |+ Common Chord |- ! Song ! Performer/Composer ! Approximate Location ! Lyric Marker ! Chordal Context ! Citation |} {| class="wikitable" |+ Common Tone |- ! Song ! Performer/Composer ! Approximate Location ! Lyric Marker ! Chordal Context ! Citation |} {| class="wikitable" |+ Chromatic |- ! Song ! Performer/Composer ! Approximate Location ! Lyric Marker ! Chordal Context ! Citation |} {| class="wikitable" |+ Chromatic 3rd |- ! Song ! Performer/Composer ! Approximate Location ! Lyric Marker ! Chordal Context ! Citation |- |[[Wikipedia:Tears in Heaven|"Tears in Heaven"]] |[[Wikipedia:Eric Clapton|Eric Clapton]] |After 2nd Chorus |"Time can bring you down" |''A: C G/B'' | |} {| class="wikitable" |+ Direct |- ! Song ! Performer/Composer ! Approximate Location ! Lyric Marker ! Chordal Context ! Citation |} {| class="wikitable" |+ Sequential |- ! Song ! Performer/Composer ! Approximate Location ! Lyric Marker ! Chordal Context ! Citation |} {| class="wikitable" |+ Pump up modulation ("Truck driver's gear change") ! Song ! Performer/Composer ! Approximate Location ! Lyric Marker ! Chordal Context ! Citation |} == Neapolitan chords == {| class="wikitable" |+ Neapolitan 6th ! Song ! Performer/Composer ! Approximate Location ! Lyric Marker ! Chordal Context ! Citation |} == Augmented sixth chords == {| class="wikitable" |+ German +6 ! Song ! Performer/Composer ! Approximate Location ! Lyric Marker ! Chordal Context ! Citation |- |[[Wikipedia:Oh! Darling"|Oh! Darling"]] |[[Wikipedia:The Beatles|The Beatles]] |Bridge |"Fell down and..." |A: E7 F7 E7 |<ref>http://www.icce.rug.nl/~soundscapes/DATABASES/AWP/od.shtml</ref> |} {| class="wikitable" |+ French +6 ! Song ! Performer/Composer ! Approximate Location ! Lyric Marker ! Chordal Context ! Citation |} {| class="wikitable" |+ Italian +6 ! Song ! Performer/Composer ! Approximate Location ! Lyric Marker ! Chordal Context ! Citation |} [[Category:Harmony]] towagwiypewfz4jep20otq3lu91h1k3 Complex analysis in plain view 0 171005 2812847 2812735 2026-06-05T14:00:54Z Young1lim 21186 /* Geometric Series Examples */ 2812847 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.20260605.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]] sy2lqqrw56e6hy1ujzcsl9vwti0077f 2813276 2812847 2026-06-06T09:31:23Z Young1lim 21186 /* Geometric Series Examples */ 2813276 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.20260606.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]] h0yts4wqbejcyw1nluucyye71vodab4 Category:Music styles 14 206085 2812864 1499490 2026-06-05T15:37:23Z Kirby - Electrotechnics 3074947 changed to 'category:Music genres and styles' instead of category:Music 2812864 wikitext text/x-wiki [[Category:Music genres and styles]] qhimz008596xnk5444wagy28xmwcppl Category:Harmony 14 208497 2812971 1528543 2026-06-05T22:05:17Z Kirby - Electrotechnics 3074947 changed to 'category:Music theory' instead of category:Music 2812971 wikitext text/x-wiki [[Category:Music theory]] sxoiuy3uh9mowerc6l2cwxuz2atgy8j Music and Songwriting 0 211847 2812967 2455099 2026-06-05T21:57:29Z Kirby - Electrotechnics 3074947 Removed from the main "Category:Music" 2812967 wikitext text/x-wiki == Otherwise Known as the Will Hunting School of Music == “You wasted $150,000 on an education you coulda got for $1.50 in late fees at the public library.” ― Matt Damon, Good Will Hunting == Intro == Welcome to the collaborative school for musical do-how! The goal of this project is to create an easily accessible body of knowledge on how to learn and perform any and all kinds of music. We are also aiming to build a community of musicians, everyone from amateurs to professionals to visionaries, who can support each other and create a whole that is greater than the sum of its parts. == Collaborative Learning Projects == [[/20 Hours of Film]] [[/Fingerstyle Improvisation - John Mayer Style]] [[/Collaborative Songwriting]] [[/Dub FX Looping Style]] == Matrix of Musical Technique == [[/The Swift Equation - A Guide To Songwriting and Song Improvisation]] [[/The Songwriter's Grooves Project - A Guide to Singing and Playing Guitar]] [[/The Infinite Grid - A Guide to Melody and Harmony]] [[/Impulse Geometry - A Guide to Musical Rhythm]] [[/Street Religion - A Guide to Lyric writing and improvisation]] [[/Man/Machine - A Guide to Gear and Music Technology]] [[/The Question Box - A Guide to Performing and Exploring Musical Emotion and Communication]] [[/Elevator Etudes - How to make practice sound musical]] == JMS's Guide to Songwriting == [[/Balance in Songwriting|Balance in Songwriting]] [[/Freedom from your Brain Children]] == Sources for Free Music Courses Online == [https://soundfly.com/courses Soundfly] [https://www.edx.org/school/berkleex Berklee EdX] YouTube == Unfiled Notes == === Student Note Files === :[[/Jazz Chord Mining]] :[[/Bryan Wang]] :[[/Noah Israel]] :Kevin McFadden :Micah Boughton :[[/Buck]] :[[/Max Fram-Schwartz]] :[[/Jason Chinn]] :[[/Dylan Gonzalez]] :[[/Caleb Ryland]] :[[/Matt Green]] :[[/Charles Dennis]] :[[/Janine Gedmin]] Singing through a straw https://www.youtube.com/watch?v=COgiz4O4tRw Rapid Songwriting Andrew Huang and Rob Scallon (10 Hours) https://www.youtube.com/watch?v=-QT0Bvf2FIQ ==See also== * [[Music production]] {{CourseCat}} i3k4vdy8iiukf6uizzzwct4pacsbyoz Category:Music/Lectures 14 213036 2812968 1581561 2026-06-05T21:59:00Z Kirby - Electrotechnics 3074947 changed to 'category:Music education' instead of category:Music 2812968 wikitext text/x-wiki [[Category:Lectures]] [[Category:Music education]] mkv2qel7m2wq2lc0o45ckkyf6vu0bcy Category:Nursery Rhymes and Songs 14 215826 2813014 1796612 2026-06-06T00:35:12Z MathXplore 2888076 added [[Category:Songs]] using [[Help:Gadget-HotCat|HotCat]] 2813014 wikitext text/x-wiki [[Category:Pre-school Education]] [[Category:Songs]] [[Nursery rhymes and songs]] are an important component of [[Pre-school education]]. slr9t86ktl32zqaye6qk4kdjjqf8qcw Theological Studies and Culture 0 218376 2812965 2583425 2026-06-05T21:55:15Z Kirby - Electrotechnics 3074947 changed to 'category:Musicology' instead of category:Music 2812965 wikitext text/x-wiki == Theological Research Interest 1 (Christianity): The Possible identity of Santa Claus stated in New Testament - The visit of the Magi (in MATHEW 2 and LUKE 2) == === Research Background === Reviewing some articles introducing 'Father Christmas', or 'Santa Claus', or 'Christkind' in Wikipedia and their talks following , we can find a great amount of debates are existing about the true identity of Santa Claus. (Wikipedia introductive articles: https://en.wikipedia.org/wiki/Santa_Claus ; https://en.wikipedia.org/wiki/Father_Christmas; https://en.wikipedia.org/wiki/Christkind) Regarding with these statements, we can also find some elements summarizing religion beliefs, local customs, folk legends and diverse cultural differences, from Europe to America, and then all around the worlds, have been inputted, which made this topic in Christmas holidays spread the magical, beautiful and vivid impressions attacking not only adults, but young teenagers and children's interests. If it comes to the traditions of Christianity, yes, Saint Nicholas 'was famous for his generous gifts to the poor' was of a great possibility to be trusted the right one by many places, which was also taught by my grandma in the course of Global Bible Study in Newcastle. (We used to present a small discussion, which deeply open my horizon and enlighten my mind-seed in thinking about this topic) Besides him, some other possibilities, such as the import of god Wodan from other ancient religion in Northern Europe then into Christianity, can also take a seat to be Santa (Father Christmas) in Christmas. However, if it was in Christianity, one fact we can ensure is Father Christmas's existing must content the relationship with Jesus's birth on Christmas Eve. <ref>Wikipedia (2016) '''Wiki-entry''' 'Saint Nicholas' Searched from ''Wikipedia''. ，Commons Attribution-ShareAlike License: Wikimedia Foundation, Inc. Available at: https://en.wikipedia.org/wiki/Saint_Nicholas. Accessed: 15th, December, 2016 </ref> === Methodology: Documentary research of Bible - New Testament === If further deeply applying this topic into the category of theological academy, in order to recover some lost facts, documentary collections of words from Bible for matching custom phenomena & historical experiences with what have already been taught in holly book is still a good principle. Following it, another possibility can be extracted out: === Participant-observing words in Bible === '''Collections from MATTEW 2:''' 'The visit of Magi: After Jesus was born in Bethlethem in Judea, during the time of King Herod, Magi from the east came to Jerusalem and asked, ' Where is the one who has been born King of the Jews? We saw his star in the east and have come to worship him' ... 'Then Herod called the Magi secretly and found out from them the exact time the star had appeared... After they had heard the king, they went on their way ahead of them until it stopped over the place where the child was. When they saw the star, they were overjoyed. On coming to the house, they saw the child with his mother Mary, and they bowed down and worshipped him. Then, '''''they opened their treasures and presented him with gifts of gold and of incense and of myrrh ...''''' ' <ref>The GIDEONS INTERNATIONAL in the British Isles, Global Bible Study and NTV (2006)' The visit of the Magi', ''Mathew 2'', Collected from ''New Testament & Psalms''. Website: www.gideons. org. uk </ref> 'Visitors from the East: ...And so they left, and on their way they saw the same star they had seen in the East ... They went into the house, and when they saw the child with his mother Mary, they knelt down and worshiped him. '''''They brought out their gifts of gold, frankincense, and myrrh, and presented them to him...'''''' <ref> United Bible Societies (eds.) (1975, 1979,1995, and modified as the 3rd edition - 2016) ' Visitors from the East', ''Mathew 2'', Collected from ''The New Testament''. English text and all English helps for the readers: American Bible Society in New York USA; and National TSPM & CCC of mainland China. Local print: JSE - 0001371 </ref> '''Collections from LUKE 2:''' The Shepherds and the Angels And there were shepherds living out in he fields near by, keeping watch over their flocks at night. An angel of the Lord appeared to them, and the glory of the Lord shone around them, and they were terrified. But '''''the angel said to them,''''' 'Do not be afraid. I bring you good news of great joy that will be for all the people. Today in the town of David a Saviour has been born to you; he is Christ the Lord ... 'Glory to God in the highest, and on earth peace to men on whom his favour rests ... So they hurried off and found Mary and Joseph, and the baby, who was lying in the manger. When they had seen him, they spread the word concerning what had been told them about this child, and all who heard it were amazed at what the shepherds said to them. But Mary '''''treasured up all these things''''' and pondered them in her heart ... <ref>The GIDEONS INTERNATIONAL in the British Isles, Global Bible Study and NTV (2006)' The birth of Jesus', ''LUKE 2'', Collected from ''New Testament & Psalms''. Website: www.gideons. org. uk </ref> === Analysis for researching Santa's possible identity === In these words collected, we can find: no mater spiritual messages or solidified treasures, as gifts, 'Magi from East (other ancient religion area )', 'angel (sent by Lord)' and 'Shepherds (sending message-gifts to Mary)' have all played the role of Father Christmas (Santa), of whom 'Magi from East' is of the most likelihood to be Father Christmas (Or, we can say Christmas man indicated by Father). Further more, histories taught us: 'Magi', from East, have been researching Celestial stem and Earthly Branches who could have explored the top of which Christmas tree baby Christ's birth star would land upon and what it stood for. ( Wikipedia introductions: Celestial stem [https://en.wikipedia.org/wiki/Celestial_stem]; Earthly Branches [https://en.wikipedia.org/wiki/Celestial_stem]). After many readings and participant-observations, the identity of Christmas Father has an increasing possibility to be one 'Magi' from East. Following, another question was about Christmas Father's Identity. We can make further considerations that Santa's historical functions were including: 1. Calming down Mary's emotional instability in that dangerous situation; 2. Bringing this message - Christ was born, of full joyfulness, to all the worlds that Christ was born; 3. Sending prepared gifts to baby Christ and his mother Mary (I can also remember Jesus has taught that only children or someone with child mind - as the innocent lamb can come to heaven, which is also reflected his own gift received from Eastern Magi indicated by Lord, when being a baby). Final part of this case: Wishing you can enjoy a VERY MERRY CHRISTMAS in the coming CHRISTMAS for 2017 NEW YEAR and 2018 NEW YEAR! Best Wishes! == Musical research and collection, from local applications for Theological Worship == 1. 'Prelude' from Bach's 'Prelude and Fugue in C major, BWV 846', after spreading to China, and other local cultures from Europe, was usually seen, symbolized and applied for Christianity usage of imagining and worshiping Jesus's Holly Mother - Virgin Marry. When analysing this work, we can find Broken Root-Chordal Progressions (vertical) and their solutions (horizontally) were strictly produced in music theoretical tonality, motif by motif and wave by wave. It can be compared with visualized imageries of Mother's Holly Spirituality (love) to her son and his 'not-easy life experiences in mortal world' (both sufferings and happiness). When playing it, the keyboard-touching techniques of dynamics levels' controllability need to be specially trained, in each of which the attached long time-value root-notes of every group (heads of each slur) need to be softly kept and gently down-pressed (released) the force in each 'central weights' (created by two long-notes: a minim and a semi-quaver-cut minim following ) and let it wave up (in tendency) from one end of left hand to the other end of right head naturally. It needs to be matched and heard as a whole in one hand through energetic Wrist-rotating. '''In this procedure, your prayers to Peace, Health and Love to people surrounding you - your parents, friends, important persons in heart, Lord Trinity God, Jesus, His Holly Mother and so on, can be input alongside forces-pressing and love-energy's spreading... ''' As paragraph's going-on, the depth of timbre has been enlarged, and some 'unhappy sufferings' might be thoughtfully reflected out with colour's grey and emotional pains, which can be seen as Jesus's mortal sufferings in Mother's Eyes. Finally, it's holly mother's love carrying the spirituality, wave by wave, transformed to be composer's musicality that drove away those unhappy memories and make broken Chordal Progressions threaded together, to be peaceful impression of 'universal harmony' to all. Comment: In some cultures, when, making the self-reflexivity of European music culture and Baroque Era, this peaceful piece has always been collected in piano's or other instrumental praying list for worshiping Virgin Mary's contributions of love to Baby Jesus. (Yes, though St. Virgin Mary wasn't the most important one, her merciful love associated Jesus in touching away 'paints of hurts' brought by mortal world...) The most hard technique is broken chordal progressions' threading should be made through not only physical forces, but spiritual worships or relative imaginations of musicality. I tried to make one piece in piano classroom, my heart if true and would like to share in Wikiversity. However, for more professional information and expertise edition, please review Wikipedia introductions (Virgin Marry: https://en.wikipedia.org/wiki/Mary,_mother_of_Jesus; Prelude and Fugue in C major, BWV 846: https://en.wikipedia.org/wiki/Prelude_and_Fugue_in_C_major,_BWV_846), especially the harpsichord edition -(Performed on a French harpsichord tuned in equal temperament by Robert Schröter) Even without theological purpose, we can also make a self-reflexivity of musicality to imagine mother's peaceful love towards us. [[File:Prelude C major.ogg|thumb|Classroom plays of Sir Johann Sebastian Bach's Prelude of 'Prelude and Fugue in C major, BWV 846' in piano edition for Educational purpose of cultural exploration (from a local piano classroom), techniques discussion and theological learning. In the second edition, after I got a perfect & beloved paper Piano-Educational Manuscript-Researching Book from America, imported by SMPH, Shang Hai to China in 2017 - «J.S.BACH WELL_TIMPERED CLAVIER VOLUME I» (Willand a. Palmn edited; Zhe, Tang Bilingual translated) by Alfred Publishing Co., Inc., I have paid much time and energy to research out possible piano techniques in my local classrooms to realize those notation-requirements which summarized many Sir Bach's Hand-manuscripts and his fellows' researching results.]] As from heart-respects and piano-researching nature, I need to make a reference of this beloved manuscript paper-book accordingly mentioned in the second edition of our background music and our Portal (You knew, it's sweet-smelling and texture when in hands) - <ref> J.S.BACH WELL_TIMPERED CLAVIER VOLUME I (eds.in times: 1st edition in April of 2017; Now-11st edition in March of 2017) (eds. in authors: Willand a. Palmn edited; Zhe, Tang Bilingual translated) (2017) 'Part 2: Words In terms of J.S.BACH WELL_TIMPERED CLAVIER VOLUME I: pp.5, pp.16-17 - Ornamentation (trill.), pp.27-Dynamics Signals, and pp.28- Sustaining pedal ', ''Part 3: Notation-manuscript: pp.29- BNV846 and its notes''. Imported: SMPH; Copyright at: MMIV by Alfred Publishing Co., Inc. Local print: {{ISBN|978-7-80667-986-9}}/Jˑ942 </ref> '''2. Swan's Praying - Artistic Perfection across Death to peaceful door of heaven:''' (When playing, you would feel two 'rough hands' in body, but reflected a pure soul facing Heaven-door) [[File:The Dying Swan - black angel JMC Han.ogg|thumb|It's a piano edition of Camille Saint-Saëns's 'The Swan', or we can say 'The Dying Swan', submitted up as a remembrance of Traditional Tomb-sweeping Festival of China. The inspiration of playing this piano edition was from a movie named Black Swan (Wikipedia introduction: https://en.wikipedia.org/wiki/Black_Swan_(film) ) deeply attracted my artistic appreciation & enthusiast. Where is the artistic perfection? Maybe, it's only in dreaming heaven... however, after one character can manage two totally & extremely opposite roles - a death-breathing black swan deluded by Satan (Wikipedia introduction: https://en.wikipedia.org/wiki/Satan) and a holly-dreaming white swan guided by Angel (Wikipedia introduction: https://en.wikipedia.org/wiki/Angel); afterwards, she could really get her Artistic Perfection from life but crazily make a repayment of death. Meanwhile,Le cygnet Performed by John Mitchel did give me some direct feeling of violin-piano dialogue (Wikipedia introduction: https://en.wikipedia.org/wiki/The_Dying_Swan) ]] '''3. White Peace Dove - Praying for Lord's Love of Peace and individual love for life:''' [[File:Dove With Spanish Sense in Piano JMC.Han.ogg|thumb|La Paloma: In English, it's called 'No More' composed by Sebastián Iradier (later Yradier) in the 1850s, and with a funny 1930 Cartoon, just known through reading and learning in Wikipedia: https://en.wikipedia.org/wiki/La_PalomaIn my life journey coming here, the story (Cultural Phonomenon) should be like: Since Childhood, I have heard this song through musical learning and loud KTV-singing (one type of karaoke room with TV in the middle for singing some pop songs and watching MTV movies). After I learnt English, the memory usually taught me: its musical imagery should be matched with a White Peace 'Dove' for 'Love'; but on each of two sides, only a pair of feet were kissed by waves, and only the 'Missing' travels across a deep blue sea... Then, tenderly, with a song sounds from the depth of heart-sea... Jason, why did you play it with a deep emotional expression? Did you have a lover, or some lovers? No...no... I am not quite sure... about Love. From my culture, I should keep it implicitly as my own heart's secret, and wait... You knew, if conditions and fate didn't permit, how can your things be going to be. (Heart-beating fast) Just, just...my piano-teaching classroom gave me an educational task - to realize it through hands. And, I began to pick it up and research large octave-chords' 'smooth bridging-way' as to breathe in sentences.However, there was a real deep blue sea over there...; and in Bible, there was also a Whilte Peace Dove who picked up Olive Branch for 'Flood's Down', Peace's coming, and Lord's Love all around the world. Those were what I knew as knowledge... Hi, Jason, we still would like to know about you? Oh, forgive me... I didn't know when I could be sure (If I can, I should... someday)Smiling, watching the deep blue sea, and listening toA Dove]] '''3. Mozart, as God's son, for the discovery and the peacefulness:''' [[File:KV279 2nd Movement Shading Andante hollowed-out by Sunlights.ogg|thumb|KV279 2nd Movement Shading Andante hollowed-out by Sunlights]] '''4. Simple minor-etude, but with the Sky's Colours (hard working) - to watch out its changes and pray peacefully through church's window:''' [[File:Cramer Etudes- &#039;d minor Etude&#039;, Player JMC, Han.ogg|thumb|Cramer Etudes- &#039;d minor Etude&#039;, Player JMC, Han]] '''5. Tears In Heaven (Praying song for the peacefulness and love)''' [[Category:Theology]][[Category:Musicology]] dullw5bcr4v0ifrovhy3mrgzvkgnz2e Category:Music Theory 14 219302 2812961 1648585 2026-06-05T21:50:53Z Kirby - Electrotechnics 3074947 Now defunct page. 2812961 wikitext text/x-wiki phoiac9h4m842xq45sp7s6u21eteeq1 Template:Demo 10 219699 2813169 1651876 2019-01-02T19:54:33Z CambridgeBayWeather 24544 Protected "[[Template:Demo]]": [[WP:High-risk templates|Highly visible template]] ([Edit=Require template editor access] (indefinite) [Move=Require template editor access] (indefinite)) 1651876 wikitext text/x-wiki {{#invoke:Demo|main}}<noinclude>{{documentation}}</noinclude> qo8uxqtbhlvri8ihi5293by4l1j4soc 2813170 2813169 2019-01-02T19:54:35Z CambridgeBayWeather 24544 Adding {{pp-template}} ([[WP:TW|TW]]) 2813170 wikitext text/x-wiki <noinclude>{{pp-template|small=yes}}</noinclude>{{#invoke:Demo|main}}<noinclude>{{documentation}}</noinclude> feoy20irdt5nh9t4j3rmmbmiyvqgkdk 2813171 2813170 2019-03-22T06:54:17Z JJMC89 2085686 handled by {{documentation}} 2813171 wikitext text/x-wiki {{#invoke:Demo|main}}<noinclude>{{documentation}}</noinclude> qo8uxqtbhlvri8ihi5293by4l1j4soc 2813172 2813171 2024-03-31T14:46:33Z CambridgeBayWeather 24544 Changed protection settings for "[[Template:Demo]]" ([Edit=Require extended confirmed access] (indefinite) [Move=Require extended confirmed access] (indefinite)) 2813171 wikitext text/x-wiki {{#invoke:Demo|main}}<noinclude>{{documentation}}</noinclude> qo8uxqtbhlvri8ihi5293by4l1j4soc 2813173 2813172 2026-06-06T08:30:21Z Jtneill 10242 4 revisions imported from [[:w:Template:Demo]] 2813171 wikitext text/x-wiki {{#invoke:Demo|main}}<noinclude>{{documentation}}</noinclude> qo8uxqtbhlvri8ihi5293by4l1j4soc Escuela de Lenguas UNLP/About Argentina/Argentinean Music 0 223010 2812986 2480514 2026-06-05T22:26:28Z Kirby - Electrotechnics 3074947 changed to 'category:Music history' instead of category:Music 2812986 wikitext text/x-wiki == Musical Genres == === Tango === ==== Origin and development ==== Tango was created on both sides of the Río de la Plata between 1850 and 1890. At the beginning of the 19th century, with its popular acceptance worldwide, the dance evolved to its current form. This dance, that originated in the Port of Buenos Aires and quickly spread to the southern neighborhoods, such as San Telmo, Monserrat and Pompeya, had a parallel growth to that of the Argentine society, formed by European immigrants, who contributed to it with many of their elements. ==== Instruments and sources ==== Around 1860 it was danced loose and with music that was based on the fandango and the candombe of the black people. At that time, there was still no Tango, as a dance style. The sound of the bandoneon (of German origin) was incorporated as something essential to pianos, creole guitars, double basses and violins. ==== An underground dance ==== The "tango arrabalero" arose in the neighboroods that was danced in the suburbs. Men and women with bodies tightly embraced scandalized the society of the time. It was condemned by the church and banned by the police for inciting to scandal. Its prohibition forced people to dance in hidden places until the nineteenth century. At that time, only the humble social strata, those of the suburb, cultivated that dance. The Tango emerged in brothels, ranches and clubs. The brothels encouraged tango in order to get male and female bodies closer. Tango was conceived as "vulgar" by the most conservative strata, marginalized socially for seeking sensuality and pleasure. ==== Tango in the XXth century ==== In 1910 the tango was danced in Paris, rapidly expanding its popularity around the world. His glamour conquered the highest sectors of society and it was danced in almost all European capitals. The people in charge of presenting the tango in the salons of the old world were the young sons of traditional families, who had long frequented the places of Buenos Aires where tango was danced, to the disgust of the aristocratic society, which still viewed it with bad eyes for its background. Its final acceptance took a while longer until it was considered an example of popular expression. ==== Origin of the word 'tango' ==== Among the many mysteries that tango contains, the first one is that of its own denomination. The tango voice is found in African, Hispanic and colonial cultures. According to some theories, "tango" would derive from "tang", which, in one of the languages ​​spoken in the black continent, means “to touch and approach”. Tango, in Spanish, is considered to be a voice derived from "tangir", a word that in ancient Spanish means "to play an instrument" and from "tangere", which in Latin means "to touch". At the time of the colony, "tango" was the name that black people gave to their percussion drums. They pronounced the word "tango" accentuating it on the last syllable. And "tango" was also the dances organized by the Africans who came to the Río de La Plata. ==== A 'sad thought' that is danced ==== The tango was born as a dance, as a different way of dancing that was known until then as: "habaneras", "mazurcas", "milongas"...  It was a spontaneous creation of men and women in brothels, where it was possible to embrace the couple, stick to your partner´s body: face to face, chest to chest, belly to belly, thigh to thigh, pulse to pulse. For Horacio Ferrer, the tango of the beginning was "the most torn or improvised and sudden choreographic creation for a couple that will surpass, in these respects, the creations of the popular Spanish baroque or the high German waltz." Dancers proposed to the musicians of that time, usually guitar, flute and violin players, the intensity of the rhythm with a changing display of figures that were created on the on fly: "the cut", "the eight", “the half-moon”... They were the inventors of the dance. The story will speak of that tango of the suburbs. Then the "canyengue" and the "lounge tango" will appear, which are, according to Enrique Santos Discépolo <ref>Santos Discépolo, Enrique. (Buenos Aires, March 27, 1901 - Buenos Aires, December 23, 1951). He was a composer, musician, dramatist and Argentine filmmaker.</ref>, different ways of dancing "a sad thought". Also, Argentine painters, enhanced through their paintings a universe that is a true reflection of the tango spirit. In his works,  Sigfredo Pastor<ref>Pastor, Sigfredo. (1912 - 1994). He was an Argentine painter, draftsman and engraver. His favorite subject was tango and the universe of popular music.</ref> affirms the vigor of the intertwined dance, forming or fusing the couple almost in a single figure, in fulfillment of a magnificent appearance of sculptural plasticity and that seems determined to fix indefinitely in time. == Famous Musicians == === Mercedes "La Negra" Sosa === Haydée Mercedes Sosa (her real name) was born on 9th July 1935 in a humble home in San Miguel de Tucumán. She began to sing when she was 15 year sold. In 1963 she founded with other artists, the "movimiento del Nuevo Cancionero", a group whose goal was to put into value, create and develop popular art. Her career was led by those values. She was forced to leave the country during the military dictatorship of the seventies in our country. She was really appreciated by many international artists. She also sang with a lot of young Argentinean artists, such as Charly Garcia, Victor Heredia and Leon Gieco. She died on 4thOctober 2009whenshewas 74 year sold. She was known as "THE AMERICAN VOICE" and is still popular by that name all over the world. === Atahualpa Yupanqui === Atahualpa Yupanqui (which in Quechua means "he that comes from afar to say something”) is the pseudonym used by Hector Roberto Chavero Aramburo, who was born in Pergamino, Province of Buenos Aires, on January 31st, 1908 and died in Nimes, France, on 23rd May 1992. "Don Ata", as he used to be called, was a singer, guitarist, poet and writer. He is considered the most important Argentinian folklore musician. His songs have been sung by great artists such as Mercedes Sosa, Jorge Cafrune, Alfredo Zitarrosa, José Larralde, Víctor Jara, Angel Parra, Jairo, Andrés Calamaro, Divididos, Marie Laforêt and Mikel Laboa, among many others. And they continue to be part of the repertoire of countless artists in Argentina and around the world. His works express the love for the land where he was born and they reflect the culture of Río de La Plata. His poetry conveys his acute vision of nature and human beings. His guitar, as he frequently said, had been with him "a few hours after his birth, playing melodies that helped him sleep". Atahualpa Yupanqui is one of the most influential Argentinian creators of all time. === Palito Ortega === Ramón Bautista Ortega, called “Palito “ is a famous Argentinian singer.He is not a very virtuous singer but he obtained the applause of all the public through his simple songs. However his career has not been easy and his life has taken him in many different directions. Palito Ortega was born in 1941 in the province of Tucuman. He started to do many jobs when he was a child and when was a teenager too. He had a very poor and hard childhood; since he was a little boy he went to work to help his father in the breeding of his six brothers. He worked selling coffee, newspapers and shining shoes. At the age of 16 came to try his luck in Buenos Aires. His contact with the artistic world was in Belgrano radio where he became known selling coffee, later he was carrying musical instruments of the orchestras, and then he started to sing. His musical consagration was in his early twenties, when he worked in the T.V. program “El club del clan”. He also worked as an actor in that show. His songs toured the world and were translated into several languages. Also he was a director of many movies. After those jobs in his middle-ages, he decided to became a businessman. He bought his own offices and contracted Frank Sinatra to sing in the Luna Park. This resulted in a financial crash because the dollar became very expensive and he had a lot of debts. Palito could not stand being a businessman, then he came back to his previous jobs, so he went to USA (in particular Miami ) and he worked in TV there for about five years. After that experience in his early fifties, he returned to Argentina and decided to be a politician. From October, 1991 to October, 1995 he was the governor of Tucuman, then, he was a senator for four years. After that he was candidate to vice-president of Argentina in the years 2000, but he lost it, then he retired from the politics. Now in his elderly-age, Palito keeps his best job: Singer! === Sandro === Sandro was born in Buenos Aires on August 19th, 1945. His real name was Roberto Sanchez Ocampo. His father was Vicente Sanchez and his mother was Nydia Ocampo. He grew up in Valentín Alsina and he learned to play the guitar when he was a child. His paternal grandfather was from Hungary, because of that, he was known as “Gitano”. He was a musician, an actor and a famous singer. He admired and imitated Elvis Presley. In 1961, he created the musical group “Sandro y los de Fuego” which gained popularity on the TV show called “Sábados Circulares”. Some famous songs were “Rosa Rosa”, ”Quiero llenarme de ti”, ”Penumbras”, “Porque yo te amo,” among others. Sandro was the first latino Singer that sang at Madison Square Garden in U.S five times during 1970 and he was also the first artist that sang at Luna Park in Buenos Aires with great success in 1972. He also was the main actor in 16 films. The news that he suffered from enphisema was made public in 1998, causing great concern among his followers from around the world. He was addicted to tobacco, which was the main cause of the disease. On November 20th, 2009, Sandro received a heart-lung transplant in Mendoza, Argentina. On January 4th, 2010, forty five days after that and after many complications, he died. [[Category:Argentine culture]] [[Category:Music history]] 0746fo5sbiaprzb0joxruifqlp60n7b Escuela de Lenguas UNLP/Academic Writing/Music/Music in the Renaissance 0 224330 2812985 2459708 2026-06-05T22:25:14Z Kirby - Electrotechnics 3074947 changed to 'category:Music history' instead of category:Music 2812985 wikitext text/x-wiki At the end of the Middle Ages, Western Europe witnessed the emergence of Renaissance humanism, a movement of philosophical, scientific and artistic reform. Artists who adhered to this school of thought longed to exalt beauty, human faculties and worldly life as the ancient Greeks did. Renaissance artists literally shaped their world according to the Classical-era ideals (mainly in literature, architecture, painting and music). Coetaneous historical events, such as the decline of the Byzantine Empire, the invention of Gutenberg’s printing press, voyages of discovery and colonization, the Protestant Reformation and the development of nationalisms, encouraged the expansion of Renaissance humanism throughout Europe together with a strong sense of self-determination and individualism. == The Idea of Rebirth in 15th and 16th-Century Music == [[File:Monodic and polyphonic compositions.png|thumb|A Greek song from Seikilos (above) and a Renaissance madrigal composed by Francisco Guerrero (below).]] The need to restore the old techniques and to repudiate the achievements conquered during the Middle Ages reinforced the Renaissance ideals of the humanist movement. However, 15th-century musicians were not willing to abandon all musical innovations behind those rebirth ideals. For example, they preferred the polyphonic composition from the late Middle Ages (two or more simultaneous melodies) over the monodic Greek singing (a single melodic line). This issue suggests that the idea of ​​''rebirth'' in music would have responded to an ‘inspiration’ in the classical world rather than its recovery. == Musical Styles of the Renaissance == Guided by this new tendency, 15th and 16th-century composers built a vast vocal and instrumental repertoire that responded to specific sociocultural requirements. According to this criterion, it is possible to distinguish between ‘sacred’ and ‘secular’ Renaissance pieces. The first ones were conceived for religious purposes, written mostly in Latin and usually performed ''a cappella'' or with a harmonic accompaniment (e.g., organ) inside sacred enclosures. Instead, pieces of the second group described worldly situations in vernacular languages, frequently combined several kinds of instruments and resonated in courtly, domestic and public places. During the Renaissance, several musical forms were created in many languages (Latin, Italian, English, Spanish, etc.), structures (strophic, repetitive, rondo, etc.) and styles of musical composition (fauxbourdon, imitation, variation, etc.). One of the most important sacred forms or the Renaissance was the motet. Its voices were usually developed from a previous liturgic melody and its text was extracted from passages of the Bible. Some famous motet composers were Josquin des Prez, Giovanni Pierluigi da Palestrina, John Dunstaple and Tomás Luis de Victoria. In the field of secular music, on the other hand, the most widespread form was the madrigal, which reached its maximum splendor in the 16th century. This form declaimed amatory, philosophical, moralizing and descriptive themes, and its pieces were usually performed with voices, instruments or both in small spaces that promote an intimate atmosphere. Jacques Arcadelt, William Byrd, Juan Vázquez, Orlando di Lasso and Claudio Monteverdi were recognised as some of the greatest madrigalists of their time. == References == Atlas, A. W. (1998). ''Renaissance Music: Music in Western Europe, 1400-1600''. New York: Norton. Haar, J. (2014). ''European Music, 1520-1640''. Woodbridge: Boydell Press. Perkins, L.L. (1999). ''Music in the Age of the Renaissance''. New York: W. W. Norton & Company. Reese, G. (1959). ''Music in the Renaissance''. New York: Dent. Suárez Urtubey, P. (2004). ''Historia de la Música''. Buenos Aires: Claridad. Taruskin, R. (2005). ''The Oxford History of Western Music: Music from the Earliest Notations to the Sixteenth Century'' (Vol. 1). Oxford MS: Oxford University Press. [[Category:Music history]] 9xtxz3jt6e2evrxy2nxo7fnxt2mt236 Escuela de Lenguas UNLP/Academic Writing/Music/Rondo 0 224683 2813006 2459709 2026-06-06T00:29:30Z Kirby - Electrotechnics 3074947 changed to 'category:Music theory' instead of category:Music 2813006 wikitext text/x-wiki == Rondo == The Rondo is a musical form based on the alternation of a main theme or refrain with secondary sections, called episodes or ''couplets''. The rondo is generally used as the final movement of classical sonatas, concertos or symphonies. In this musical form, the refrain is always the first section to appear and it defines the tonal centre of the piece. If it is used as the final movement of the sonata is called sonata rondo, and its formal schema is A B A C A B’ A. This schema presents some parallelisms with the sonata form. The A B A sections correlates with the exposition of the sonata, episode C can be related with the development section and A B’ A sections with the reexposition. In the sonata rondo, each episode is related with the refrain since they prepare each new entrance of this section. Other rondo forms present shorter alternated schemas, such as A B A C A (called five-part rondo) or A B A. === The rondo form in classical music === The rondo has been used, usually as the final movement of multi-movements works, by many classical composers such as Haydn, Mozart or Beethoven. We can find some examples in the last movements of Mozart’s Piano Sonata No.16 in C major K.545 and Clarinet Concerto in A major K.622, Beethoven’s Piano Sonata No.8 Op.13 and No.15 Op.28, and Haydn’s String Quartet in C major Hob.III:39. === History === The term “rondo” develops first from the ''rondeau'', a form of medieval French music usual in the songs of the troubadours. This is considered one of the most common French chanson forms between the 13th and 15th centuries. The musical structure of the ''rondeau'' is generally AB-aAab-AB, where the capital letters refer to the refrain verses, and the small letters to the secondary verses. The rondo form also develops from the rondeau of the French clavecinists. From a structural point of view, these pieces are long rondos that alternate a refrain section A with a series of ''couplets''. The formal schema of this rondeau is A B A C A D A (…) A. The rondo form makes this scheme shorter by reducing the number of ''couplets'' to three in the sonata rondo. === Bibliography === Kühn, C. (1992) Tratado de la forma musical. Barcelona: Editorial Labor. Randel, D. M. (2003). The Harvard dictionary of music. Cambridge, Mass: Belknap Press of Harvard University Press. Rosen, Charles (1988) [1980]. Sonata Forms. Nueva York: W. W. Norton. [[Category:Music theory]] 635n5i9gq8jnv7yyhthfqinhvkd5aga Template:Sandbox other 10 230890 2813127 2147404 2025-08-05T04:52:35Z Theknightwho 2768786 Make substable. 2813127 wikitext text/x-wiki <onlyinclude>{{safesubst:<noinclude/>#if:{{safesubst:<noinclude/>#ifeq:{{safesubst:<noinclude/>#invoke:String|sublength|s={{safesubst:<noinclude/>SUBPAGENAME}}|i=0|len=7}}|sandbox|1}}{{safesubst:<noinclude/>#ifeq:{{safesubst:<noinclude/>SUBPAGENAME}}|doc|1}}{{safesubst:<noinclude/>#invoke:String|match|{{safesubst:<noinclude/>PAGENAME}}|/sandbox/styles.css$|plain=false|nomatch=}}|{{{1|}}}|{{{2|}}}}}</onlyinclude><!-- -->{{documentation}} qglmfuqapld2v7m2c8ehvpm6ympoljf 2813128 2813127 2026-06-06T08:30:16Z Jtneill 10242 1 revision imported from [[:w:Template:Sandbox_other]] 2813127 wikitext text/x-wiki <onlyinclude>{{safesubst:<noinclude/>#if:{{safesubst:<noinclude/>#ifeq:{{safesubst:<noinclude/>#invoke:String|sublength|s={{safesubst:<noinclude/>SUBPAGENAME}}|i=0|len=7}}|sandbox|1}}{{safesubst:<noinclude/>#ifeq:{{safesubst:<noinclude/>SUBPAGENAME}}|doc|1}}{{safesubst:<noinclude/>#invoke:String|match|{{safesubst:<noinclude/>PAGENAME}}|/sandbox/styles.css$|plain=false|nomatch=}}|{{{1|}}}|{{{2|}}}}}</onlyinclude><!-- -->{{documentation}} qglmfuqapld2v7m2c8ehvpm6ympoljf Research in programming Wikidata/Musical Compositions 0 231409 2812974 2590069 2026-06-05T22:10:07Z Kirby - Electrotechnics 3074947 changed to 'category:Music composition' instead of category:Music 2812974 wikitext text/x-wiki This article is about research of musical compositions using the knowledge base of international project called Wikidata. With the help of SPARQL queries for items classified as "musical compositions" the following were received: the list of all musical compositions, the list of musical compositions that has a composer, the bubble diagram for composers that shows composers with most compositions. Moreover, the task of searching music gaps in public domain was done and completeness of Wikidata was evaluated. == List of musical compositions == * Item: [[d:Q207628|musical composition (Q207628)]]. * Property: [[d:Property:P31|instance of (P31)]]. Let's build a list of all musical compositions. <syntaxhighlight lang="SPARQL">#List of all musical compositions SELECT ?composition ?compositionLabel WHERE { ?composition wdt:P31 wd:Q207628. SERVICE wikibase:label { bd:serviceParam wikibase:language "ru". } }</syntaxhighlight> [https://query.wikidata.org/#%23List%20of%20all%20musical%20compositions%0ASELECT%20%3Fcomposition%20%3FcompositionLabel%20%0AWHERE%20%7B%0A%20%20%3Fcomposition%20wdt%3AP31%20wd%3AQ207628.%0A%20%20SERVICE%20wikibase%3Alabel%20%7B%20bd%3AserviceParam%20wikibase%3Alanguage%20%22ru%22.%20%7D%0A%7D SPARQL query], 5494 records. <span style="color:green">👍</span>>The most complete and well-developed musical compositions on Wikidata are [[d:Q5064| The Magic Flute]], [[d:Q11980|Für Elise]], [[d:Q207875|Mozart's Requiem]], [[d:Q12025|Eine kleine Nachtmusik]]. <span style="color:red">👎</span>>Almost empty and uninformative musical compositions were [[d:Q1342275|Flight of the Bumblebee]], [[d:Q763716|Romeo and Juliet]], [[d:Q1845909|Iron Foundry]], [[d:Q28807544|Binks’ Waltz]], [[d:Q28803595|The Rose-bud March]], [[d:Q28804177|Leola]]. == Search music gaps in public domain == The task is to find musical works the composers of which passed away more than 70 years ago and audios of which are absent from Wikimedia Commons. The list of compositions must be sorted the in ascending order by publication date. This script can be used to find musical compositions that need to be digitized and then uploaded to Wikimedia Commons. <syntaxhighlight lang="SPARQL"> #Search music gaps in public domain SELECT ?composition ?compositionLabel ?publication WHERE { ?composition wdt:P31 wd:Q207628. # instance of compostion ?composition wdt:P86 ?composer. # composition has a composer ?composition wdt:P577 ?publication. # composition has a publication date ?composer wdt:P570 ?death. # composer has a date of death MINUS {?composition wdt:P51 []}. # compositions without audio FILTER(?death < "1947-01-01T00:00:00Z"^^xsd:dateTime) # composers that passed away more than 70 years ago FILTER(?publication < "1947-01-01T00:00:00Z"^^xsd:dateTime) # compositions that were published more than 70 years ago SERVICE wikibase:label { bd:serviceParam wikibase:language "ru". } } ORDER BY ASC(?publication) </syntaxhighlight> [https://query.wikidata.org/#%23Search%20music%20gaps%20in%20public%20domain%0ASELECT%20%3Fcomposition%20%3FcompositionLabel%20%3Fpublication%0AWHERE%20%7B%0A%20%20%3Fcomposition%20wdt%3AP31%20wd%3AQ207628.%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20instance%20of%20compostion%0A%20%20%3Fcomposition%20wdt%3AP86%20%3Fcomposer.%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20composition%20has%20a%20composer%0A%20%20%3Fcomposition%20wdt%3AP577%20%3Fpublication.%20%20%20%20%20%20%20%20%20%20%23%20composition%20has%20a%20publication%20date%0A%20%20%3Fcomposer%20wdt%3AP570%20%3Fdeath.%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20composer%20has%20a%20date%20of%20death%0A%20%20MINUS%20%7B%3Fcomposition%20wdt%3AP51%20%5B%5D%7D.%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20compositions%20without%20audio%20%0A%20%20FILTER%28%3Fdeath%20%3C%20%221947-01-01T00%3A00%3A00Z%22%5E%5Exsd%3AdateTime%29%20%20%20%20%20%20%20%20%23%20composers%20that%20passed%20away%20more%20than%2070%20years%20ago%0A%20%20FILTER%28%3Fpublication%20%3C%20%221947-01-01T00%3A00%3A00Z%22%5E%5Exsd%3AdateTime%29%20%20%23%20compositions%20that%20were%20published%20more%20thatn%2070%20years%20ago%0A%20%20SERVICE%20wikibase%3Alabel%20%7B%20bd%3AserviceParam%20wikibase%3Alanguage%20%22ru%22.%20%7D%0A%7D%0AORDER%20BY%20ASC%28%3Fpublication%29 SPARQL query], 140 records. == Completeness of Wikidata == Let's analyze the completeness of Wikidata . According to Grove Dictionary of Music and Musicians{{sfn|Grove Dictionary of Music and Musicians|2001}} there are 20374 composers. According to the category [[w:ru:Категория:Композиторы_по_алфавиту|"List of composers by name"]] of Russian Wikipedia there are 6130 composers. According to the category [[w:en:List_of_composers_by_name|"List of composers by name"]] of English Wikipedia there are 4685 composers. The number of musical compositions with filled property [[d:Property:P86|"composer (P86)"]] equals 3862, which is shown in [https://query.wikidata.org/#SELECT%20%3Fcomposition%20%3FcompositionLabel%20%3Fcomposer%20%3FcomposerLabel%20%0AWHERE%20%7B%0A%20%20%3Fcomposition%20wdt%3AP31%20wd%3AQ207628.%0A%20%20%3Fcomposition%20wdt%3AP86%20%3Fcomposer.%0A%20%20SERVICE%20wikibase%3Alabel%20%7B%20bd%3AserviceParam%20wikibase%3Alanguage%20%22%5BAUTO_LANGUAGE%5D%2Cen%22.%20%7D%0A%7D%0AORDER%20BY%20DESC%28%3Fcomposer%29 SPARQL query], and that's if you take into account the fact that one composer could have written several musical compositions. For example [[w:en:Wolfgang_Amadeus_Mozart|Wolfgang Amadeus Mozart]] is the composer of 95 compositions, which decreases the number of unique composers. The number of 3862 is lower than the amount of composers from both Russian and English Wikipedia and substantially lower than the amount of composers from Grove Dictionary of Music and Musicians which confirms the incompleteness of Wikidata. [https://query.wikidata.org/#SELECT%20%3Fcomposition%20%3FcompositionLabel%20%3Fcomposer%20%3FcomposerLabel%20%0AWHERE%20%7B%0A%20%20%3Fcomposition%20wdt%3AP31%20wd%3AQ207628.%0A%20%20%3Fcomposition%20wdt%3AP86%20%3Fcomposer.%0A%20%20%7B%20%3Fcomposition%20wdt%3AP495%20wd%3AQ34266%20%7D%20UNION%20%20%23%20Russian%20Empire%0A%20%20%7B%20%3Fcomposition%20wdt%3AP495%20wd%3AQ15180%20%7D%20UNION%20%20%23%20Soviet%20Union%0A%20%20%7B%20%3Fcomposition%20wdt%3AP495%20wd%3AQ159%20%7D%20%20%20%20%20%20%20%20%20%20%23%20Russia%0A%20%20SERVICE%20wikibase%3Alabel%20%7B%20bd%3AserviceParam%20wikibase%3Alanguage%20%22%5BAUTO_LANGUAGE%5D%2Cen%22.%20%7D%0A%7D%0A SPARQL query] for compositions with filled property [[d:Property:P86|"composer (P86)"]] and property [[d:Property:P495|"country of origin (P495)"]] with value of [[d:Q34266|"Russian Empire (Q34266)"]], [[d:Q15180|"USSR (Q15180)"]] or [[d:Q159|"Russia (Q159)"]], gave us only 8 compositions, which means that it's impossible to analyze Russian musical compositions due to lack of data. Let's build the bubble diagram for composers of musical compositions. <syntaxhighlight lang="SPARQL"> #composers of musical compositions #defaultView:BubbleChart SELECT ?composer ?form (COUNT(*) AS ?count) WHERE { ?composition wdt:P31 wd:Q207628. # instance of composition ?composition wdt:P86 ?composer. # composition has a composer OPTIONAL { ?composer rdfs:label ?form. FILTER((LANG(?form)) = "en") } } GROUP BY ?composer ?form ORDER BY DESC(?count) ?form </syntaxhighlight> [https://query.wikidata.org/#%23composers%20of%20musical%20compositions%0A%23defaultView%3ABubbleChart%0ASELECT%20%3Fcomposer%20%3Fform%20%28COUNT%28%2a%29%20AS%20%3Fcount%29%20WHERE%20%7B%0A%20%20%3Fcomposition%20wdt%3AP31%20wd%3AQ207628.%20%23%20instance%20of%20composition%0A%20%20%3Fcomposition%20wdt%3AP86%20%3Fcomposer.%20%20%23%20composition%20has%20a%20composer%0A%20%20OPTIONAL%20%7B%0A%20%20%20%20%3Fcomposer%20rdfs%3Alabel%20%3Fform.%0A%20%20%20%20FILTER%28%28LANG%28%3Fform%29%29%20%3D%20%22en%22%29%0A%20%20%7D%0A%7D%0AGROUP%20BY%20%3Fcomposer%20%3Fform%0AORDER%20BY%20DESC%28%3Fcount%29%20%3Fform%0A SPARQL query], 773 records. [[File:Composers of musical compositions bubble diagram.png|thumb|center|900px|Musical compositions grouped by composer]]<br style="clear: both;"> Size of a bubble tells us about the amount of musical compositions. This diagram shows us that some composers have more compositions that the others. Top 5 includes [[w:en:Niels_Gade|Niels Gade]] (173 compositions), [[w:en:Johann_Sebastian_Bach|Johann Sebastian Bach]] (155 compositions), [[w:en:Christian_Sinding|Christian Sinding]] (125 compositions), [[w:en:Johan_Halvorsen|Johan Halvorsen]] (121 compositions), [[w:en:Alan_Hovhaness|Alan Hovhaness]] (108 compositions). == Filling of Wikidata == The decision was made to fill [[d:Property:P86|"composer (P86)"]] property for [[d:Q207628|"musical composition (Q207628)"]] items to get better results while performing the query for searching music gaps in public domain. Let's build a list of all musical compostions with filled property [[d:Property:P86|"composer (P86)"]]. <syntaxhighlight lang="SPARQL"> #Lists of compositions that has a composer in Russian SELECT ?composition ?compositionLabel ?composer ?composerLabel WHERE { ?composition wdt:P31 wd:Q207628. # instance of composition ?composition wdt:P86 ?composer. # composition has a composer SERVICE wikibase:label { bd:serviceParam wikibase:language "ru". } } }</syntaxhighlight> [https://query.wikidata.org/#SELECT%20%3Fcomposition%20%3FcompositionLabel%20%3Fcomposer%20%3FcomposerLabel%20%0AWHERE%20%7B%0A%20%20%3Fcomposition%20wdt%3AP31%20wd%3AQ207628.%20%23%20Q207628%20-%20composition%0A%20%20%3Fcomposition%20wdt%3AP86%20%3Fcomposer.%20%20%23%20has%20a%20composer%0A%20%20SERVICE%20wikibase%3Alabel%20%7B%20bd%3AserviceParam%20wikibase%3Alanguage%20%22ru%22.%20%7D%0A%7D SPARQL query], 3864 records at 30/10/2017, 10:51. [https://query.wikidata.org/#SELECT%20%3Fcomposition%20%3FcompositionLabel%20%3Fcomposer%20%3FcomposerLabel%20%0AWHERE%20%7B%0A%20%20%3Fcomposition%20wdt%3AP31%20wd%3AQ207628.%20%23%20Q207628%20-%20composition%0A%20%20%3Fcomposition%20wdt%3AP86%20%3Fcomposer.%20%20%23%20has%20a%20composer%0A%20%20SERVICE%20wikibase%3Alabel%20%7B%20bd%3AserviceParam%20wikibase%3Alanguage%20%22ru%22.%20%7D%0A%7D SPARQL query], 3965 records at 30/10/2017, 12:47. == Future work == # Find a list of musical compositions that were created during [[w:en:Classical_period_(music)|The Age of Classicism]] (XVII—XVIII centuries). <br>Property: [[d:Property:P571|"inception (P571)"]]. # Find a composer that had written more symphonies than the others. <br>Properties: [[d:Property:P31|"instance of (P31)"]], [[d:Property:P86|"composer (P86)"]]. # Build a histogram that displays the amount of songs by [[w:en:The_Beatles|The Beatles]] by the year of publication. <br>Properties: [[d:Property:P175|"performer (P175)"]], [[d:Property:P577|"publication date (P577)"]]. == Tasks == <quiz display=simple> {Which one of the following composers had written more compositions? |type="()"} - [[w:en:Wolfgang_Amadeus_Mozart|Wolfgang Amadeus Mozart]] - [[w:en:Igor_Stravinsky|Igor Stravinsky]] - [[w:en:Johann_Sebastian_Bach|Johann Sebastian Bach]] + [[w:en:Niels_Gade|Niels Gade]] {Which of the following compositions had been created in Russian Empire? |type="[]"} - [[w:en:The_Magic_Flute|The Magic Flute]] + [[w:en:The_Firebird|The Firebird]] + [[w:en:Flight_of_the_Bumblebee|Flight of the Bumblebee]] - [[w:en:Requiem_(Mozart)|Requiem ]] + [[w:en:Romeo_and_Juliet_(Tchaikovsky)|Romeo and Juliet]] {Character of which opera by [[w:en:Richard_Wagner|Richard Wagner]] is displayed on the picture? <br style="clear: both;"> [[File:August_von_Heckel_Lohengrin.jpg|300px|]] |type="{}"} } { Lohengrin } {Which of the following musical compositions has more parts than others? |type="()"} - [[w:en:The_Haunted_Manor|The Haunted Manor]] + [[w:en:Il_trovatore|Il trovatore]] - [[w:en:Mathis_der_Maler_(opera)|Mathis der Maler]] - [[w:en:Turandot|Turandot]] {In what century were the following compositions written? |type="()"} | 20th| 21st +- [[w:en:King_Roger|King Roger]] -+ [[w:en:Sleep_(Eric_Whitacre_song)|Sleep]] -+ [[w:en:The_Hope_(Magle)|The Hope]] +- [[w:en:Chamber_Symphony_No._2_(Schoenberg)|Chamber Symphony No.2]] { Some compositions consist of many parts. Place the following compositions by the amount of parts: [[w:en:The_Well-Tempered_Clavier|The Well-Tempered Clavier]], [[w:en:The_Magic_Flute|The Magic Flute]], [[w:en:Turandot|Turandot]], [[w:en:Il_trovatore|Il trovatore]]. |type="()"} |1st,|2nd,|3rd,|4th +--- [[File:Das Wohltemperirte Clavier titlepage 2.jpg|The Well-Tempered Clavier|150px|]] ---+ [[File:Zauberflöte-Theaterzettel1791.jpg|The Magic Flute|150px|]] --+- [[File:Poster Turandot.jpg|Turandot|150px|]] -+-- [[File:Il Trovatore.jpg|Il trovatore|150px|]] </quiz> SPARQL query with answers: *[https://query.wikidata.org/#%23composers%20of%20musical%20compositions%0ASELECT%20%3Fcomposer%20%3Fform%20%28COUNT%28%2a%29%20AS%20%3Fcount%29%20WHERE%20%7B%0A%20%20%3Fcomposition%20wdt%3AP31%20wd%3AQ207628.%0A%20%20%3Fcomposition%20wdt%3AP86%20%3Fcomposer.%0A%20%20OPTIONAL%20%7B%0A%20%20%20%20%3Fcomposer%20rdfs%3Alabel%20%3Fform.%0A%20%20%20%20FILTER%28%28LANG%28%3Fform%29%29%20%3D%20%22ru%22%29%0A%20%20%7D%0A%7D%0AGROUP%20BY%20%3Fcomposer%20%3Fform%0AORDER%20BY%20DESC%28%3Fcount%29%20%3Fform%0A SPARQL query for the list of composers sorted in ascending order by the amount of compositions], 782 records. *[https://query.wikidata.org/#SELECT%20%3Fcomposition%20%3FcompositionLabel%20%3Fcomposer%20%3FcomposerLabel%20%0AWHERE%20%7B%0A%20%20%3Fcomposition%20wdt%3AP31%20wd%3AQ207628.%0A%20%20%7B%20%3Fcomposition%20wdt%3AP495%20wd%3AQ34266%20%7D%0A%20%20SERVICE%20wikibase%3Alabel%20%7B%20bd%3AserviceParam%20wikibase%3Alanguage%20%22%5BAUTO_LANGUAGE%5D%2Cen%22.%20%7D%0A%7D%0A SPARQL query for the list of musical compositions that had been created in Russian Empire], 4 records. * [https://query.wikidata.org/#%23added%202017-02%0A%23List%20of%20%60instances%20of%60%20%22musical%20composition%22%20%0ASELECT%20%3Flang%20%3FlangLabel%0AWHERE%0A%7B%0A%20%20%20%20%3Flang%20wdt%3AP31%20wd%3AQ207628.%0A%20%20%20%20SERVICE%20wikibase%3Alabel%20%7B%20bd%3AserviceParam%20wikibase%3Alanguage%20%22en%22%20%7D%0A%20%20%0A%7D SPARQL query for the list of all musical compostions], 5259 records. * [https://query.wikidata.org/#%23added%202017-02%0ASELECT%20%3FcompositionLabel%20%3Fcomposition%0AWHERE%0A%7B%0A%20%20%20%20%3Fcomposition%20wdt%3AP31%20wd%3AQ207628.%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20It%20is%20a%20%22instance%20of%22%20musical%20composition%0A%20%20%20%20%3Fcomposition%20wdt%3AP571%20%3Finception.%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20It%20is%20property%20of%20musical%20composition%20%3A%20inception%20%0A%20%20%20%20%20%20FILTER%28%20%3Finception%20%3C%20%222100-00-00%22%5E%5Exsd%3AdateTime%29.%20%20%20%20%20%20%20%20%20%20%23%20Find%20compositions%20based%20in%20the%2019th%20century%0A%20%20%20%20%20%20FILTER%28%20%3Finception%20%3E%3D%20%222000-00-00%22%5E%5Exsd%3AdateTime%29.%20%20%0A%20%20SERVICE%20wikibase%3Alabel%20%7B%20bd%3AserviceParam%20wikibase%3Alanguage%20%22en%22%20%7D%0A%7D SPARQL query for the list of compositions that were created in 21st century], 5 records. * [https://query.wikidata.org/#%23added%202017-02%0ASELECT%20%3FcompositionLabel%20%3Fcomposition%0AWHERE%0A%7B%0A%20%20%20%20%3Fcomposition%20wdt%3AP31%20wd%3AQ207628.%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20It%20is%20a%20%22instance%20of%22%20musical%20composition%0A%20%20%20%20%3Fcomposition%20wdt%3AP571%20%3Finception.%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20It%20is%20property%20of%20musical%20composition%20%3A%20inception%20%0A%20%20%20%20%20%20FILTER%28%20%3Finception%20%3C%20%222000-00-00%22%5E%5Exsd%3AdateTime%29.%20%20%20%20%20%20%20%20%20%20%23%20Find%20compositions%20based%20in%20the%2019th%20century%0A%20%20%20%20%20%20FILTER%28%20%3Finception%20%3E%3D%20%221900-00-00%22%5E%5Exsd%3AdateTime%29.%20%20%0A%20%20SERVICE%20wikibase%3Alabel%20%7B%20bd%3AserviceParam%20wikibase%3Alanguage%20%22en%22%20%7D%0A%7D SPARQL query for the list of compositions that were created in 20th century], 49 records. *[https://query.wikidata.org/#%23added%202017-02%20%20%0A%23defaultView%3AGraph%20%0ASELECT%20%3FcompositionLabel%20%3Fcomposition%20%3Fparts%20%3Fimage%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%0AWHERE%0A%7B%0A%20%20%20%20%3Fcomposition%20wdt%3AP31%20wd%3AQ207628.%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%0A%20%20%20%20%3Fcomposition%20wdt%3AP2635%20%3Fparts.%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%0A%20%20OPTIONAL%7B%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%0A%20%20%20%3Fcomposition%20wdt%3AP18%20%3Fimage%20.%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%0A%20%20%7D%0A%20%20FILTER%28%3Fimage%21%3D0%29%20.%0A%20%20%20%20%20%0A%20%20SERVICE%20wikibase%3Alabel%20%7B%20bd%3AserviceParam%20wikibase%3Alanguage%20%22ru%22%20%7D%0A%20%7D%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%0AORDER%20BY%20DESC%20%28%3Fparts%29%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%0A SPARQL query for musical compositions that contain logos], 9 records. : == Notes == {{reflist}} == References == *{{cite web |title = Grove Dictionary of Music and Musicians |url = http://www.oxfordmusiconline.com/public/book/omo_gmo |publisher = Oxford University Press |year = 2001 |ref = Grove Dictionary of Music and Musicians }} * Krizhanovsky A., Vershinin I. [http://nauchkor.ru/pubs/the-search-of-music-gaps-in-public-domain-using-wikidata-5a391c957966e11ea24d0b49 The search of music gaps in public domain using Wikidata] // Nauchkor [[Category:Research in programming Wikidata|{{SUBPAGENAME}}]] [[Category:Music composition]] bnzw2gdla7b4h21vjkir55qtizt6245 Learning bass guitar with Joseph Patrick Moore 0 233277 2812959 2182935 2026-06-05T21:49:55Z Kirby - Electrotechnics 3074947 changed to 'category:Guitar' instead of category:Music Theory 2812959 wikitext text/x-wiki '''[[w:Joseph Patrick Moore|Joseph Patrick Moore]]''' is a [[jazz]] and [[w:rock music|rock]] [[bass guitar]]ist who has free educational resources on his web site http://www.basslessons.tv/. The author has relicensed much of his content CC BY-4.0 in order to encourage reproduction of his short lessons on [[music theory]] using popular music on bass guitar. ==Videos== <gallery> 4 Cycles of Major 3rds - BassLessons.tv.webm|4 Cycles of Major 3rds&nbsp;– http://www.basslessons.tv/free-lessons/4-cycles-of-major-3rds Chromatic Scale & Chromaticism - BassLessons.tv.webm|Chromatic Scale & Chromaticism&nbsp;– http://www.basslessons.tv/free-lessons/chromatic-scale-chromaticism 3 Cycles of minor thirds - BassLessons.tv.webm|3 Cycles of Minor Thirds&nbsp;– http://www.basslessons.tv/free-lessons/3-cycles-of-minor-thirds </gallery> ==Text== *[[Learning bass guitar with Joseph Patrick Moore/How to lock in with a drummer|How to Lock in with a Drummer]]&nbsp;– http://www.basslessons.tv/free-lessons/how-to-lock-in-with-a-drummer *What Comes Around Goes Around&nbsp;– http://www.basslessons.tv/free-lessons/what-comes-around-goes-around *Whole Tone Scale and 2 Cycles of Major 2nds&nbsp;– http://www.basslessons.tv/free-lessons/whole-tone-scale-and-2-cycles-of-major-2nds *Major Chords&nbsp;– http://www.basslessons.tv/free-lessons/major-chords *How To Practice&nbsp;– http://www.basslessons.tv/free-lessons/how-to-practice *12 Tips Buying a Bass Guitar&nbsp;– http://www.basslessons.tv/free-lessons/12-tips-buying-a-bass-guitar *Time Management&nbsp;– http://www.basslessons.tv/free-lessons/time-management *Organize&nbsp;– http://www.basslessons.tv/free-lessons/organize *JPM’s Gear SetUp&nbsp;– http://www.basslessons.tv/free-lessons/jpms-gear-setup *Strap Height & Consistency&nbsp;– http://www.basslessons.tv/free-lessons/strap-height-consistency *Circle of Fifths&nbsp;– http://www.basslessons.tv/free-lessons/circle-of-fifths *Evaluate Yourself&nbsp;– http://www.basslessons.tv/free-lessons/evaluate-yourself *Improvised Bass Solo Using Chords&nbsp;– http://www.basslessons.tv/free-lessons/improvised-bass-solo-using-chords *Intervals&nbsp;– http://www.basslessons.tv/free-lessons/intervals *Hype&nbsp;– http://www.basslessons.tv/free-lessons/hype *Waterfall&nbsp;– http://www.basslessons.tv/free-lessons/waterfall-2017-version *F Major Scale 3 Octaves&nbsp;– http://www.basslessons.tv/free-lessons/f-major-scale-3-octaves *Bass Song&nbsp;– http://www.basslessons.tv/free-lessons/bass-song *Quest&nbsp;– http://www.basslessons.tv/free-lessons/quest *The Audition&nbsp;– http://www.basslessons.tv/free-lessons/the-audtion *Read Music&nbsp;– http://www.basslessons.tv/free-lessons/reading-music1 *Bb Major Scale 2 Octaves&nbsp;– http://www.basslessons.tv/free-lessons/bb-major-scale-2-octaves *F Major Scale 2 Octaves&nbsp;– http://www.basslessons.tv/free-lessons/fmajorscale2octaves *F Major Scale&nbsp;– http://www.basslessons.tv/free-lessons/f-major-scale-1-octave *Tuning Your Bass&nbsp;– http://www.basslessons.tv/free-lessons/tuning-your-bass *Datz It&nbsp;– http://www.basslessons.tv/free-lessons/datz-it *Bb Major Scale&nbsp;– http://www.basslessons.tv/free-lessons/bb-major-scale-1-octave *Left Hand Technique&nbsp;– http://www.basslessons.tv/free-lessons/left-hand-technique *Blues Bass Line #6 in a 12 Bar Blues&nbsp;– http://www.basslessons.tv/free-lessons/blues-bass-line-6-in-a-12-bar-blues *Blues Bass Line #4 in A 12 Bar Blues&nbsp;– http://www.basslessons.tv/free-lessons/blues-bass-line-4-in-a-12-bar-blues *Blues Bass Line #3 in A 12 Bar Blues&nbsp;– http://www.basslessons.tv/free-lessons/blues-bass-line-3-in-a-12-bar-blues *12 Bar Blues&nbsp;– BassLine #1&nbsp;– http://www.basslessons.tv/free-lessons/walking-bass-lines *Blues Bass Line #2 in A 12 Bar Blues&nbsp;– http://www.basslessons.tv/free-lessons/blues-bass-line-2-in-a-12-bar-blues ==External links== The following lessons include some copyrighted material that can't be reproduced here: *[[w:Going to California|"Going to California"]] – http://www.basslessons.tv/free-lessons/going-to-california-led-zeppelin-cover *Life Lessons 3 books – http://www.basslessons.tv/free-lessons/life-lessons-3-books *Qello – http://www.basslessons.tv/free-lessons/life-lessons-qellocom *Low Blow – http://www.basslessons.tv/free-lessons/low-blow-victor-bailey-cover *[[w:Fifty Ways to Leave Your Lover|"Fifty Ways to Leave Your Lover"]] – http://www.basslessons.tv/free-lessons/50-ways-to-leave-your-lover-paul-simon-cover *[[w:32 Flavors|"32 Flavors"]] – http://www.basslessons.tv/free-lessons/32-flavors-ani-difranco-cover [[Category:Guitar]] n05cmxj5hdg82ci2f0yn1w3a2yrjm2w Learning bass guitar with Joseph Patrick Moore/How to lock in with a drummer 0 233295 2812960 1819652 2026-06-05T21:50:24Z Kirby - Electrotechnics 3074947 changed to 'category:Guitar' instead of category:Music Theory 2812960 wikitext text/x-wiki Originally posted here: http://www.basslessons.tv/free-lessons/how-to-lock-in-with-a-drummer ---- The relationship between Bass and Drums is a special and unique relationship. While a typical rhythm section would consist of bass, drums, guitar, piano/keys, percussion, '''it is the bass and drums that gives music its “heartbeat” and “foundation”'''. The marriage between the bass and drums should be a happy one, if either of them are out of sync with one another, it can make for a difficult and tense musical situation. When the bassist and drummer connect as “one heartbeat”, they possess a tremendous amount of groove, power and backbone for the rest of the musical ensemble. As bassists, it is essential that we learn to trust and openly communicate with your partner in crime. The dialog between bassists and drummers should be a healthy, honest and trusting relationship. Below, I have outlined several key points that you should consider when working and locking in with your drummer. ​ ==Key Points== *The Song *Beat Placement *With the Kick, Not with the Kick *Dynamics *One with Yourself *Just Because They’re Great, Doesn’t Make It So *Listen to the Masters ==The Song== Most music will usually dictate what is needed within the rhythm section. The melody will usually tell you what type of feel or what to play or what not to play. It is important to listen to the music and let the music dictate what approach you and your drummer should take. Never be afraid to try ideas or take chances, but always be honest with the music and learn to compromise with each other. '''Remember to always "Serve the Music"'''. ==Beat Placement== Most professional bass and drum masters understand the subtleties of “Beat Placement”. Essentially there are three ways of playing “time” in any musical situation: ​ #on '''TOP''' of the beat (''slightly ahead of the metronome click'') #in the '''MIDDLE''' of the beat (''dead center with the metronome click'') #'''BEHIND''' the beat (''slightly behind the metronome click'') There will be a future video post on this concept so I won’t go into great detail now. Just understand and be aware that this subtle difference of where you place your note, makes the truly great rhythm section players stand apart from the rest. As a guideline, I tend to play straight-ahead jazz slightly ahead of the beat. With rock and country music I tend to play right in the middle of the beat. With funk, blues, hip hop and gospel I will play slightly behind the beat. Again, the tune will usually dictate where I’ll place the beat. Ultimately you and your drummer need to understand this concept and openly communicate about this. Try experimenting with these three different beat placements to see how it effects the groove of your music. ==With the Kick, Not with the Kick== Most pop tunes require that the bass and the kick drum are playing in sync and in unison together to create a full, cohesive and powerful sound. However there are two schools of approaches that bassists and drummers subscribe to: #Playing '''with the kick''' drum. #Playing '''off the kick''' and around it :''(not necessarily playing in unison with every kick)''. You should be aware of this concept and try experimenting where to place those unison notes together. Listen to the masters for how they approach this subtle yet powerful ingredient. Yet again, the composition will usually dictate what is needed and where to place those strategic notes and accents. ==Dynamics== Soft, loud and all points in between. The bass and drums should aspire and utilize dynamics as a rhythm section, thus propelling and lifting the musical ensemble to higher levels of tension and release. When there are moments of opportunities to rise and fall with the music, de- fine these sections and work them out dynamically. Often times, this will expose how well you and the drummer are locking in and how much you are really listening to each other. This seems like a given, yet dynamics are often overlooked by many performers. Try just getting together with your drummer and have a bass/drum rehearsal. Find these moments in the song that you can add dynamics. ==One with Yourself== While the goal in any musical situation is to create a powerful unified sound, don’t forget to be aware of your own internal clock and don’t solely rely on the drummer (or any other musician) to keep “time” for you. If you do and the drummer were to stop playing, your sense of “time” or lack of it, could show its ugly head. Make sure you're always concentrating and striving to be the best “time keeper” yourself, at the same time learning to work together with others to create that ultimate groove. ==Just Because They're Great Doesn't Make It So== Just because you may be playing with a masterful drummer doesn’t always guarantee that your groove and pocket will result in a locked and cohesive sound. When you are playing with great musicians or musicians better than yourself, it will often propel you to greater heights. However, there is one important often overlooked word—“'''chemistry'''”. I have seen and I have first hand experienced of playing with incredible drummers, but sometimes its a struggle to connect and make that unified sound. If the chemistry, communication and trust factor aren’t present between you and the drummer, this will destroy any hopes of making beautiful music together. When you find a unique camaraderie and chemistry with other musicians, you should cherish and treasure this secret ingredient. Listen to the great bass and drum combinations ==Listen to the Masters== If you want to learn how to lock in with a drummer, there’s no substitute then to listen to the master bass and drum combinations. There are hundreds if not thousands of great bass and drum teams that you should check out. Listen to the way they work together, off of each other and how they communicate as a rhythm section as well as how they communicate with the rest of the ensemble. ​In order to “'''lock in'''” and a have a stronger cohesive sound with your drummer, you need to create a healthy, honest and trusting relationship. Practice implementing some of these '''KEY POINTS'''. The drummer is and should be “the quarterback” and the bassist should be “the receiver”. Working together they can create a winning combination. ==JPM’s Favorite Bass and Drum Combinations== *James Jamerson and Benny Benjamin (“The Funk Brothers” The motown sound) *Paul Chambers and Philly Joe Jones (many CD and touring groups) *George Porter Jr and Joseph Zigaboo Modeliste (The Meters) *Paul McCartney and Ringo Starr (The Beatles) *Billy Cox and Buddy Miles (Jimi Hendrix) *Noel Redding and Mitch Mitchell (Jimi Hendrix) *Ron Carter and Tony Williams (Miles Davis) *John Entwistle and Keith Moon (The Who) *Duck Dunn and Al Jackson Jr. (Stax) *Bob Hurst and Jeff Tain Watts (many CD and touring groups) *John Paul Jones and John Bonham (Led Zeppelin) *Dave Holland and Jack Dejohnette (many CD and touring groups) *Adam Clayton and Larry Mullins Jr. (U2) *Jimmy Garrison and Elvin Jones (John Coltrane) *Michael Anthony and Alex Van Halen (Van Halen) *Robbie Shakespeare and Sly Dunbar (freelance reggae masters) *Eddie Gomez and Steve Gadd (many different cd and touring projects) *Geddy Lee and Neil Pert (Rush) *Victor Wooten with JD Blair and Futureman (Victor W. solo cds and Bela Fleck and the Flektones) *Sting and Stewart Copeland (The Police) *Jymie Merritt and Doug Watkins with Art Blakey (Art Blakey) *Mark Egan and Danny Gottlieb (Elements) *Jaco Pastorius and Peter Erskine (many different cd projects) *Steve Rodby and Paul Wertico (Pat Methany) *Paul Jackson and Mike Clark (Herbie Hancock) *George Morrow and Max Roach (Clifford Brown/Max Roach Quintet) [[Category:Guitar]] s6unmlooh2xqi5jxo1h5xxdjpyqlk44 Music production 0 234048 2812995 2723405 2026-06-06T00:19:14Z Kirby - Electrotechnics 3074947 changed to 'category:Music technology' instead of category:Music 2812995 wikitext text/x-wiki Music production is a parallel activity in music that aligns ''art with science'' to provide a better message and improve sound quality for an ideal product. This product is destined to the listener (i.e the ''public, i.e the consumer)'', either in a live performance, or contained in a sound recording (formally known as a phonogram, i.e sound in a physical mean). Universities and product developers test diverse methods in order to expand the knowledge, resources and technologies for this activity, not only for the delivery of a better product, but also to contribute for the evolution of music (in contemporary civilization) and better audio restoration/reproduction (of music from the past centuries). {{RightTOC}} ==Discussion question== * How can music positively impact the lives of listeners? * What are some different styles of music production? * What are different methods of music production? * Why is it that music can elicit emotions? ==Readings== * [[Rhyme schemes by set partition]] - in creating lyrics (whether rap or singing) rhyming can be helpful sometimes ===Wikipedia=== * [[Wikipedia:Audio engineer|Audio engineer]] * [[Wikipedia:Hip hop production|Hip hop production]] * [[Wikipedia:Ableton Live|Ableton Live]] * [[Wikipedia:List of music software|List of music software]] * [[Wikipedia:Digital audio workstation|Digital audio workstation]] ==Learning videos== * [https://www.youtube.com/watch?v=1IjDGFQLH8g How To Turn Loops Into Full Songs] * [https://www.youtube.com/watch?v=sH5OM1bEQrA Making a Traditional Hip-Hop Beat | FL Studio Tutorial] ==External links== * [https://www.reddit.com/r/makinghiphop/ Discuss Hiphop music production] * [https://www.reddit.com/r/WeAreTheMusicMakers/ Discuss music production] * [https://www.reddit.com/r/musictheory/ Discuss music theory] * [http://www.cras.edu/music-production/ Basic info on music production] [[Category:Music technology]] sc24iagufo9eja4je73bt8tabr0i4bd Digital Media Concepts/Trinh Cong Son (Vietnamese songwriter) 0 239916 2812987 2305694 2026-06-05T22:27:11Z Kirby - Electrotechnics 3074947 changed to 'category:Musicians' instead of category:Music 2812987 wikitext text/x-wiki == Trinh Cong Son == {{Infobox |name = Trinh Cong Son |image = |title = Songwriter Trinh Cong Son |headerstyle = background:#ccf; |labelstyle = background:#ddf; |label1 = |data1 = |label2 = Nationality |data2 = {{{item_one| Vietnamese}}} |label3 = Born |data3 = {{{item_two|Feb 28, 1939}}} |label4 = Died |data4 = {{{item_three|April 1, 2001}}} |label5 = Place of birth |data5 = {{{item_Four|Buon Me Thuot, Vietnam}}} |label6 = Place of death |data6 = {{{item_Five|Saigon, Vietnam}}} }} Trinh Cong Son was born on February 28, 1939, in [[wikipedia: Buôn Ma Thuột|Buôn Ma Thuột, Vietnam]], and he passed away on April 1, 2001, in[[wikipedia: Ho Chi Minh City| Saigon, Vietnam]]. He had written many anti-war songs<ref>{{Cite news|url=https://www.nytimes.com/2001/04/05/world/trinh-cong-son-62-stirred-vietnam-with-war-protest-songs.html|title=Trinh Cong Son, 62; Stirred Vietnam With War Protest Songs|last=Mydans|first=Seth|access-date=2018-09-21|language=en}}</ref> and love songs<ref>{{Cite web|url=https://www.tcs-home.org/english/articles/writing-ageless-songs-of-love-and-peace-1|title=Writing ageless songs of love and peace — Trinh Cong Son|website=Trinh Cong Son|language=vi|access-date=2018-09-21}}</ref> during the Vietnam war. After the [[wikipedia: Reunification Day|Reunification]] in 1975, he was sent to prison by the [[wikipedia: Communist Party of Vietnam|communist government]]. But, he was honored by the authority and many people sent their respects with floral tributes. == Education == Trinh Cong Son attended elementary schools and studied at French high school<ref>{{Cite web|url=http://www.phutranvan.com/en/from_one_experience_a_way.html|title=FROM ONE EXPERIENCE, A WAY|website=www.phutranvan.com|access-date=2018-09-24}}</ref> in Hue. Unfortunately, his father, [[wikipedia: Trịnh Xuân Thanh| Trinh Xuan Thanh]], who worked for the [[wikipedia: August Revolution|revolutionary movement]], died in 1955 when he crashed his [[wikipedia: Vespa|Vespa]] in [[wikipedia: Quang Tri|Quang Tri]]. Between 1956-1957, Son studied at the School run by the [[wikipedia: Roman Catholic Archdiocese of Huế|Catholic diocese in Hue]]. After receiving his [[wikipedia: baccalaureate|bachelor degree]], Son moved to Saigon where he studied philosophy at Le Quy Don High School<ref>{{Cite news|url=http://en.skydoor.net/entry/Historic_schools_of_Saigon/302|title=Historic schools of Saigon|work=SkyDoor|access-date=2018-09-23}}</ref> to avoid the draft. == Career == Trinh Cong Son was a teacher in [[wikipedia: Huế|Hue]] for many years in the mid-1950. However, he decided to give up teaching and became a songwriter. Son's songs were performed by famous singers such as [[wikipedia: Khanh Ly|Khanh Ly]] and [[wikipedia: Hong Nhung|Hong Nhung]]. Lastly, he had been the author of more than 600 songs<ref>{{Cite news|url=https://www.wired.com/2001/04/passage-trinh-cong-son-62/|title=Passage: Trinh Cong Son, 62|work=WIRED|access-date=2018-09-18|language=en-US}}</ref> which were popular with Vietnamese youth. == Noble work and award == In 2004, Son was a great influential songwriter of the anti-war movement <ref>{{Cite news|url=https://www.history.com/topics/vietnam-war/vietnam-war-protests|title=Vietnam War Protests|work=HISTORY|access-date=2018-09-21|language=en}}</ref> and he was one of the sixth honored songwriters at the Second Annual World Peace Music Award<ref>{{Cite web|url=https://www.countryjoe.com/musicawards.htm|title=Country Joe McDonald, World Peace Music Awards|website=www.countryjoe.com|access-date=2018-09-19}}</ref> concert in [[wikipedia: Hanoi|Hanoi]]. Especially, his peace song Lullaby<ref>{{Cite web|url=https://www.tcs-home.org/english/songs-en/songs/lullaby-of-the-artillery|title=Lullaby of the artillery — Trinh Cong Son|website=Trinh Cong Son|language=en|access-date=2018-09-19}}</ref> was the best hit in Asia during 1969. At this time, Son was sick, but his sister, Trinh Vinh Trinh<ref>{{Cite web|url=https://www.encyclopedia.com/history/encyclopedias-almanacs-transcripts-and-maps/son-trinh-cong|title=Son, Trinh Cong {{!}} Encyclopedia.com|website=www.encyclopedia.com|language=en|access-date=2018-09-25}}</ref> was at the [[wikipedia: United Nations|United Nations]] in New York when the award was announced. == Personal life == According to [http://news.bbc.co.uk/2/hi/entertainment/1260527.stm BBC news], Son's family successfully fled to America. After the war ended, some of Son's songs were banned in Vietnam. Son had owned a popular [[wikipedia: Cabaret|Cabaret]] in Saigon and he was a heavy drinker through his adult life. Unfortunately, Son was diagnosed with [[wikipedia: Diabetes mellitus|Diabetes mellitus]] and died in Sai Gon. Finally, Son had left his written composed songs to his sister, so that she would continue his career of writing songs. == Legacy == Trinh Cong Son's songs have influenced other songwriters in Vietnam such as [[wikipedia: Hoàng Thị Thơ|Hoang Thi Tho]], [[wikipedia: Phạm Duy|Pham Duy]], and [[wikipedia:Văn Cao|Van Cao]] about the power of love and the patriotism of his own country<ref>{{Cite journal|last=Vo|first=Van Dinh|title=Aesthetic Principles in the Music of Trinh Cong Son|url=http://www.academia.edu/14163397/Aesthetic_Principles_in_the_Music_of_Trinh_Cong_Son|language=en}}</ref>. They believed that the beauty and the patriotism of Son's songs will create a better world. Furthermore, Son's songs has performed many times by an Italian saxophonist, Fulvio Albano<ref>{{Cite book|url=https://books.google.com/books?id=EHuQAgAAQBAJ&pg=PA139&lpg=PA139&dq=fulvio+albano+jazz&source=bl&ots=mEtIIrmAPE&sig=uTINGq921WJA2JXBUw4eh0m75zk&hl=en&sa=X&ved=2ahUKEwi3lZHgldzdAhVSoFsKHY4gDxg4ChDoATAAegQICRAB#v=onepage&q=fulvio%20albano%20jazz&f=false|title=Popular Music of Vietnam: The Politics of Remembering, the Economics of Forgetting|last=Olsen|first=Dale A.|date=2008-06-30|publisher=Routledge|isbn=9781135858506|language=en}}</ref>. Finally, Son's songs have involved the beauty of the Vietnamese culture and the power of the Vietnamese language that influenced Fulvio<ref>{{Cite news|url=https://vietnamnews.vn/sunday/features/270308/writing-ageless-songs-of-love-and-peace.html#pp9fcLKAPVj9icpZ.97|title=Writing ageless songs of love and peace|work=vietnamnews.vn|access-date=2018-09-29|language=en}}</ref>. == External links == https://www.washingtonpost.com/archive/local/2001/04/03/vietnamese-songwriter-trinh-cong-son-dies/4566885c-b359-4a2e-945b-e3f5591bc614/ https://www.voanews.com/a/a-13-a-2004-02-04-21-vietnam-67338877/382266.html https://www.bbc.com/news/world-asia-pacific-16567315 http://www.who.int/mediacentre/factsheets/fs138/en/ https://www.tcs-home.org/english/songs-en/songs == Reference == [[Category: Musicians]] [[Category: Digital Media Concepts]] [[Category: Culture]] [[Category: Assignments]] rtfnuo8fpzdrqjnp8jauyhjadr7arz8 Digital Media Concepts/BROCKHAMPTON 0 239989 2812988 2588808 2026-06-05T22:27:48Z Kirby - Electrotechnics 3074947 changed to 'category:Musicians' instead of category:Music 2812988 wikitext text/x-wiki ='''BROCKHAMPTON (Band)'''= {{Infobox |name = BROCKHAMPTON |image = [[File:Brockhampton 2018.jpg|thumb|Brockhampton 2018]] |title = BROCKHAMPTON |headerstyle = background:#ccf; |labelstyle = background:#ddf; |label1 = Origin |data1 = San Marcos, Texas, U.S. |label2 = Genre |data2 = Alternative Hip Hop, Rap, R&B |label3 = Years Active |data3 = 2014-Present |label4 = Label |data4 = Question Everything, RCA |label5 = Website |data5 = http://brckhmptn.com/ |label6 = Members |data6 = Kevin Abstract Merlyn Wood Joba Matt Champion Dom McLennon Bearface Romil Hemnani Jabari Manwa Kiko Merley Henock "HK" Sileshi Ashlan Grey Robert Ontenient Jon Nunes Kevin Doan |label7 = Past Members |data7 = Ameer Vann Rodney Tenor }} BROCKHAMPTON is a musical group who call themselves an "all American boyband"[https://www.youtube.com/watch?v=7mKLT3KGUFQ] who formed in San Marcos, Texas in 2015[https://www.thefader.com/2017/07/13/brockhampton-members-kevin-abstract-interview-saturation]. The group rose to fame in 2017 with the help of their three albums titled SATURATION I which was released on June 9, 2017, SATURATION II on August 25, and SATURATION III on December 15. This Saturation trilogy is loosely based from their youtube show Billy Star, a story that follows the events of the main character Helmet Boy.[https://www.youtube.com/watch?v=DlCFNmqaS3Y&t=98s] These three albums were praised by critics as it focused on different types of genres such as Hip-Hop, Alternative R&B, Pop, and Soul. ==Group Formation== Kevin Abstract, one of the founding members of BROCKHAMPTON, in 2010 posted to the online forum KanyeToThe that was titled “Anybody wanna make a band?” [https://www.kanyetothe.com/forum/index.php?topic=51240.0] This post gained a some attention from people and would be the staple for the start of their group. The band was first named Alice Since Forever which had about 30 members altogether. They released their first EP in 2013 named the ASF EP [https://officialasf.bandcamp.com/] but broke up in 2014, and instead decided to rebrand their name BROCKHAMPTON which consisted of 17 people such as, Kevin Abstract Ameer Vann, Merlyn Wood, Dom McLennon, Matt Champion, etc. On March 24, 2016 they released their debute mixtape called All-American Trash, which had the group learning what worked and what didn't work for them. Eventually they decided if they wanted to start making even greater music they would have to work full-time, and the group all moved to LA.[https://www.thefader.com/2017/07/13/brockhampton-members-kevin-abstract-interview-saturation] ==Kevin Abstract and Viceland== Kevin Abstract focused on some of his solo work and eventually released his album "American Boyfriend: A Suburban Love Story"[https://pitchfork.com/reviews/albums/22689-american-boyfriend-a-suburban-love-story/] which was the beginning of the story of Billy Star. The album talks about Kevin Abstract's problems he has to be face while being gay and living in a conservative area where you don't feel accepted by people. The album received good reviews which made Kevin have a following. The big following Kevin received took interest from the TV program Viceland and decided to give Kevin Abstract his own TV show. He used this opportunity to give BROCKHAMPTON an outlet and decided to have the show focus on the group itself. The show would appear on June 8, 2017 and would be called American Boyband.[https://www.viceland.com/en_us/topic/brockhampton] The show consisted of eight episodes and showed the group and their process in making their debut album Saturation I. ==SATURATION Trilogy and Rise To Fame== The release of the SATURATION Trilogy was the start of something big for the boyband. Critic reception from this album was praised upon since they are known to be very a very consistent group, even though they are constantly changing their sound. What critics love about them though is all the members have such unique personalities and portray themselves differently. Some examples include from Kevin Abstract who talks about his sexuality, family, and dreams, Ameer Vann who talks struggles with his past and drugs, Dom Mclennon who talks about his constant drive to reach the top of the music industry, Merlyn Wood who is shown to be something as simple as a "hype man" but it is shown he has hidden layers through him. The stories they tell are unfiltered and blunt which is what drives the narrative of the story. This becomes interesting since they did all of this without signed to a record label and were completely independent. The group didn't have any money to go by as one of the group members says “They had no money,” their manager Kelly Clancy tells me. “Their diets consisted entirely of corner store stuff. We’d have morning meetings and dudes would be eating ice cream for breakfast.”[https://www.billboard.com/articles/columns/hip-hop/8465993/brockhampton-rise-hip-hop-boy-band] It was certainly until a YouTuber by the name of Anthony Fantano, had released his review for SATURATION I that the group had started to gain more and more attraction[https://www.youtube.com/watch?v=iBHjkRZQbv8] as he praised the album for its uniqueness. The group, as a special thanks to him, decide to put his name in one of their songs, ZIPPER.[https://www.youtube.com/watch?v=7lKl_YvTizw] This huge following BROCKHAMPTON had gotten had proven how today it is possible to gain attraction towards their music by using the internet to help them, instead of signing with a record deal to help them gain more popularity. ==External Links== https://www.youtube.com/watch?v=7mKLT3KGUFQ https://www.thefader.com/2017/07/13/brockhampton-members-kevin-abstract-interview-saturation https://www.youtube.com/watch?v=DlCFNmqaS3Y&t=98s https://www.kanyetothe.com/forum/index.php?topic=51240.0 https://officialasf.bandcamp.com/ https://pitchfork.com/reviews/albums/22689-american-boyfriend-a-suburban-love-story/ https://www.viceland.com/en_us/topic/brockhampton https://www.billboard.com/articles/columns/hip-hop/8465993/brockhampton-rise-hip-hop-boy-band https://www.youtube.com/watch?v=iBHjkRZQbv8 https://www.youtube.com/watch?v=7lKl_YvTizw ==References== <ref>''[https://www.youtube.com/watch?v=7mKLT3KGUFQ BROCKHAMPTON Talk 'Saturation III' & Redefining 'Boyband' | MTV News]'', </ref> <ref>''[https://www.thefader.com/2017/07/13/brockhampton-members-kevin-abstract-interview-saturation Get To Know The Members Of Brockhampton, A New Kind Of American Boy Band]'', </ref> <ref>''[https://www.youtube.com/watch?v=DlCFNmqaS3Y&t=98s Billy Star]'', </ref> <ref>''[https://www.kanyetothe.com/forum/index.php?topic=51240.0 Anybody wanna make a band?]'', </ref> <ref>''[https://officialasf.bandcamp.com/ The ASF EP]'', </ref> <ref>''[https://pitchfork.com/reviews/albums/22689-american-boyfriend-a-suburban-love-story/ Kevin Abstract American Boyfriend: A Suburban Love Story]'', </ref> <ref>''[https://www.viceland.com/en_us/topic/brockhampton BROCKHAMPTON]'', </ref> <ref>''[https://www.billboard.com/articles/columns/hip-hop/8465993/brockhampton-rise-hip-hop-boy-band How Hip-Hop Boy Band Brockhampton Scored a $15 Million Deal -- and Weathered a #MeToo Scandal]'', </ref> <ref>''[https://www.youtube.com/watch?v=iBHjkRZQbv8 Brockhampton - Saturation ALBUM REVIEW]'', </ref> <ref>''[https://www.youtube.com/watch?v=7lKl_YvTizw ZIPPER]'', </ref> {{reflist}} [[Category:Musicians]] [[Category:Culture]] [[Category:Hip hop]] dps9ue3tf2wvgxhoagurb6r8py529cu Digital Media Concepts/Love of my Life (Queen song) 0 245556 2813015 2239020 2026-06-06T00:35:33Z MathXplore 2888076 added [[Category:Songs]] using [[Help:Gadget-HotCat|HotCat]] 2813015 wikitext text/x-wiki {{Infobox | name = Queen's: "Love of My Life" | caption = Song by Queen from the Album ''A Night at the Opera'' | label1 = Released | data1 = November 21, 1975 | label2 = Length | background = 3:39 | data2 = 3:39 | label3 = Recorded | data3 = 1975 at various studios | label4 = [[:w:Record label|Record Label]] | data4 = [[:w:EMI Records|EMI]], [[:w:Elektra Records|Elektra]] | label5 = [[:w:Songwriter|Songwriter]] | data5 = [[:w:Freddie Mercury|Freddie Mercury]] | label6 = [[:w:Record producer|Producer(s)]] | data6 = [[:w:Queen (band)|Queen]], [[:w:Roy Thomas Baker|Roy Thomas Baker]] }} One the most famous songs, "'''Love of my Life'''" was created by the unparalleled [[:w:British rock music|British rock band]], famously known as "'''[[:w:Queen (band)|Queen]]'''. The [[:w:Sentimental ballad|ballad]] <ref>{{cite web|url=http://ultimateclassicrock.com/freddie-mercury-queen-songs/|title=Top 10 Freddie Mercury Queen Songs|website=Ultimate Classic Rock}}</ref> originated from the album ''[[:w:A Night at the Opera (Queen album)|A Night at the Opera]]'' and turned out to be a huge success towards many countries worldwide. The ballad was performed by all members of the British rock band consisting of lead [[:w:Vocal music|vocalist]] [[:w:Freddie Mercury|Freddie Mercury]], lead guitarist [[:w:Brian May|Brian May]], lead [[:w:Drummer|drummer]] [[:w:Roger Taylor (Queen drummer)|Roger Taylor]], and lead [[:w:Bassist|bassist]] [[:w:John Deacon|John Deacon]]. Love of my Life was first written on the [[:w:Piano|piano]] and [[:w:Guitar|guitar]] by the lead vocalist, Freddie Mercury. Brian May then touched up upon the ballad with an acoustic [[:w:Twelve-string guitar|12-string guitar]] solely for live performances. Brian May also occasionally added guitar phrases to the original recording and then played the harp. Followed by pasting together multiple takes of single chords altogether in the end. Queen’s “Love of My Life” is one classic tribute to love. [[File:Queenalbumfont.png]] ==Background Behind the Song== The intention of this song is dedicated to Freddie Mercury's ex-[[:wikt:fiancée|fiancée]], best friend, and the true love of his life, Mary Austin<ref>{{cite web|url=https://www.thevintagenews.com/2017/07/18/mary-austin-the-woman-who-inspired-freddie-mercury-as-a-muse-and-stood-by-him-at-the-end/amp/|title=Mary Austin: The woman who inspired Freddie Mercury as a muse and stood by him at the end|publisher=}}</ref>. Mercury's true passion for Mary Austin is clear in every performance of this song by Queen. Written entirely by Freddie Mercury himself, this song is a confession towards his unconditional love and affection towards Mary Austin. But who exactly is Mary Austin? Mary Austin was just an ordinary girl, born in 1951 in [[:w:London|London, UK]]. Her first encounter with a future Queen band member was first, Brian May during college. Mary Austin befriended Brian May and even dated for a little awhile, and since then Austin has tagged along with May into the upbringing of a forming a rag-tag band in college with Roger Taylor. After Brian May and Roger Taylor's lead singer departed from them, they were introduced to Freddie Mercury who then later on changed the group band name famously known to ‘Queen.’ This then began Mary Austin and Freddie Mercury’s relationship, where in 1973 they moved in together, and even got engaged. Unfortunately, they never got officially married due to their relationship fallout. Mercury and Austin resided together for seven years. As revealed to in an interview with Freddie’s mother Jer Bulsara by The Telegraph, Mary Austin labored two children named Richard and Jamie, not biologically related to Freddie. Mercury became the godfather of the two children of Mary. And even after Mercury and Austin's fallout with their relationship, they both remained very close friends until Freddie’s death in 1991. When Freddie tragically passed away due to [[:w:HIV/AIDS|AIDS]], Mary Austin was passed on Mercury’s massive mansion in [[:w:West London|West London]]. ==Accolades== [[File:Museum RockArt, Queen grootste hits.JPG|thumb|Museum RockArt, Queen grootste hits]] '''"Love of my life"''' is deemed to be one of Queen’s best works put forth of their mixed portfolio. Queen’s lead singer, Freddie Mercury takes a step back from the usual formula of Rock and writes and performs a new [[:w:Serenade|serenading song]] with his showing of his unbelievable vocal expertise. “Love of My Life” is also apart of one of the biggest and top selling Rock albums of all time, which is ‘A Night at the Opera’ by Queen from the year 1975. The album as well is also the home to one of the most well-respected Rock songs ever created “[[:w:Bohemian Rhapsody|Bohemian Rhapsody.]]” "Love of my life" was such a hit song worldwide that it was coincided with Queen's other hit songs into a highly requested, full pledged CD of [[:w:Greatest Hits (Queen album)|''Queen's Greatest Hits.'']] ==Band Members== [[File:Queen - montagem.png|thumb|Queen's band members: [[:w:Freddie Mercury|Freddie Mercury]], [[:w:Brian May|Brian May]] [[:w:Drummer|drummer]] [[:w:Roger Taylor (Queen drummer)|Roger Taylor]] [[:w:John Deacon|John Deacon]] * [[:w:Freddie Mercury|Freddie Mercury]] – [[:w:Vocal music|Vocals]] & [[:w:Piano|piano]] * [[:w:Brian May|Brian May]] – [[:w:Acoustic music|Acoustic]] & [[:w:Electric guitar|Electric Guitar]], [[:w:Harp|Orchestral Harp]] * [[:w:Roger Taylor (Queen drummer)|Roger Taylor]] – [[:w:Cymbal|Cymbals]] * [[:w:John Deacon|John Deacon]] – [[:w:Bass guitar|Bass Guitar]] ==Notable Queen Quotes== Some notable quotes by Queen band members regarding the ballad, "Love of my Life": {{cquote|''Interviewer: Did you learn to play harp for "Love Of My Life"?'' <br/>Learning would be too strong a word. I did it chord by chord. Actually, it took longer to tune the thing than to play it. It was a nightmare because every time someone opened the door, the temperature would change and the whole thing would go out. I would hate to have to play a harp on stage. I just figured out how it worked – the pedals and everything – and did it bit by bit.<ref>[http://www.queenarchives.com/index.php?title=Brian_May_-_XX-XX-1982_-_On_the_Record Queen Archives – Brian May Interview, ''On the Record'' 1982]. Queenarchives.com. Retrieved on 2018-11-06.</ref>|25px|25px|Brian May, 1982}} {{cquote|I hate this thing about trying to recreate the albums on stage. We sometimes deliberately change. Look at 'Love Of My Life', for instance. On record I play piano on that but on stage it's just Brian playing the guitar and me singing it, because it works on stage better that way.<ref>[http://www.queenpedia.com/index.php?title=Love_Of_My_Life Love Of My Life – Queenpedia.com – Freddie Mercury, Brian May, Roger Taylor, John Deacon, Discography, Bibliography, Charts]. Queenpedia.com (2009-08-19). Retrieved on 2015-10-21.</ref>|25px|25px|Freddie Mercury, 1984}} {{cquote|"Love Of My Life" is adapted on stage for guitar, but it was written on the piano. I've totally forgotten the original and if you asked me to play that now, I couldn't. Sometimes, I have to go back to the music sheet, and I can't read that well either.<ref>[http://www.queenarchives.com/index.php?title=Freddie_Mercury_-_05-02-1981_-_Melody_Maker Queen Archives – Freddie Mercury Interview, ''Melody Maker'' 05-02-1981]. Queenarchives.com. Retrieved on 2018-11-06.</ref>|25px|25px|Freddie Mercury, 1981}} ==External Links== * Official [[YouTube]] videos: [https://www.youtube.com/watch?v=sUJkCXE4sAA Original Music Video], [https://www.youtube.com/watch?v=2o2RwOWhhJY Love of my Life Remastered], [https://www.youtube.com/watch?v=jO9C7frk2ss Live Performance at Wembley], [https://www.youtube.com/watch?v=vNsEEvRMtGI Live Performance at Montreal], [https://www.youtube.com/watch?v=MIK4n3TbQQ0 Live Performed by Brian May w/ Freddie Mercury Video] * [http://www.queenonline.com/en/the-band/discography/night-opera/#lyrics Lyrics at Queen official website] * [http://justrandomthings.com/2018/11/06/queen-love-of-my-life-lyrics-review-and-song-meaning/ Lyric Review & Song Meaning] ==References== [[Category:Songs]] 1o0hccsp5s42083wez75p3r9pe3807l 2813016 2813015 2026-06-06T00:35:40Z MathXplore 2888076 Added {{[[Template:BookCat|BookCat]]}} using [[User:1234qwer1234qwer4/BookCat.js|BookCat.js]] 2813016 wikitext text/x-wiki {{Infobox | name = Queen's: "Love of My Life" | caption = Song by Queen from the Album ''A Night at the Opera'' | label1 = Released | data1 = November 21, 1975 | label2 = Length | background = 3:39 | data2 = 3:39 | label3 = Recorded | data3 = 1975 at various studios | label4 = [[:w:Record label|Record Label]] | data4 = [[:w:EMI Records|EMI]], [[:w:Elektra Records|Elektra]] | label5 = [[:w:Songwriter|Songwriter]] | data5 = [[:w:Freddie Mercury|Freddie Mercury]] | label6 = [[:w:Record producer|Producer(s)]] | data6 = [[:w:Queen (band)|Queen]], [[:w:Roy Thomas Baker|Roy Thomas Baker]] }} One the most famous songs, "'''Love of my Life'''" was created by the unparalleled [[:w:British rock music|British rock band]], famously known as "'''[[:w:Queen (band)|Queen]]'''. The [[:w:Sentimental ballad|ballad]] <ref>{{cite web|url=http://ultimateclassicrock.com/freddie-mercury-queen-songs/|title=Top 10 Freddie Mercury Queen Songs|website=Ultimate Classic Rock}}</ref> originated from the album ''[[:w:A Night at the Opera (Queen album)|A Night at the Opera]]'' and turned out to be a huge success towards many countries worldwide. The ballad was performed by all members of the British rock band consisting of lead [[:w:Vocal music|vocalist]] [[:w:Freddie Mercury|Freddie Mercury]], lead guitarist [[:w:Brian May|Brian May]], lead [[:w:Drummer|drummer]] [[:w:Roger Taylor (Queen drummer)|Roger Taylor]], and lead [[:w:Bassist|bassist]] [[:w:John Deacon|John Deacon]]. Love of my Life was first written on the [[:w:Piano|piano]] and [[:w:Guitar|guitar]] by the lead vocalist, Freddie Mercury. Brian May then touched up upon the ballad with an acoustic [[:w:Twelve-string guitar|12-string guitar]] solely for live performances. Brian May also occasionally added guitar phrases to the original recording and then played the harp. Followed by pasting together multiple takes of single chords altogether in the end. Queen’s “Love of My Life” is one classic tribute to love. [[File:Queenalbumfont.png]] ==Background Behind the Song== The intention of this song is dedicated to Freddie Mercury's ex-[[:wikt:fiancée|fiancée]], best friend, and the true love of his life, Mary Austin<ref>{{cite web|url=https://www.thevintagenews.com/2017/07/18/mary-austin-the-woman-who-inspired-freddie-mercury-as-a-muse-and-stood-by-him-at-the-end/amp/|title=Mary Austin: The woman who inspired Freddie Mercury as a muse and stood by him at the end|publisher=}}</ref>. Mercury's true passion for Mary Austin is clear in every performance of this song by Queen. Written entirely by Freddie Mercury himself, this song is a confession towards his unconditional love and affection towards Mary Austin. But who exactly is Mary Austin? Mary Austin was just an ordinary girl, born in 1951 in [[:w:London|London, UK]]. Her first encounter with a future Queen band member was first, Brian May during college. Mary Austin befriended Brian May and even dated for a little awhile, and since then Austin has tagged along with May into the upbringing of a forming a rag-tag band in college with Roger Taylor. After Brian May and Roger Taylor's lead singer departed from them, they were introduced to Freddie Mercury who then later on changed the group band name famously known to ‘Queen.’ This then began Mary Austin and Freddie Mercury’s relationship, where in 1973 they moved in together, and even got engaged. Unfortunately, they never got officially married due to their relationship fallout. Mercury and Austin resided together for seven years. As revealed to in an interview with Freddie’s mother Jer Bulsara by The Telegraph, Mary Austin labored two children named Richard and Jamie, not biologically related to Freddie. Mercury became the godfather of the two children of Mary. And even after Mercury and Austin's fallout with their relationship, they both remained very close friends until Freddie’s death in 1991. When Freddie tragically passed away due to [[:w:HIV/AIDS|AIDS]], Mary Austin was passed on Mercury’s massive mansion in [[:w:West London|West London]]. ==Accolades== [[File:Museum RockArt, Queen grootste hits.JPG|thumb|Museum RockArt, Queen grootste hits]] '''"Love of my life"''' is deemed to be one of Queen’s best works put forth of their mixed portfolio. Queen’s lead singer, Freddie Mercury takes a step back from the usual formula of Rock and writes and performs a new [[:w:Serenade|serenading song]] with his showing of his unbelievable vocal expertise. “Love of My Life” is also apart of one of the biggest and top selling Rock albums of all time, which is ‘A Night at the Opera’ by Queen from the year 1975. The album as well is also the home to one of the most well-respected Rock songs ever created “[[:w:Bohemian Rhapsody|Bohemian Rhapsody.]]” "Love of my life" was such a hit song worldwide that it was coincided with Queen's other hit songs into a highly requested, full pledged CD of [[:w:Greatest Hits (Queen album)|''Queen's Greatest Hits.'']] ==Band Members== [[File:Queen - montagem.png|thumb|Queen's band members: [[:w:Freddie Mercury|Freddie Mercury]], [[:w:Brian May|Brian May]] [[:w:Drummer|drummer]] [[:w:Roger Taylor (Queen drummer)|Roger Taylor]] [[:w:John Deacon|John Deacon]] * [[:w:Freddie Mercury|Freddie Mercury]] – [[:w:Vocal music|Vocals]] & [[:w:Piano|piano]] * [[:w:Brian May|Brian May]] – [[:w:Acoustic music|Acoustic]] & [[:w:Electric guitar|Electric Guitar]], [[:w:Harp|Orchestral Harp]] * [[:w:Roger Taylor (Queen drummer)|Roger Taylor]] – [[:w:Cymbal|Cymbals]] * [[:w:John Deacon|John Deacon]] – [[:w:Bass guitar|Bass Guitar]] ==Notable Queen Quotes== Some notable quotes by Queen band members regarding the ballad, "Love of my Life": {{cquote|''Interviewer: Did you learn to play harp for "Love Of My Life"?'' <br/>Learning would be too strong a word. I did it chord by chord. Actually, it took longer to tune the thing than to play it. It was a nightmare because every time someone opened the door, the temperature would change and the whole thing would go out. I would hate to have to play a harp on stage. I just figured out how it worked – the pedals and everything – and did it bit by bit.<ref>[http://www.queenarchives.com/index.php?title=Brian_May_-_XX-XX-1982_-_On_the_Record Queen Archives – Brian May Interview, ''On the Record'' 1982]. Queenarchives.com. Retrieved on 2018-11-06.</ref>|25px|25px|Brian May, 1982}} {{cquote|I hate this thing about trying to recreate the albums on stage. We sometimes deliberately change. Look at 'Love Of My Life', for instance. On record I play piano on that but on stage it's just Brian playing the guitar and me singing it, because it works on stage better that way.<ref>[http://www.queenpedia.com/index.php?title=Love_Of_My_Life Love Of My Life – Queenpedia.com – Freddie Mercury, Brian May, Roger Taylor, John Deacon, Discography, Bibliography, Charts]. Queenpedia.com (2009-08-19). Retrieved on 2015-10-21.</ref>|25px|25px|Freddie Mercury, 1984}} {{cquote|"Love Of My Life" is adapted on stage for guitar, but it was written on the piano. I've totally forgotten the original and if you asked me to play that now, I couldn't. Sometimes, I have to go back to the music sheet, and I can't read that well either.<ref>[http://www.queenarchives.com/index.php?title=Freddie_Mercury_-_05-02-1981_-_Melody_Maker Queen Archives – Freddie Mercury Interview, ''Melody Maker'' 05-02-1981]. Queenarchives.com. Retrieved on 2018-11-06.</ref>|25px|25px|Freddie Mercury, 1981}} ==External Links== * Official [[YouTube]] videos: [https://www.youtube.com/watch?v=sUJkCXE4sAA Original Music Video], [https://www.youtube.com/watch?v=2o2RwOWhhJY Love of my Life Remastered], [https://www.youtube.com/watch?v=jO9C7frk2ss Live Performance at Wembley], [https://www.youtube.com/watch?v=vNsEEvRMtGI Live Performance at Montreal], [https://www.youtube.com/watch?v=MIK4n3TbQQ0 Live Performed by Brian May w/ Freddie Mercury Video] * [http://www.queenonline.com/en/the-band/discography/night-opera/#lyrics Lyrics at Queen official website] * [http://justrandomthings.com/2018/11/06/queen-love-of-my-life-lyrics-review-and-song-meaning/ Lyric Review & Song Meaning] ==References== [[Category:Songs]] {{BookCat}} 9zjia18qrki0pz8ew6e03kn4kt4y1vx Template:Shortcut/styles.css 10 253044 2813174 2289424 2022-09-12T16:29:37Z Izno 21264 delete margin inside mbox, it already has a padding 2813174 sanitized-css text/css /* {{pp-template}} */ .module-shortcutboxplain { float: right; margin: 0 0 0 1em; border: 1px solid #aaa; background: #fff; padding: 0.3em 0.6em 0.2em 0.6em; text-align: center; font-size: 85%; } .module-shortcutboxleft { float: left; margin: 0 1em 0 0; } .module-shortcutlist { display: inline-block; border-bottom: 1px solid #aaa; margin-bottom: 0.2em; } .module-shortcutboxplain > ul { font-weight: bold; } .module-shortcutanchordiv { position: relative; top: -3em; } li .module-shortcutanchordiv { float: right; /* IE/Edge in list items */ } .mbox-imageright .module-shortcutboxplain { padding: 0.4em 1em 0.4em 1em; line-height: 1.3; margin: 0; } 6e6xiz2nxulblepkebq48z8dj8ux4mf 2813175 2813174 2022-12-13T23:35:56Z Izno 21264 move this to module, remove the > for the ul which is overly defensive in this template 2813175 sanitized-css text/css /* {{pp-template}} */ .module-shortcutboxplain { float: right; margin: 0 0 0 1em; border: 1px solid #aaa; background: #fff; padding: 0.3em 0.6em 0.2em 0.6em; text-align: center; font-size: 85%; } .module-shortcutboxleft { float: left; 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} .module-shortcutlist { display: inline-block; border-bottom: 1px solid #aaa; margin-bottom: 0.2em; } .module-shortcutboxplain ul { font-weight: bold; } .module-shortcutanchordiv { position: relative; top: -3em; } li .module-shortcutanchordiv { float: right; /* IE/Edge in list items */ } .mbox-imageright .module-shortcutboxplain { padding: 0.4em 1em 0.4em 1em; line-height: 1.3; margin: 0; } nxv0gudbcek6se7teobafngr84wwn3s 2813177 2813176 2022-12-13T23:39:45Z Izno 21264 Adding {{pp-template}} 2813177 sanitized-css text/css /* <noinclude>{{pp-template|small=yes}}</noinclude> *//* {{pp-template}} */ .module-shortcutboxplain { float: right; margin: 0 0 0 1em; border: 1px solid #aaa; background: #fff; padding: 0.3em 0.6em 0.2em 0.6em; text-align: center; font-size: 85%; } .module-shortcutboxleft { float: left; margin: 0 1em 0 0; } .module-shortcutlist { display: inline-block; border-bottom: 1px solid #aaa; margin-bottom: 0.2em; } .module-shortcutboxplain ul { font-weight: bold; } .module-shortcutanchordiv { position: relative; top: -3em; } li .module-shortcutanchordiv { float: right; /* IE/Edge in list items */ } .mbox-imageright .module-shortcutboxplain { padding: 0.4em 1em 0.4em 1em; line-height: 1.3; margin: 0; } t5fiufyugggb9kr4u3eq2fikfhkyl78 2813178 2813177 2022-12-13T23:39:56Z Izno 21264 gibberish 2813178 sanitized-css text/css /* {{pp-template}} */ .module-shortcutboxplain { float: right; margin: 0 0 0 1em; border: 1px solid #aaa; background: #fff; padding: 0.3em 0.6em 0.2em 0.6em; text-align: center; font-size: 85%; } .module-shortcutboxleft { float: left; margin: 0 1em 0 0; } .module-shortcutlist { display: inline-block; border-bottom: 1px solid #aaa; margin-bottom: 0.2em; } .module-shortcutboxplain ul { font-weight: bold; } .module-shortcutanchordiv { position: relative; top: -3em; } li .module-shortcutanchordiv { float: right; /* IE/Edge in list items */ } .mbox-imageright .module-shortcutboxplain { padding: 0.4em 1em 0.4em 1em; line-height: 1.3; margin: 0; } nxv0gudbcek6se7teobafngr84wwn3s 2813179 2813178 2023-03-14T12:20:54Z TheDJ 21679 clear shortcuts on smaller screens 2813179 sanitized-css text/css /* {{pp-template}} */ .module-shortcutboxplain { float: right; 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} .module-shortcutlist { display: inline-block; border-bottom: 1px solid #aaa; margin-bottom: 0.2em; } .module-shortcutboxplain ul { font-weight: bold; } .module-shortcutanchordiv { position: relative; top: -3em; } li .module-shortcutanchordiv { float: right; /* IE/Edge in list items */ } .mbox-imageright .module-shortcutboxplain { padding: 0.4em 1em 0.4em 1em; line-height: 1.3; margin: 0; } h2yezvi2gjx939dl5xy8oj426gp3r7c 2813182 2813181 2024-04-28T16:35:22Z JWBTH 2944618 use border color from the style guide: https://doc.wikimedia.org/codex/latest/design-tokens/border.html 2813182 sanitized-css text/css /* {{pp-template}} */ .module-shortcutboxplain { float: right; margin: 0 0 0 1em; border: 1px solid #a2a9b1; background: #fff; color: inherit; /* support dark mode */ padding: 0.3em 0.6em 0.2em 0.6em; text-align: center; font-size: 85%; } .module-shortcutboxleft { float: left; margin: 0 1em 0 0; } .module-shortcutlist { display: inline-block; border-bottom: 1px solid #a2a9b1; margin-bottom: 0.2em; } .module-shortcutboxplain ul { font-weight: bold; } .module-shortcutanchordiv { position: relative; top: -3em; } li .module-shortcutanchordiv { float: right; /* IE/Edge in list items */ } .mbox-imageright .module-shortcutboxplain { padding: 0.4em 1em 0.4em 1em; line-height: 1.3; margin: 0; } 5gsmhoyto3c2ht4gpqet7q4uslia1j4 2813183 2813182 2024-05-25T05:10:31Z Izno 21264 use the bg-color 2813183 sanitized-css text/css /* {{pp-template}} */ .module-shortcutboxplain { float: right; margin: 0 0 0 1em; border: 1px solid #a2a9b1; background-color: var(--background-color-base, #fff); padding: 0.3em 0.6em 0.2em 0.6em; text-align: center; font-size: 85%; } .module-shortcutboxleft { float: left; margin: 0 1em 0 0; } .module-shortcutlist { display: inline-block; border-bottom: 1px solid #a2a9b1; margin-bottom: 0.2em; } .module-shortcutboxplain ul { font-weight: bold; } .module-shortcutanchordiv { position: relative; top: -3em; } li .module-shortcutanchordiv { float: right; /* IE/Edge in list items */ } .mbox-imageright .module-shortcutboxplain { padding: 0.4em 1em 0.4em 1em; 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margin: 0 0 0 1em; border: 1px solid var(--border-color-base, #a2a9b1); background-color: var(--background-color-base, #fff); padding: 0.3em 0.6em 0.2em 0.6em; text-align: center; font-size: 85%; } .module-shortcutboxleft { float: left; margin: 0 1em 0 0; } .module-shortcutlist { display: inline-block; border-bottom: 1px solid var(--border-color-base, #a2a9b1); margin-bottom: 0.2em; } .module-shortcutboxplain ul { font-weight: bold; } .module-shortcutanchordiv { position: relative; top: -3em; } li .module-shortcutanchordiv { float: right; /* IE/Edge in list items */ } .mbox-imageright .module-shortcutboxplain { padding: 0.4em 1em 0.4em 1em; line-height: 1.3; margin: 0; float: initial; } 0sj5gx0epg27cws08ifhyc5binoawf6 2813186 2813185 2025-10-03T21:27:48Z Izno 21264 shorterhand padding 2813186 sanitized-css text/css /* {{pp-template}} */ .module-shortcutboxplain { float: right; margin: 0 0 0 1em; border: 1px solid var(--border-color-base, #a2a9b1); background-color: var(--background-color-base, #fff); padding: 0.3em 0.6em 0.2em 0.6em; text-align: center; font-size: 85%; } .module-shortcutboxleft { float: left; margin: 0 1em 0 0; } .module-shortcutlist { display: inline-block; border-bottom: 1px solid var(--border-color-base, #a2a9b1); margin-bottom: 0.2em; } .module-shortcutboxplain ul { font-weight: bold; } .module-shortcutanchordiv { position: relative; top: -3em; } li .module-shortcutanchordiv { float: right; /* IE/Edge in list items */ } .mbox-imageright .module-shortcutboxplain { padding: 0.4em 1em; line-height: 1.3; margin: 0; float: initial; } iy0abv0s7wkq06bjcild5vh6fo8vjrr 2813187 2763686 2026-06-06T08:30:22Z Jtneill 10242 13 revisions imported from [[:w:Module:Shortcut/styles.css]] 2763686 sanitized-css text/css .module-shortcutboxplain { float: right; margin: 0 0 0 1em; border: 1px solid var(--border-color-base, #aaa); background: var(--background-color-base, #fff); color: inherit; padding: 0.3em 0.6em 0.2em 0.6em; text-align: center; font-size: 85%; } .module-shortcutboxleft { float: left; margin: 0 1em 0 0; } .module-shortcutlist { display: inline-block; border-bottom: 1px solid var(--border-color-base, #aaa); margin-bottom: 0.2em; } .module-shortcutboxplain > ul { font-weight: bold; } .module-shortcutanchordiv { position: relative; top: -3em; } li .module-shortcutanchordiv { float: right; /* IE/Edge in list items */ } .mbox-imageright .module-shortcutboxplain { padding: 0.4em 1em 0.4em 1em; line-height: 1.3; } tr5ej6due7szvc428hxl7wi6q9dcre1 Theory of Musical Equilibration 0 260947 2812964 2536679 2026-06-05T21:53:32Z Kirby - Electrotechnics 3074947 changed to 'category:Music theory' instead of category:Music 2812964 wikitext text/x-wiki The Theory of Musical Equilibration<ref>[https://www.willimekmusic.de/music-and-emotions.pdf See Daniela Willimek, Bernd Willimek:] ''Music and Emotions. Research on the Theory of Musical Equilibration.'' 2013.</ref> (in German: die Strebetendenz-Theorie) creates a psychological paradigm which explains the emotional effects of musical harmonies. The theory was designed by music theorist Bernd Willimek and developed together with his wife Daniela Willimek to today's version. It emerges form [[w:Ernst Kurth|Ernst Kurth's]] teaching in music psychology and states that in contrast to previous hypotheses, music does not directly describe emotions: instead; it evokes processes of the will which the listener identifies with, and relating to these processes gives music its emotional content. The theory was first presented at the Karlsruhe University of Music and the University of Rostock in 1997 and then published in 1998. == The fundamentals of the Theory of Musical Equilibration == The basic statement of the Theory of Musical Equilibration is as follows: emotions conveyed or evoked by tonal music such as anger, grief, loneliness or longing can be explained by identifications with abstract will content. Conversely, emotions that cannot be explained by identifications with abstract will content, such as envy, jealousy, hatred, embarrassment, boredom, pity, contempt, shame or disgust, cannot be conveyed or evoced by tonal music. In contrast to Ernst Kurth, the Theory of Musical Equilibration interprets the perception of leading tones as a phenomenon in which the listeners identify with the desire not to change the leading notes. Because of its leading tone, a major chord, for example, is something listeners generally identify with the message “I want to!” whereas a minor chord conveys the desire “No more!” The volume at which a minor chord is played determines whether it is perceived as sorrow or anger. == The emotional characters of musical harmonies == The following descriptions of sound characters corresponded to the derivation from the Theory of Musical Equilibration: *Major tonic: a feeling of sober-minded contentment.<ref>See Deryck Cooke: ''The Language of Music.'' Oxford, Oxford University Press 1959, P. 50.</ref> *Minor tonic: grief (when quietly), anger (when loudly).<ref>See Deryck Cooke: ''The Language of Music.'' Oxford, Oxford University Press 1959, P. 50.</ref> *Natural minor: courage, tension. *Dominant: a feeling of motion.<ref name="Cooke">See Deryck Cooke: ''The Language of Music.'' Oxford, Oxford University Press 1959, P. 90.</ref> *Dominant of a minor tonic: assumes the chracter of the tonic. *Seventh chord: resistance. *Secondary dominant: being painfully moved. *Major subdominant: lighttheartedness. *Major subdominant with a major seventh: longing, wistfulness.<ref>See Deryck Cooke: ''The Language of Music'', Oxford: Oxford University Press 1959, P. 90.</ref><ref>[http://emusicology.org/article/view/4534/4151 See Imre Lahdelma, Tuomas Eerola]: ''Theoretical Proposals on How Vertical Harmony May Convey Nostalgia and Longing in Music.'' Empirical Musicology Review, 10 (3).</ref> *Subdominant with added sixth in major: warmth, security. *Subdominant with added sixth in minor: heartbreak, loneliness. *Neapolitan sixth chord: pain, death. *Diminished seventh chord: fright, despair.<ref>See Arnold Schoenberg: ''Harmonielehre.'' Wien, Universal Edition 1922, P. 288.</ref> *Augmented chord: astonishment, magic. *Whole-tone scale: weightlessness.<ref>See Miranda Wilson: ''Cello Practise, Cello Performance.'' Lanham, Rowman and Littlefield 2015, P. 119.</ref> *Minor sixth: fear.<ref>See Deryck Cooke: ''The Language of Music'', Oxford, Oxford University Press 1959, P. 90.</ref> *Fifth: ghostly.<ref>See Friedrich Herzfeld: ''Ullstein Musiklexikon.'' Berlin et. al., Ullstein 1965, P. 431.</ref> *Tritone: devilish, inimical forces.<ref>See Deryck Cooke: ''The Language of Music'', Oxford, Oxford University Press 1959, P. 90.</ref> == Lectures on the Theory of Musical Equilibration at international conferences == *Sempre Met 2018: ''[https://www.educamus.cl/wp-content/uploads/2018/05/2018-MET-Actas_oki.pdf Theory of Musical Equilibration and the Use of Stimulus Chords in Romantic Lieder.]'' bei der ''Sempre MET Researching Music, Education, Technology.'' University of London, 2018. *Apscom 6: ''[http://apscom2017.org/pdf/Oral.pdf Music and Emotion.]'' bei der ''6th Conference of the Asia-Pacific Society for the Cognitive Sciences of Music (APSCOM 6).'' Women's University Kyoto, 2017.<br> *Tags 2017: ''[http://tags2017.bangor.ac.uk/documents/TAGS_2017_Bangor_Programme.pdf Music and Emotions: Research on the Theory of Musical Equilibration.]'' bei der "Theory & Analysis Graduate Students' Conference (TAGS)". Bangor University, 2017. *Guap 2017: ''[http://lmac.hf-guap.ru/en/2017/03/08/program-of-second-international-workshop-on-language-music-and-computing/ Music as the Language of Emotions.]'' beim ''Second International Workshop on Language, Music and Computing.'' St. Petersburg, 2017. *IMS 2016: ''[http://media.wix.com/ugd/510480_4e3aa5de255a4868b0a86de92b7fd15e.pdf Israel Musicological Society 2016 Music and Emotions - Research on the Theory of Musical Equilibration.]'' bei der "Winterkonferenz der Israel Musicological Society". Tel Aviv, 2016. == Literature == *Daniela Willimek, Bernd Willimek: ''[https://www.tandfonline.com/doi/abs/10.1080/25742442.2023.2185064 Revealing the Mystery of Emotions in Sounds The Theory of Musical Equilibration Explains the Impact of Ordered Sounds as the Listener’s Identification with Processes of Will.'' In: Timothy L. Hubbard (Ed.): ''Auditory Perception & Cognition.'', London, Taylor & Francis 2023, 6:1-2, DOI: 10.1080/25742442.2023.2185064, p. 128-153. *Bernd Willimek, Daniela Willimek: ''[https://www.educamus.cl/wp-content/uploads/2018/05/2018-MET-Actas_oki.pdf The Theory of Musical Equilibration and the Use of Stimulus Chords in Romantic Lieder.]'' In: Evangelos Himonides, Andrew King, Francisco Cuadrado (Hrsg.): ''Proceedings of the Sempre MET 2018. Researching Music, Education, Technology.'' London, International Music Education Research Centre 2018, ISBN: 978-1-905351-37-4, S. 45. *Daniela Willimek, Bernd Willimek: ''[https://www.cambridgescholars.com/music-on-stage-volume-iii Sound and Emotions. The Theory of Musical Equilibration.]'' In Fiona Schopf (Hrsg.): ''Music on Stage, Vol III.'', Cambridge, Cambridge Scholars Publishing 2018, ISBN: 978-1-5275-1161-3, S. 202-223. *Daniela Willimek, Bernd Willimek: ''[http://www.apscom2017.org/program.html Music and Emotions. Research on the Theory of Musical Equilibration.]'' In Iwao Yoshino: ''APSCOM 6: Proceedings of the 6th Conference of the Asia-Pacific Society for the Cognitive Sciences of Music.'' Kyoto 2017, S. 34. == External Links == *Daniela Willimek, Bernd Willimek: [https://theoryofmusic.wordpress.com/2015/11/14/the-theory-of-music-equilibration/ ''The Theory of Musical Equilibration.''] In: ''Theory of Music.'' November 2015. Retrieved on 21 March 2020. *Bernd Willimek, Daniela Willimek: [https://www.scirp.org/journal/paperinformation.aspx?paperid=73939 ''Feelings Which Strike a Chord, and Chord Which Strike a Feeling.''] In: ''Scientific Research.'' March 2017. Retrieved on 21 March 2020. *Catherine Meeson: [https://www.cyclicdefrost.com/2014/03/review-of-laura-russells-translation-2013-of-research-on-the-theory-of-musical-equilibration-die-strebetendenz-theorie/ Review of Laura Russells Translation (2013) of ''Research on the Theory of Musical Equilibration (die Strebetendenz-Theorie)''] In: ''Cyclic Defrost.'' March 2014. Retrieved on 21 March 2020. *Luiz Gazzola: ''[https://operalively.com/forums/showthread.php/2226-Theory-of-Musical-Equilibration-emotional-effects-of-music Theory of Musical Equilibration. Emotional Effects of Music]'' In: ''Opera Lively.'' September 2014. Retrived on 24 March 2020. *Bernd Willimek: [https://www.researchgate.net/publication/271102416_Why_do_Minor_Chords_Sound_Sad_The_Theory_of_Musical_Equilibration_and_the_Emotions_of_Chords ''Why do Minor Chords Sound Sad? The Theory of Musical Equilibration and the Emotions of Chords.''] In: ''ResearchGate.'' January 2014. Retrieved on 21 March 2020. == References == [[Category:Music theory]] tl3acnpylq71u1h1t7ur1mvnz9x2nt5 Category:Motivation and emotion/Book/Autism 14 267278 2812831 2596429 2026-06-05T12:04:43Z Jtneill 10242 added [[Category:Motivation and emotion/Book/Psychopathology]] using [[Help:Gadget-HotCat|HotCat]] 2812831 wikitext text/x-wiki [[Category:Motivation and emotion/Book]] [[Category:Autism]] [[Category:Motivation and emotion/Book/Psychopathology]] tom17r9u0msh3r4yoj3dxmxgeo95kmy 2812832 2812831 2026-06-05T12:04:59Z Jtneill 10242 added [[Category:Motivation and emotion/Book/Neurodiversity]] using [[Help:Gadget-HotCat|HotCat]] 2812832 wikitext text/x-wiki [[Category:Motivation and emotion/Book]] [[Category:Autism]] [[Category:Motivation and emotion/Book/Psychopathology]] [[Category:Motivation and emotion/Book/Neurodiversity]] dmi2ek6igxuy2lpelgthos2xpt0wuf4 Maritime Health Research and Education-NET 0 267737 2813028 2812153 2026-06-06T07:47:49Z Saltrabook 1417466 2813028 wikitext text/x-wiki == '''The John Snow Prediabetes-Remission Research Institute''' == [[File:ChatGPT Image 30 may 2026, 11 58 20 a.m.png|thumb|Prediabetes-remission research program]] [[File:ChatGPT Image 24 abr 2026, 08 16 04 a.m.png|thumb|]] Millions are at increased risk of developing metabolic syndromes with prediabetes, diabetes type 2, high blood pressure and overweight. All can lower their risks by staying physical active and eating well. For early identification of the risks we propose to register weight and height (BMI) and the fasting blood sugar in the '''Prevalence studies''' at the schools for seafarers, nurses, medical students and the kids schools.(12+ y).The 16-weeks '''intervention studies''' include learnings by short video sequences and self-monitoring of blood sugar with glucometer, and self-evaluation of diet and physical activity. Early diagnosis of prediabetes can provide both health and financial benefits.From a financial perspective, preventing or delaying diabetes can significantly lower healthcare costs. Early diagnosis of prediabetes is a cost-effective preventive strategy that can improve long-term health outcomes while helping individuals and healthcare systems avoid the substantial costs associated with diabetes and its complications.[[File:Lifestyle Medicine Pillars.png|300px|right|The focus of Lifestyle Medicine is on these 6 pillars.]] [[File:John Snow.jpg|thumb|left| John Snow in the early nineteenth century]] [[File:Cholera in London 1866.gif|thumb|250px|Map of a later cholera outbreak in London, in 1866]] [[File:Choleramaplondon1866.png|thumb|right|250px|Legend for the map above]] 1. '<nowiki/>'''Prevalence studies'''' 1.1 The-International-Maritime-Health-Database <ref>https://www.dropbox.com/scl/fi/z3cq5ciiev06y8v9duw7u/A-International-Maritime-Health-Database.docx?cloud_editor=word&dl=0&rlkey=pt0kdesvmagcxaa2wez3tmza3 </ref> 1.2 Nursing Students Health Database <ref> https://www.dropbox.com/scl/fi/tcznmmd2y3nona5e3h1ro/The-Nursing-students-health-database.docx?cloud_editor=word&dl=0&rlkey=onbjh4o8ko1lzdvgyi8nlrotk </ref> 1.3. Medical student's Health Database <ref>https://www.dropbox.com/scl/fi/f16h9b60u4gxgt56un2jf/The-Medical-students-Health-database.docx?cloud_editor=word&dl=0&rlkey=xyfqen5trdc5lniaovipl548n </ref> 1.4. School childrens Health database <ref> https://www.dropbox.com/scl/fi/u6u50c8bxwhte9t2t6ck8/The-School-children-s-Health-database.docx?cloud_editor=word&dl=0&rlkey=zlyz5wn673wf7owettq3nx3h5 </ref> 2. '''Intervention studies''' Englsh <ref>https://www.dropbox.com/scl/fi/oi6cx6tlwwvoko3ed37tn/Invitation-to-the-course-English.docx?cloud_editor=word&dl=0&rlkey=7kzg91tqfgjskxf5aji8khicx </ref> Danish <ref>https://www.dropbox.com/scl/fi/2qahc3q9hmf4skbvk77ab/Invitation-to-the-course-in-Danish.docx?cloud_editor=word&dl=0&rlkey=x63w8oqvarz284zg2btq2johv </ref> Spanish <ref> https://www.dropbox.com/scl/fi/bn71inqeeth4o4mc1fjth/Invitation-to-the-course-Spanish.docx?cloud_editor=word&dl=0&rlkey=popmr1fnodh1v951v9l7k9ezv </ref> - General research protocol draft <ref> https://www.dropbox.com/scl/fi/gau25oy5y1s57046icjt2/Research-protocol-draft.docx?cloud_editor=word&dl=0&rlkey=wat63e25ritmujwcpss8s4v0s </ref> - Health Promoting Schools <ref> https://www.dropbox.com/scl/fi/0rm7honrezbjwrcy3h3yk/Health-promoting-schools.docx?cloud_editor=word&dl=0&rlkey=673jyzcmwbfw7k9ui9nmtp0zh </ref> - John Snow Institute bylaws <ref> https://www.dropbox.com/scl/fi/lccr7jtnga1u0x75117zn/John-Snow-revision-2-March-11.doc?cloud_editor=word&dl=0&rlkey=lz2gi7mslcoay5dzygg8h6n6r </ref> 3. '''Publications and pptx''' 2016-2026 <ref>https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/The_International_Type_2_Diabetes_Mellitus_and_Hypertension_Research_Group#The_John_Snow_Institute </ref><ref name=":0"> https://www.dropbox.com/scl/fi/mw7ft423lkkpjoxywd2bf </ref> 4. '''Strategies for research and implementation''' For early identification of the risks we propose to register weight and height (Body Mass Index) and the fasting blood sugar in the '''Prevalence studies''' at the schools for seafarers, nurses, medical students and the kids` schools.. A practical strategy for prediabetes remission in low- and middle-income countries (LMICs) must assume that laboratory capacity, workforce, and financing are constrained: '''5. Minutes from meetings''' <ref> https://www.dropbox.com/t/3ZfLGngkS3pSlAQ3 </ref> 6. '''Prediabetes-Remission Research Network:''' <small>Prof. Ing. MSc. Nailet Delgado; Dr. Olaf Jensen, MD, MPH, PhD, o147248@gmail.com; MSc.Ph.D. Bishal Gyawali (SDU); MSc.PhD Vivi Just-Nørregaard; Dr. Johan Hviid Andersen MD, PhD. Prof Århus University; Prof. MSc. Agnes Flores, UMECIT, Panama; Dr. Maite, Vacamonte, Panama; Bruno Nørdam, Randers; Dr. Maite Duque, Venezuela; Dr. Indira Santos Panama; Med.Stud. Ashley Lezcano, Panama; Dr. Antonio Roberto Abaya MD Filippines; Dr. Jen Mendoza, MD, Filippines; Dr. Andra Ergle MD, Latvia; Prof. MSc. Ingrid Morató, Tarragona/Cadiz, Spain; MBA Christian Acheampong, Turkey; Dr. Alejandro Martinez, MPH, Costa Rica; Dr. Med. Sci Finn Gyntelberg; NFA.and Bispebj. Hosp. Denmark,</small> ==References== [[Category:Prediabetes ]] <references />Education 1: Research Methodology <ref>https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/EDUCATION/Education_module_links</ref> <references /> = Online Meeting , May 11, 2026 = Prediabetes – Remission in Small - and medium economy countries is the target. Keeping eyes open for applications for lifestyle medicine, sporadic supplement metformin Prepare documentation to apply for funding. Clearly define the project title, objectives, scope (countries, communities, ages), strategy (how to collect data, with what equipment, what variables), required materials, and required personnel. Meeting with Lene Daugaard dir. SIMAC Svendborg. Periodically search for organizations that could fund our project. Apply for funding when the opportunity arises. Obtain those funds. In parallel, without interruption, continue prevalence data collection and a comparative study between countries can be conducted using this collected data. Intervention study 16 weeks in one or two of the target populations. Proposed budget for 5 years:  5 mill Dkr. The first year could be collect data from two countries, Denmark and Turkey (Istanbul) compare with the data from Panama, UMIP including a short review study on similar data and an 16 week intervention study with the goal to produce a strategic model for prevalence and effectful intervention to be reported in 1-2 international  articles. Possible funding entities: Innovation Fund Denmark; EIFO; DANIDA; CROWDFOUNDING; European Commission programs; SKOV website; Lundbeckfonden: other i8yu2x5omb4mkgocgke0k2fnih5ol3r 2813029 2813028 2026-06-06T07:50:15Z Saltrabook 1417466 2813029 wikitext text/x-wiki == '''The John Snow Prediabetes-Remission Research Institute.''' == [[File:ChatGPT Image 30 may 2026, 11 58 20 a.m.png|thumb|Prediabetes-remission research program]] [[File:ChatGPT Image 24 abr 2026, 08 16 04 a.m.png|thumb|]] Millions are at increased risk of developing metabolic syndromes with prediabetes, diabetes type 2, high blood pressure and overweight. All can lower their risks by staying physical active and eating well. For early identification of the risks we propose to register weight and height (BMI) and the fasting blood sugar in the '''Prevalence studies''' at the schools for seafarers, nurses, medical students and the kids schools.(12+ y).The 16-weeks '''intervention studies''' include learnings by short video sequences and self-monitoring of blood sugar with glucometer, and self-evaluation of diet and physical activity. Early diagnosis of prediabetes can provide both health and financial benefits.From a financial perspective, preventing or delaying diabetes can significantly lower healthcare costs. Early diagnosis of prediabetes is a cost-effective preventive strategy that can improve long-term health outcomes while helping individuals and healthcare systems avoid the substantial costs associated with diabetes and its complications.[[File:Lifestyle Medicine Pillars.png|300px|right|The focus of Lifestyle Medicine is on these 6 pillars.]] [[File:John Snow.jpg|thumb|left| John Snow in the early nineteenth century]] [[File:Cholera in London 1866.gif|thumb|250px|Map of a later cholera outbreak in London, in 1866]] [[File:Choleramaplondon1866.png|thumb|right|250px|Legend for the map above]] 1. '<nowiki/>'''Prevalence studies'''' 1.1 The-International-Maritime-Health-Database <ref>https://www.dropbox.com/scl/fi/z3cq5ciiev06y8v9duw7u/A-International-Maritime-Health-Database.docx?cloud_editor=word&dl=0&rlkey=pt0kdesvmagcxaa2wez3tmza3 </ref> 1.2 Nursing Students Health Database <ref> https://www.dropbox.com/scl/fi/tcznmmd2y3nona5e3h1ro/The-Nursing-students-health-database.docx?cloud_editor=word&dl=0&rlkey=onbjh4o8ko1lzdvgyi8nlrotk </ref> 1.3. Medical student's Health Database <ref>https://www.dropbox.com/scl/fi/f16h9b60u4gxgt56un2jf/The-Medical-students-Health-database.docx?cloud_editor=word&dl=0&rlkey=xyfqen5trdc5lniaovipl548n </ref> 1.4. School childrens Health database <ref> https://www.dropbox.com/scl/fi/u6u50c8bxwhte9t2t6ck8/The-School-children-s-Health-database.docx?cloud_editor=word&dl=0&rlkey=zlyz5wn673wf7owettq3nx3h5 </ref> 2. '''Intervention studies''' Englsh <ref>https://www.dropbox.com/scl/fi/oi6cx6tlwwvoko3ed37tn/Invitation-to-the-course-English.docx?cloud_editor=word&dl=0&rlkey=7kzg91tqfgjskxf5aji8khicx </ref> Danish <ref>https://www.dropbox.com/scl/fi/2qahc3q9hmf4skbvk77ab/Invitation-to-the-course-in-Danish.docx?cloud_editor=word&dl=0&rlkey=x63w8oqvarz284zg2btq2johv </ref> Spanish <ref> https://www.dropbox.com/scl/fi/bn71inqeeth4o4mc1fjth/Invitation-to-the-course-Spanish.docx?cloud_editor=word&dl=0&rlkey=popmr1fnodh1v951v9l7k9ezv </ref> - General research protocol draft <ref> https://www.dropbox.com/scl/fi/gau25oy5y1s57046icjt2/Research-protocol-draft.docx?cloud_editor=word&dl=0&rlkey=wat63e25ritmujwcpss8s4v0s </ref> - Health Promoting Schools <ref> https://www.dropbox.com/scl/fi/0rm7honrezbjwrcy3h3yk/Health-promoting-schools.docx?cloud_editor=word&dl=0&rlkey=673jyzcmwbfw7k9ui9nmtp0zh </ref> - John Snow Institute bylaws <ref> https://www.dropbox.com/scl/fi/lccr7jtnga1u0x75117zn/John-Snow-revision-2-March-11.doc?cloud_editor=word&dl=0&rlkey=lz2gi7mslcoay5dzygg8h6n6r </ref> 3. '''Publications and pptx''' 2016-2026 <ref>https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/The_International_Type_2_Diabetes_Mellitus_and_Hypertension_Research_Group#The_John_Snow_Institute </ref><ref name=":0"> https://www.dropbox.com/scl/fi/mw7ft423lkkpjoxywd2bf </ref> 4. '''Strategies for research and implementation''' For early identification of the risks we propose to register weight and height (Body Mass Index) and the fasting blood sugar in the '''Prevalence studies''' at the schools for seafarers, nurses, medical students and the kids` schools.. A practical strategy for prediabetes remission in low- and middle-income countries (LMICs) must assume that laboratory capacity, workforce, and financing are constrained: '''5. Minutes from meetings''' <ref> https://www.dropbox.com/t/3ZfLGngkS3pSlAQ3 </ref> 6. '''Prediabetes-Remission Research Network:''' <small>Prof. Ing. MSc. Nailet Delgado; Dr. Olaf Jensen, MD, MPH, PhD, o147248@gmail.com; MSc.Ph.D. Bishal Gyawali (SDU); MSc.PhD Vivi Just-Nørregaard; Dr. Johan Hviid Andersen MD, PhD. Prof Århus University; Prof. MSc. Agnes Flores, UMECIT, Panama; Dr. Maite, Vacamonte, Panama; Bruno Nørdam, Randers; Dr. Maite Duque, Venezuela; Dr. Indira Santos Panama; Med.Stud. Ashley Lezcano, Panama; Dr. Antonio Roberto Abaya MD Filippines; Dr. Jen Mendoza, MD, Filippines; Dr. Andra Ergle MD, Latvia; Prof. MSc. Ingrid Morató, Tarragona/Cadiz, Spain; MBA Christian Acheampong, Turkey; Dr. Alejandro Martinez, MPH, Costa Rica; Dr. Med. Sci Finn Gyntelberg; NFA.and Bispebj. Hosp. Denmark,</small> ==References== [[Category:Prediabetes ]] <references />Education 1: Research Methodology <ref>https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/EDUCATION/Education_module_links</ref> <references /> = Online Meeting , May 11, 2026 = Prediabetes – Remission in Small - and medium economy countries is the target. Keeping eyes open for applications for lifestyle medicine, sporadic supplement metformin Prepare documentation to apply for funding. Clearly define the project title, objectives, scope (countries, communities, ages), strategy (how to collect data, with what equipment, what variables), required materials, and required personnel. Meeting with Lene Daugaard dir. SIMAC Svendborg. Periodically search for organizations that could fund our project. Apply for funding when the opportunity arises. Obtain those funds. In parallel, without interruption, continue prevalence data collection and a comparative study between countries can be conducted using this collected data. Intervention study 16 weeks in one or two of the target populations. Proposed budget for 5 years:  5 mill Dkr. The first year could be collect data from two countries, Denmark and Turkey (Istanbul) compare with the data from Panama, UMIP including a short review study on similar data and an 16 week intervention study with the goal to produce a strategic model for prevalence and effectful intervention to be reported in 1-2 international  articles. Possible funding entities: Innovation Fund Denmark; EIFO; DANIDA; CROWDFOUNDING; European Commission programs; SKOV website; Lundbeckfonden: other 70w2zqk0ie95920r72ztjxoh132kn70 2813278 2813029 2026-06-06T10:24:34Z Saltrabook 1417466 /* The John Snow Prediabetes-Remission Research Institute. */ 2813278 wikitext text/x-wiki == '''The John Snow Prediabetes Research Institute.''' == [[File:ChatGPT Image 30 may 2026, 11 58 20 a.m.png|thumb|Prediabetes-remission research program]] [[File:ChatGPT Image 24 abr 2026, 08 16 04 a.m.png|thumb|]] Millions are at increased risk of developing metabolic syndromes with prediabetes, diabetes type 2, high blood pressure and overweight. All can lower their risks by staying physical active and eating well. For early identification of the risks we propose to register weight and height (BMI) and the fasting blood sugar in the '''Prevalence studies''' at the schools for seafarers, nurses, medical students and the kids schools.(12+ y).The 16-weeks '''intervention studies''' include learnings by short video sequences and self-monitoring of blood sugar with glucometer, and self-evaluation of diet and physical activity. Early diagnosis of prediabetes can provide both health and financial benefits.From a financial perspective, preventing or delaying diabetes can significantly lower healthcare costs. Early diagnosis of prediabetes is a cost-effective preventive strategy that can improve long-term health outcomes while helping individuals and healthcare systems avoid the substantial costs associated with diabetes and its complications.[[File:Lifestyle Medicine Pillars.png|300px|right|The focus of Lifestyle Medicine is on these 6 pillars.]] [[File:John Snow.jpg|thumb|left| John Snow in the early nineteenth century]] [[File:Cholera in London 1866.gif|thumb|250px|Map of a later cholera outbreak in London, in 1866]] [[File:Choleramaplondon1866.png|thumb|right|250px|Legend for the map above]] 1. '<nowiki/>'''Prevalence studies'''' 1.1 The-International-Maritime-Health-Database <ref>https://www.dropbox.com/scl/fi/z3cq5ciiev06y8v9duw7u/A-International-Maritime-Health-Database.docx?cloud_editor=word&dl=0&rlkey=pt0kdesvmagcxaa2wez3tmza3 </ref> 1.2 Nursing Students Health Database <ref> https://www.dropbox.com/scl/fi/tcznmmd2y3nona5e3h1ro/The-Nursing-students-health-database.docx?cloud_editor=word&dl=0&rlkey=onbjh4o8ko1lzdvgyi8nlrotk </ref> 1.3. Medical student's Health Database <ref>https://www.dropbox.com/scl/fi/f16h9b60u4gxgt56un2jf/The-Medical-students-Health-database.docx?cloud_editor=word&dl=0&rlkey=xyfqen5trdc5lniaovipl548n </ref> 1.4. School childrens Health database <ref> https://www.dropbox.com/scl/fi/u6u50c8bxwhte9t2t6ck8/The-School-children-s-Health-database.docx?cloud_editor=word&dl=0&rlkey=zlyz5wn673wf7owettq3nx3h5 </ref> 2. '''Intervention studies''' Englsh <ref>https://www.dropbox.com/scl/fi/oi6cx6tlwwvoko3ed37tn/Invitation-to-the-course-English.docx?cloud_editor=word&dl=0&rlkey=7kzg91tqfgjskxf5aji8khicx </ref> Danish <ref>https://www.dropbox.com/scl/fi/2qahc3q9hmf4skbvk77ab/Invitation-to-the-course-in-Danish.docx?cloud_editor=word&dl=0&rlkey=x63w8oqvarz284zg2btq2johv </ref> Spanish <ref> https://www.dropbox.com/scl/fi/bn71inqeeth4o4mc1fjth/Invitation-to-the-course-Spanish.docx?cloud_editor=word&dl=0&rlkey=popmr1fnodh1v951v9l7k9ezv </ref> - General research protocol draft <ref> https://www.dropbox.com/scl/fi/gau25oy5y1s57046icjt2/Research-protocol-draft.docx?cloud_editor=word&dl=0&rlkey=wat63e25ritmujwcpss8s4v0s </ref> - Health Promoting Schools <ref> https://www.dropbox.com/scl/fi/0rm7honrezbjwrcy3h3yk/Health-promoting-schools.docx?cloud_editor=word&dl=0&rlkey=673jyzcmwbfw7k9ui9nmtp0zh </ref> - John Snow Institute bylaws <ref> https://www.dropbox.com/scl/fi/lccr7jtnga1u0x75117zn/John-Snow-revision-2-March-11.doc?cloud_editor=word&dl=0&rlkey=lz2gi7mslcoay5dzygg8h6n6r </ref> 3. '''Publications and pptx''' 2016-2026 <ref>https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/The_International_Type_2_Diabetes_Mellitus_and_Hypertension_Research_Group#The_John_Snow_Institute </ref><ref name=":0"> https://www.dropbox.com/scl/fi/mw7ft423lkkpjoxywd2bf </ref> 4. '''Strategies for research and implementation''' For early identification of the risks we propose to register weight and height (Body Mass Index) and the fasting blood sugar in the '''Prevalence studies''' at the schools for seafarers, nurses, medical students and the kids` schools.. A practical strategy for prediabetes remission in low- and middle-income countries (LMICs) must assume that laboratory capacity, workforce, and financing are constrained: '''5. Minutes from meetings''' <ref> https://www.dropbox.com/t/3ZfLGngkS3pSlAQ3 </ref> 6. '''Prediabetes-Remission Research Network:''' <small>Prof. Ing. MSc. Nailet Delgado; Dr. Olaf Jensen, MD, MPH, PhD, o147248@gmail.com; MSc.Ph.D. Bishal Gyawali (SDU); MSc.PhD Vivi Just-Nørregaard; Dr. Johan Hviid Andersen MD, PhD. Prof Århus University; Prof. MSc. Agnes Flores, UMECIT, Panama; Dr. Maite, Vacamonte, Panama; Bruno Nørdam, Randers; Dr. Maite Duque, Venezuela; Dr. Indira Santos Panama; Med.Stud. Ashley Lezcano, Panama; Dr. Antonio Roberto Abaya MD Filippines; Dr. Jen Mendoza, MD, Filippines; Dr. Andra Ergle MD, Latvia; Prof. MSc. Ingrid Morató, Tarragona/Cadiz, Spain; MBA Christian Acheampong, Turkey; Dr. Alejandro Martinez, MPH, Costa Rica; Dr. Med. Sci Finn Gyntelberg; NFA.and Bispebj. Hosp. Denmark,</small> ==References== [[Category:Prediabetes ]] <references />Education 1: Research Methodology <ref>https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/EDUCATION/Education_module_links</ref> <references /> = Online Meeting , May 11, 2026 = Prediabetes – Remission in Small - and medium economy countries is the target. Keeping eyes open for applications for lifestyle medicine, sporadic supplement metformin Prepare documentation to apply for funding. Clearly define the project title, objectives, scope (countries, communities, ages), strategy (how to collect data, with what equipment, what variables), required materials, and required personnel. Meeting with Lene Daugaard dir. SIMAC Svendborg. Periodically search for organizations that could fund our project. Apply for funding when the opportunity arises. Obtain those funds. In parallel, without interruption, continue prevalence data collection and a comparative study between countries can be conducted using this collected data. Intervention study 16 weeks in one or two of the target populations. Proposed budget for 5 years:  5 mill Dkr. The first year could be collect data from two countries, Denmark and Turkey (Istanbul) compare with the data from Panama, UMIP including a short review study on similar data and an 16 week intervention study with the goal to produce a strategic model for prevalence and effectful intervention to be reported in 1-2 international  articles. Possible funding entities: Innovation Fund Denmark; EIFO; DANIDA; CROWDFOUNDING; European Commission programs; SKOV website; Lundbeckfonden: other 7lxc3pm5gjyq27x75aaemrlcq6qzqeu Online music jam 0 269528 2812854 2283229 2026-06-05T15:14:48Z Kirby - Electrotechnics 3074947 changed to 'category:Music technology' instead of category:Music 2812854 wikitext text/x-wiki Online Music Jam is a learning resouces addresses concepts to practice music online when the musicians where not able to meet in person. The learning resource was create in the context of the development of the [[COVID-19/Workflow Transformation|COVID-19 Workflow Transformations]]. Learning resources could support cultural live under the epidemiological requirements and constraints. == Introduction == This learning resource was created in the context of [[COVID-19]] pandemics. Singing and joined music was affected through epidemiological constraints. Support for small bands and choirs in establishing online rehearsal to support joint activities of musicians. To create also an operational environment, where musician can experience online pratice and rehearsal, searching for Open Source software, that supports this concept was the first step. [https://github.com/corrados/jamulus Jamulus] was identified as OpenSource software to support the learning resource. On the other hand this learning resource will guide you into bit into the computer science background and the requirements and constraints of an Online rehearsal room. * What is missing in comparison to gathering of the musicians in a rehearsal room? * What is possible with such kind of infrastructure? * What are possible further developments, that could improve online rehearsal and how can these new feature be implemented? (see [[Open Innovation Ecosystem]]) == Learning Objectives == With this learning resource you will * learn about the underlying principles to facilitat online jaming of musicians. * You are guided to setup an Open Source Online Rehearsal rooms<ref>Volker Fischer (2015) Case Study: Performing Band Rehearsals on the Internet WithJamulus (Date: 2015/02/24) URL: https://jamulus.io/PerformingBandRehearsalsontheInternetWithJamulus.pdf - (accessed 2020/11/08)</ref>. * You will get a basic understanding how to exchange audio and MIDI signals between intruments and hardware and the clients of musicians that are connected to joined server, that distributes the signals to the server. * Understand the challenges for the latency of signal transmissions to generate almost synchronous perceived audio and MIDI streams and distributed the audio and MIDI streams to the musicians. The learning resources is build on the [[Open Community Approach]] so that performing the learning tasks is possible without the need to buy any software. == Learning Tasks == * '''(MIDI / Audio)''' Decribe the difference between MIDI and audio streams and explain the benefits and drawbacks of the different approaches to encode music in a digital form ([[Open Innovation Ecosystem|Open Innovation Remark]] - Open Source adaptation required to add a Jamaulus Midi exchange layer to the already available Audio Layer). ** Compare why keyboard, drum kit, bass, digital wind instruments and singing/voice as tracks for jamming online with other musician. Which instruments are appropriate for transmission with a MIDI signal. Which tracks would you transmit with MIDI and which track would you share as audio? Please keep in mind that MIDI exchange is currently not available in the OpenSource software Jamulus. ** What Soundfonts and how do they work in conjunction with a MIDI signal coming from a keyboard, digital drum kit or digital wind instrument? ** Play around with the OpenSource software LMMS and use a MIDI track and create a audio with a [[/Software synthesizer/]] * '''(Jamulus)''' Explore the [https://github.com/corrados/jamulus/wiki Open Source Software Jamulus] and describe which type of data streams are exchanged between the musicians. * '''([[Latency]])''' Explain the term latency and describe how latency affects the online practice of musicians. How can latency be reduced on the hardware side and on the software side? (e.g. LAN-Cable instead of Wifi, analyse the limitation of Upstream DSL in comparison to Downstream DSL at router, perform a bottleneck analysis for the client server * '''(Artificial Intelligence and Pattern Recognition)''' Currently the latency in the network is a challenge for practicing jointly over the internet. How can artificial intelligence be used to train e.g. the melody of a solo and explain how the train neural network can be used to fill missing fragments due to latency problems? (Feature not implemented in Jamulus - it is just part of the learning resource) ** '''(Drum Kit)''' Start with a digital drum create pattern/rythm and use [[w:en:Hopfield_network|Hopefield Network]] to train the pattern. Now provide a [[w:en:Hopfield_network|Hopefield Network]] a drum pattern or digital sequence that has missing beats. Explain how the [[w:en:Hopfield_network|Hopefield Network]] is able to reconstruct the missing beats pattern in pattern! See also Hydrogen * '''(Password Protection Online Session)''' Why is it important to add a password protection to e.g. Jamulus? What are use cases in which the current Jamulus server access without password protection for the online rehearsal room is useful (e.g. meet with people in open environment)? * '''([[Video conferencing|Combination with Videoconference]])''' Assume we would integrate Jamulus Server with a [[w:en:BigBlueButton|BigBlueButton open source video conferencing server]]. What are the benefit of seeing the other musicians and what are drawbacks of such an approach: ** (Benefits - Implementation) Access code for meeting rooms (password protection) is already integrated in [[w:en:BigBlueButton|BigBlueButton]]. ** (Benefits - Implementation) Jamulus Server could replace the audio communication framework in [[w:en:BigBlueButton|BigBlueButton]] or could be set in the room settings of [[w:en:BigBlueButton|BigBlueButton]] to define if the standard audio oder the Jamulus audio will be used for communication. ** (Benefits - User) the users may see each other and interact by gesture. Musicians could interact visually with each other, e.g. providing a gesture with the guitar, that indicates that the solo finished. ** (Drawbacks - Implementation) the integration is not part of Jamulus Server into the OpenSource [[w:en:BigBlueButton|BigBlueButton]] does not exist and first it necessary to analyse the programming efforts for an plugin or interface to use Jamulus within [[w:en:BigBlueButton|BigBlueButton]]. ** Analyse the room concept [[w:en:BigBlueButton|BigBlueButton]] and the [https://bigbluebutton.org/2018/07/09/greenlight-2-0/ Greenlight Front End for BigBlueButton]. How can the room management of BigBlueButton be transfered to a Jamulus Server infrastructure to host more than one rehearsal rooms and have also the access management from https://bigbluebutton.org/2018/07/09/greenlight-2-0/ BBB and Greenlight] inherited to the Jamulus Infrastructure. == Submodules == * [[/Software synthesizer/]] - to understand the MIDI concept in conjunction with audio generation. * [[/Jamulus Server/]] * [[/Drum Computer/]] * [[/Global Metronome/]] == See also == * [[COVID-19/Workflow Transformation]] * [[w:en:MuseScore|MuseScore]] to generate MIDI tracks of sheet music. * [[w:en:Hopfield_network|Hopefield Network]] Reconstruction of missing MIDI events in a drum kit sequence. * [[Latency]] * [[Audio-Video-Compression]] == External Resources == * [https://github.com/corrados/jamulus Jamulus Open Source Software] * [http://jchordbox.sourceforge.net/ JChordBox] - JCordBox used in conjunction with Jamulus to create backing tracks for rehearsal or online jam session * [https://sourceforge.net/projects/jamulus-os/ Jamulus Operating System (OpenSource)] - Having your OpenSource online rehearsal room on a USB thumb drive. * [https://github.com/grundic/jamulus-docker Jamulus Docker Container (OpenSource)] - Start a full configured Jamulus Server Rehearsal Room in 15min in a Docker Container * [https://bigbluebutton.org/2018/07/09/greenlight-2-0/ Greenlight Front End - for Room Management (OpenSource)] - Technical Information for Developers - use the Room Management System of Greenlight to manage Jamulus Rehearsal Rooms ([[Open Innovation Ecosystem|Open Innovation Remark]]) * [https://lmms.io/wiki/index.php?title=Installing_LMMS LMMS Digital Audio Workstation (OpenSource)] - Use a Digital Audio Workstation in Conjunction with MIDI e.g. to provide a backingtrack for a jam session or as a software developer use LMMS as DAW plugin for Jamulus Midi Layer exchange in addition to existing audio layer exchange of musicians. * [http://hydrogen-music.org/ Hydrogen MIDI Drumkit (OpenSource)] - use in conjuction with Jamulus for drum patterns (see [http://hydrogen-music.org/documentation/manual/manual_en.html Hydrogen Manual]). Hydrogen is a software synthesizer which can be used standalone, for creating drum machine based on patterns, or via an external MIDI keyboard/sequencer software to generate a drum audio track. * [https://github.com/pantherb/setBfree setBfree Software Organ Synth (OpenSource)] - see also [https://x42-plugins.com/x42/setBfree MIDI Video Demo Plugin in Action] == References == [[Category:Music technology]] [[Category:Internet audio and video]] 9iei3a87u37omgdf3burkpl4ddn4xqe Online music jam/Software synthesizer 0 269529 2812853 2472366 2026-06-05T15:13:18Z Kirby - Electrotechnics 3074947 changed to 'category:Music technology' instead of category:Music 2812853 wikitext text/x-wiki == Learning Task == This learning resource look at the interface between a MIDI signal and sound generation. You will learn to create "key pressed" events with the keyboard or an Open Source Digital Audio Workstations DAW (like LMMS) and create the corresponding sound with software synthesizer on a PC. The example is performed on the OpenSource Linux Operating system with OpenSource software only and is designed in way that musicians can learn about the underyling concepts without buying software or hardware. Furthermore learners should be able to reproduce the learning example with available software for different Operating systems (Jamulus, Musescore is available for different Operating systems). In the first section the examples are elaborated for Linux operating system and could be transfered to other Operating systems in a similar way. Current operating systems are able to work with USB Midi Controllers. If not just used an existing MIDI file to create the MIDI events for this learning example. === Requirements === If you do not have a MIDI keyboard, digital drum kit or digital wind instrument to create live MIDI events, then it would be helpful to have a * an Open Source digital audio workstation like LMMS, * a freely available MIDI song e.g. with drum track a bass track and keyboard track. Choose a MIDI track according to your "taste of music". This is file serves just as a method to create MIDI events without having a digital instrument available. An other option is to use the OpenSource notation software * e.g. the OpenSource software [[w:en:MuseScore|Musescore]] and create a few notes, * export the notation to a MIDI file and * assign different sounds to track with [[w:en:MuseScore|Musescore]] The following example assumes you have * MIDI Keyboard, * Software Synthesizer and a * Online Jam Software (here Open Source Jamulus is used as an example) for distributed practicing of musicians located at different places == History of this Subpage == This contribution<ref>Engelbert Niehaus (2020) GitHub Wiki Contribution to the Jamulus Github Repository https://github.com/corrados/jamulus maintained by Volker Fischer - Former Location: https://github.com/corrados/jamulus/wiki - Current Location: https://github.com/jamulussoftware/jamulussoftware.github.io/blob/main/wiki/en/en-Software-Synth.md </ref> was made to the [https://github.com/corrados/jamulus Jamulus Github Repository by Volker Fischer]. This learning focuses on the underlying concepts that are used by Open Source software Jamulus and goes beyond the current implementation of available Open Source software to optimize the implementation Online Music Jam Software. The learning resource follows the [[Open Community Approach]] to support musicians for keeping alive their joint cultural activities durch [[COVID-19]]. == Linux == We will elaborate the concept of playing a software synth on Linux as example and transfer that to other operating systems (e.g. Windows, MacOSX). * '''(Hammond Organ Synth)''' As an example we talk open source software synth [https://github.com/pantherb/setBfree setBfree], which is a hammond organ emulator that outputs through JACK on Linux. * '''(MIDI Keyboard)''' The MIDI keyboard is used to generate the key pressed events, that are transmitted through a USB-MIDI interface to Linux system and the Synthesizer [https://github.com/pantherb/setBfree setBfree] generates the corresponding tone for the pressed key. In general the MIDI keyboard can also be replaced by a digital wind midi controller, that does the same but the tone is dependent on how the musician blows air into the digital wind controller. * '''(Connector to Jamulus)''' You may hear the audio output of your Linux system on your speakers but the audio output may not be use as input for your Jamulus session. So another step is to connect the audio out to Jamulus client (with QJackCtl). === General Information about ALSA, JACK and MIDI === First of all we explain a few basic concepts on Linux to use MIDI and Audio together. * ALSA is the basic to connection to the hardware, i.e. the soundcard and the midi controller for the keyboard. * JACK is build on top of ALSA and uses ALSA for the audio output. It provides a adequat environment for music production on Linux. * Synthesizer like the [https://github.com/pantherb/setBfree Hammond Organ emulator setBfree] connect to JACK for audio output. * Jamulus connects to JACK. === Requirements for Using Software Synths in Jamulus on Linux === For ALSA2JACK MIDI (<code>a2jmidid</code>) connection it is necessary to install the a2j package. Furthermore we use in this example a simple Hammond organ emulator <code>setBfree</code> in this example. Replace <code>setBfree</code> by your favorite Linux synthesizer. Both packages <code>a2jmidid</code> and <code>setbfree</code> can be installed via your package manager or e.g. on Ubuntu/Mint with <code>apt-get</code> commands: <syntaxhighlight lang="shell"> sudo apt-get install a2jmidid sudo apt-get install setbfree</syntaxhighlight> === Connecting ALSA, JACK and MIDI for a Synth for Jamulus === Now we look an the workflow to use the whole infrastructure in Jamulus. * '''(Jamulus Installation and JACK)''' Use the [https://github.com/corrados/jamulus/wiki/Linux-Client-Install-Script install script] for Jamulus on Linux or [https://github.com/corrados/jamulus/wiki/Installation-for-Linux install Jamulus with the manual in the wiki]. With this installation we have JACK and QJackCtl installed for Jamulus. * '''(ALSA to JACK - MIDI)''' Next we must create a connection from ALSA to JACK to route the MIDI input from the keyboard to the [https://github.com/pantherb/setBfree Hammond Emulator setBfree]. ** '''''(Command Line: <code>qjackctl</code>)''''' For this step you can start JACK-Control (QJackCtl) from the command line by <code>qjackctl</code>, ** '''''(Command Line: <code>a2jmidid -e</code>)''''' Then you launch the ALSA2JACK MIDI connector with command: <code>a2jmidid -e</code>. As long as this connector runs in the shell the <code>a2j</code> connection interface is visible in <code>QJackCtl</code>. ** '''''(QJackCtl: <code>ALSA-MIDI</code>)''''' Connect your USB MIDI as input to <code>MIDI through</code> in the ALSA-MIDI tab of QJackCtl. Now the midi events from your keyboard reach JACK and are ready to connect. ** '''''(Start Synth)''''' Start the Hammond Synthesizer too e.g. from the command line with <code>setBfree</code>. You will see the <code>setBfree</code> graphical user interface GUI (see http://setbfree.org/gui_3d). Press with your mouse on the visualised organ keyboard to test if the synth generates an organ sound on your speakers. ** '''''(QJackCtl: <code>JACK-MIDI</code>)''''' Now we can connect <code>a2j</code> in JACK-MIDI as input to the synth <code>setBfree</code>, that generates the sound as output for each key pressed on the keyboard. Keep in mind that you can connect synths if they are started and visible for QJackCtl. So you can connect after starting the [https://github.com/pantherb/setBfree Hammond Emulator setBfree], i.e. connect <code>a2j</code> to the <code>setbfree</code> if you also see the Hammond Synth SetBFree in QJackCtl. ** '''''(Test Software Synth with MIDI Keyboard)''''' If you have speakers attached to your Linux Computer you would hear the emulated Hammond sound on your Linux PC for all the keys your press on your MIDI keyboard. If you just want to play the software synth with your keyboard you are ready to play from here. The only remaining step is to connect the audio to Jamulus. * '''(Synth to Jamulus - Audio)''' In the previous step we connected the MIDI signals from your keyboard to the MIDI input of JACK via a2j - MIDI through connection. Now JACK the MIDI which tell JACK which keys are pressed on the keyboard. Now create an audio connection in JACK (i.e. the generated Hammond sound of the synth). We connect the [https://github.com/pantherb/setBfree Hammond Emulator setBfree] output to Jamulus. Therefore Jamulus must be started so that you see also Jamulus in the audio connection of QJackCtl. Now you can the audio output of the [https://github.com/pantherb/setBfree Hammond Emulator setBfree] to the audio input of Jamulus in JACK. In general <code>QJackCtrl</code> is used similar operations like for plugging in instruments in mixer and the mixer into an amplifier and the amplifier again to the speakers. Have fun with your other OpenSource synths like QSynth, [https://sourceforge.net/projects/zynaddsubfx/ ZynAddSubFX] than can be installed on your Linux system and/or used within LMMS. == References == [[Category:Music technology]] [[Category:Internet audio and video]] 8cxaplmze1q43d2nmais7cyen3bnisq Category:Music and Songwriting 14 269871 2812851 2237447 2026-06-05T15:08:20Z Kirby - Electrotechnics 3074947 changed to 'category:Music composition' instead of category:Music 2812851 wikitext text/x-wiki [[Category:Music composition]] 2tnagzfj3gvt3j0uh6wcauusszlew8f Irish Language/Words and Music/Peata Beag 0 270513 2812991 2517338 2026-06-05T22:30:57Z Kirby - Electrotechnics 3074947 changed to 'category:Songs' instead of category:Music 2812991 wikitext text/x-wiki {|class="wikitable" style="float:right;text-align:center" |'''Peata Beag''' |- |Listen to This Song:<br> [https://open.spotify.com/track/3BfnO7KK58bA3xLf0C1O2N?si=u4dnAouJSJ-K3FT_Jg6atg Dervish (on Spotify)] |} ==About the Song== '''Peata Beag''' (Little Pet) is a traditional song collected by Dublin artist and antiquary George Petrie. According to Petrie: {{Quote |text=The following playful melody, with its words, was obtained in the course of the summer of 1853 from the blind county of Clare peasant, Teige MacMahon[.] The words, though of no high poetic merit, are not without interest, from their natural simplicity, and as an illustration of the thoughts of Irish peasant life. }}<ref>The Petrie Collection of the Ancient Music of Ireland (1855)</ref> The dialect would originally have been Clare Irish, closest to Munster of the surviving modern dialects. Petrie gives two verses (verses 1 and 3 below). The second verse appears in P. W. Joyce's "Irish Music and Song" (1901). The last verse appears in the most popular recording, by Dervish, and most other modern recordings. It was not recorded in either Petrie or Joyce and Dervish credits the song as "traditional".<ref>Dervish, ''At The End of the Day''. Whirling Discs (IRE) 1996.</ref> The title given in early English sources is usually Petrie's "I wish the shepherd's pet were mine." Other translations include "I wish I had the shepherd's lamb." Modern editions and recordings call it "Peata Beag". ==Lyrics== {{clear}} ===Verse 1=== {|class="wikitable" style="float:right" |- !colspan=2|vocabulary |- |trua |pity (n. f.)<br>lean (of meat) (adj.) |- |gan |without |- |peata |pet (n. m.) |- |maor |warden; supervisor (n. m.) |- |caora |sheep (n. f.) |- |beag |small (adj.) |- |bán |white (adj. & n. m.) |} Is trua gan peata an mhaoir agam<br> Is trua gan peata an mhaoir agam<br> Is trua gan peata an mhaoir agam<br> Is na caoire beaga bána ====notes==== *The basic structure here and in each of the later verses is the same: "Is trua" (it is a pity that) "gan ... agam" (I have no ...). *In the first three lines, "an" is contracted or elided to "'n" and attached to the last syllable of "peata"; in performance it sounds more like "peat-an" *The beginning of the last line ("Is na") is contracted or elided to sound like "'Sna" in performance. *"Caoire" is a Munster dialect variant of "caoirigh", sheep (plural). *"Peata" is usually translated "pet," but may also mean "lamb." {{clear}} ===Chorus=== {|class="wikitable" style="float:right" |- !colspan=2|vocabulary |- |gair |call, acclaim (v.) |- |grá |love, beloved (n. m.) |- |croí |heart (n. m.) |- |cealg |guile (n. f.) |- |máthair |mother (n. f.) |} Is ó gairim, gairim thú,<br> Is grá mo chróí gan cheilg thú,<br> Is ó gairim, gairim thú,<br> Is tú peata beag do mháthar ====notes==== *Here and in verses 2, 3, and 4, the first three lines start with "is", the copula; the last line also starts with "is," but the shortened form of "agus", meaning "and". {{clear}} ===Verse 2=== {|class="wikitable" style="float:right" |- !colspan=2|vocabulary |- | -ín |diminutive suffix |- |maol |bald (adj.) |- |maoilín |hornless cow (n. m.) |- |maoin |gift, property (n.) |- |maoinín |little treasure (n.) |- |bán |white (adj.) |- |fáilte |welcome (n.) |- |geal |bright; white (n./adj.) |} Is trua gan maoilín bán agam<br> Is trua gan maoilín bán agam<br> Is trua gan maoilín bán agam<br> Is fáilte ó mo ghrá geal ====notes==== *The fourth word in this verse is sometimes given as "maoilín" and sometimes "maoinín": **"Maoilín" is derived from "maol", meaning bald or a bald or bare object. With the addition of the diminutive "-ín" it literally means something like "the little bald-topped one" and refers to a calf or hornless cow. **"Maoinín" is a diminutive of "maoin," meaning gift or property, and means "little treasure." *"Grá geal" (literally "bright love") is a common term of endearment. {{clear}} ===Verse 3=== {|class="wikitable" style="float:right" |- !colspan=2|vocabulary |- |bólacht |cattle (n.) |- |bainne |milk (n.) |- |Cáit |Kate (given name) |- |cat |cat (n.) |- |máthair |mother (n.) |} Is trua gan bólacht bainne agam<br> Is trua gan bólacht bainne agam<br> Is trua gan bólacht bainne agam<br> Is Cáitín ó a máthair ====notes==== *"Bólacht bainne" refers to a cow in milk. *Different sources render the last line as either Cáitín or caitín: **"Cáitín," a diminutive version of the given name Cáit, meaning something like "Little Kate". **or "caitín," the diminutive form of "cat" (meaning cat) and meaning something like "kitty" or "catkin". * The original version has "ó na máthair"; the n- in "na" is likely a dialect-specific prothesis. {{clear}} ===Verse 4=== {|class="wikitable" style="float:right" |- !colspan=2|vocabulary |- |gabhar |goat (n.) |- |buí |yellow (adj.) |- |stór |wealth, treasure; storehouse (n.) |} Is trua gan gabhairín bhuí agam<br> Is trua gan gabhairín bhuí agam<br> Is trua gan gabhairín bhuí agam<br> Is thabharfainn do mo stór í ====notes==== *"Gabhairín" is another diminutive, this time for "little goat". *"Thabharfainn" is the first person singular of "tabhair", give, in the ''modh coinníollach'', conditional mood, meaning "I would give". *"Stór" literally refers to abundance or riches, but here is used figuratively to mean "darling" or "beloved". [[Category:Irish|Words and Music/Peata Beag]] [[Category:Songs]] c5cgck948et3o99fgnf05powkut925i Musical instruments 0 270790 2812882 2486418 2026-06-05T15:55:23Z Kirby - Electrotechnics 3074947 changed to 'category:Music instruments' instead of category:Musical instruments 2812882 wikitext text/x-wiki [[File:Attributes of Music.jpg|right|220px|Musical instruments]] A musical instrument is a device created or adapted to make musical sounds. In principle, any object that produces sound can be considered a musical instrument—it is through purpose that the object becomes a musical instrument.<ref>[[Wikipedia: Musical instrument]]</ref> == Resources == {{Subpages/List}} == See also == * [[Wikipedia: Musical instrument]] == References == {{Reflist}} [[Category:Music instruments]] h3i9t0sdf477b9kr7cty9t8n8793jvl Category:Neurodiversity Movement 14 271464 2812839 2251695 2026-06-05T12:34:13Z MathXplore 2888076 added [[Category:Neurodiversity]] using [[Help:Gadget-HotCat|HotCat]] 2812839 wikitext text/x-wiki [[Category:Activism]] [[Category:Recovery psychology]] [[Category:Psychopathology]] [[Category:Social movements]] [[Category:Neurodiversity|Movement]] oqjgr0jx36tol5vml349k8uoopyweac Template:Template link 10 274917 2813040 2669959 2025-01-21T19:19:39Z Ahecht 1329974 Reduce [[WP:PEIS]] 2813040 wikitext text/x-wiki <span class="nowrap">&#123;&#123;</span>[[Template:{{{1}}}|{{{1}}}]]<span class="nowrap">&#125;&#125;</span><noinclude> {{documentation}} <!-- Categories go on the /doc subpage and interwikis go on Wikidata. --> </noinclude> kkhg4xclyf9r0bcl7ypdd0yuh41mfu9 2813041 2813040 2026-06-06T08:30:11Z Jtneill 10242 1 revision imported from [[:w:Template:Template_link]] 2813040 wikitext text/x-wiki <span class="nowrap">&#123;&#123;</span>[[Template:{{{1}}}|{{{1}}}]]<span class="nowrap">&#125;&#125;</span><noinclude> {{documentation}} <!-- Categories go on the /doc subpage and interwikis go on Wikidata. --> </noinclude> kkhg4xclyf9r0bcl7ypdd0yuh41mfu9 Sing free/Songs 0 281945 2813012 2477696 2026-06-06T00:34:48Z MathXplore 2888076 added [[Category:Songs]] using [[Help:Gadget-HotCat|HotCat]] 2813012 wikitext text/x-wiki (See [[../Students/Guy vandegrift]]) __TOC__ ==Instructions== All songs should be in the key of C (or A minor). If you want to transpose, first copy it to your student page. Try to write songs so they are easily edited in this way. For example: [[Free singing lessons/Students]] <pre> %%% EDIT BEGIN (Modify only: \transpose, \tempo, and Staff.midiInstrument.) \transpose c c'' {\set Staff.midiInstrument = #"lead 2" \key c\major \time 4/4 \tempo "" 4 = 80 %%% EDIT END </pre> ==Songs== <pre> %DEF \language "english" \relative c\transpose c c'' {\set Staff.midiInstrument = #"reed organ" \key c\major \time 4/4 \tempo "" 4 = 80 %END </pre> ===Twinkle=== <score sound="1" lang="lilypond">\language "english" \relative c %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \transpose c, c %%% Edit with default \transpose c, c %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% {\set Staff.midiInstrument = #"reed organ" \key c\major \time 4/4 %%% Or: #"reed organ" | #"reed organ" | #"lead 2" %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \tempo 4 = 80 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% c4 c4 g g a a g2 f4 f e e d d c2 g4 g f f e e d2 g4 g f f e e d2 c4 c4 g g a a g2 f4 f e e d d c2\fermata \bar "|."} \addlyrics {Twin -- kle twin -- kle lit -- tle star, How I won -- der what you are. Up a -- bove the world so high. Like a dia -- mond in the sky. Twin -- kle twin -- kle lit -- tle star, How I won -- der what you are.} </score> ==end== [[Category:Songs]] e9ykj89jjfgy2jfak6gupybzk4w3vwj Sing free 0 281971 2813003 2574271 2026-06-06T00:27:06Z Kirby - Electrotechnics 3074947 changed to 'category:Music performance' instead of category:Music 2813003 wikitext text/x-wiki {{share}} ----- &nbsp; *I highly recommend '''[https://trainer.thetamusic.com/en/content/workouts trainer.thetamusic.com]''' ::I haven't had time to finish it, but their free introductory course looks fantastic. I know nothing about the prices or quality of their other courses, but [https://trainer.thetamusic.com/en/content/service-and-contact-information this page] looks promising. &nbsp; ----- ==Partially completed songs== Every song needs to be improved and we need a more songs. If you wish to contribute, leave a message at {{font|size=115%|[[User talk:Guy vandegrift]]}} ;Songs :[[/Twinkle/]] :[[/Red River Valley/]] :[[/Amazing Grace/]] :[[/Jingle Bells/]] :[[/Three blind mice/]] ;Scales and exercises :[[/Major sixth M6/]] :[[/Prelude and Fugue in C major (ear training)/]] :[[/Great_Gate_of_Kyiv_(ear_training)/]] UNDER CONSTRUCTION ;For editors of this project :[[Draft:Sing free/Tables and images]] :[[Draft:Sing free/Transclusions]] ==All you need to "sing free" is ...== *A computer with [[w:Loudspeaker|speaker]] and [[w:Microphone|microphone]]. [[w:Headphones|Headphones]] are recommended but not required. *Acces to a tuner. {{#section:Draft:Sing free/Transclusions|tuners}} *When this resource is more developed, we need to supplement this page with something like [[w:Audacity|Audacity]]. [[File:LilyPond wikimedia Playback speed.png|thumb|LilyPond wikimedia Playback speed]] ===How to change tempo (fast/slow)=== Many people like to learn a song by practicing it at different [[w:tempo|tempo]]s. The easiest way to do this is to click on the three vertical dots, as shown to the right. Then select "Playback speed". For those interested in editing or creating songs, it is generally not difficult modify the script for the [[mw:Extension:Score|Score extension]]. Generally speaking, this should always be done by first copying into your sandbox, or creating a version of the song in the [[Draft:Sing free]]. ===Two ways to change the key (pitch) of a song=== #Request a key change of any song by leaving a message at [[user talk:Guy vandegrift]]. #Modify the {{font|\transpose|font=Georgia|size=16px|color=green}} command in the source code as shown below: <score vorbis="1">\language "english" \transpose c c \relative c'' {\key c \major a4 <d, f a>2 } </score> <div class="thumb tright"><div style="width:370px;"> <syntaxhighlight lang="text"> <score vorbis="1">\language "english" \transpose c c \relative c'' {\key c \major a4 <d, f a>2 } </score> </syntaxhighlight></div></div> {{clear}} <score vorbis="1">\language "english" \transpose c f \relative c'' {\key c \major a4 <d, f a>2 } </score> <div class="thumb tright"><div style="width:370px;"> <syntaxhighlight lang="text"> <score vorbis="1">\language "english" \transpose c f \relative c'' {\key c \major a4 <d, f a>2 } </score> </syntaxhighlight></div></div> ==Appendix== '''Help:''' [[w:Help:Score]] | [[s:Help:Sheet music]] | [https://lilypond.org/doc/v2.22/Documentation/learning/aligning-lyrics-to-a-melody.html lyrics] | [https://lilypond.org/doc/v2.21/Documentation/notation/chord-mode chords] | [https://lilypond.org/doc/v2.23/Documentation/notation/midi-instruments instruments] *https://itsybitsykidsmusic.com/index.php/portfolio/free-lead-sheet-with-chords-twinkle-twinkle-little-star/ *[[Ear training]] *[[w:User:Guy vandegrift/Lectures/Just and equal tuning]] *[[w:Vocal_range]] and [[w:Piano_key_frequencies#List]] *[[Music/Software/Lilypond]] and [[Music/Software/Lilypond/MIDI]] ;Subpages of [[Sing free]]: {{Special:Prefixindex/Sing free/|hideredirects=0|stripprefix=1}} ;Subpages of [[Draft:Sing free]]: {{Special:Prefixindex/Draft:Sing free/|hideredirects=1|stripprefix=1}} [[Category:Music performance]] r7wfj0lvb328lwik52sdte61zywptra C language in plain view 0 285380 2812845 2812733 2026-06-05T13:55:39Z Young1lim 21186 /* Applications */ 2812845 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.20260605.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> g1dqm28ojh18ap7wb114wulsym46vby 2813274 2812845 2026-06-06T09:23:19Z Young1lim 21186 /* Applications */ 2813274 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.20260606.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> n0rshx1uk8dc8w90nbslqsb220jq7yh Motivation and emotion/Book/2022/ADHD and motivation 0 286738 2812834 2730067 2026-06-05T12:06:11Z Jtneill 10242 Adjust categories 2812834 wikitext text/x-wiki {{title|ADHD and motivation:<br>How does ADHD impact on motivation and what can be done about it?}} {{MECR3|1=https://www.youtube.com/watch?v=SXkKUkTtaCU}} __TOC__ ==Overview== [[File:ADHD Word Cloud in Blue.png|thumb|324x324px|''Figure 1.'' Common positive and negative stigmas associated with ADHD]] Attention Deficit Hyperactivity Disorder (ADHD) is a common psychological disorder that stems from impairments and deficits in the human brain. This affects behaviour, thoughts and emotions (ADHD Australia, 2019), which can cause challenges with [[motivation]]. Due to [[wikipedia:Social_stigma|stigmas]] and [[wikipedia:Stereotype|stereotypes]], people can mistake these motivation challenges for laziness, carelessness, or a lack of dedication (see Figure 1). Providing education about ADHD and its impact on motivation is an effective way to make treatment more accessible, combat stereotypes and stigmas, and create lasting change in the lives of those with ADHD (Retz et al., 2021). Extensive literature has developed in more recent years that investigates the causes of ADHD and treatments and interventions. This chapter focuses on reviewing current literature about ADHD and motivation. Based on this research this chapter's content aims to provide the reader with an understanding of what ADHD is, and how it affects motivation and to explore solutions and treatments to help reduce motivation challenges for those with ADHD. {{Robelbox|theme={{{theme|13}}}|title= Case study: Meet Frank and his mum}} <div style="{{Robelbox/pad}}"> This case study is used throughout the chapter to assist readers in understanding and applying the following chapter's content. Meet Frank. Frank is a teenager in Year 12 who has ADHD. Frank experiences challenges at college and at home, {{gr}} he finds himself zoning out during class and struggling to complete his assignments. At home, Frank feels like he disappoints his mum when he can't seem to bring himself to clean and maintain his room. These challenges and behaviours frustrate Frank as he doesn't understand why he can't just 'do what he needs to do'. Meet Frank's mum. She loves Frank very much and tries to be patient with him but sometimes she just 'can't take it anymore!'. Frank forgets her instructions or tasks he needs to do, and can't motivate himself to do his assignments or clean his room. She finds herself wishing he would just 'stop being lazy' and 'get his act together'. </div> {{Robelbox/close}} == ADHD explained == [[File:PSYCHOLOGY BOOK.jpg|thumb|Figure 2. The DSM-5 used to diagnose ADHD ]] ADHD is a common mental health disorder that is typically characterised by patterns of impulsive and sometimes hyperactive behaviour, inattentiveness, and impaired emotion regulation. Figure 3 includes a list of common ADHD symptoms. Many who struggle with the disorder will experience deficits in areas or tasks that involve memory, recalling information, planning, self-monitoring and self-soothing and problem-solving{{fact}}. These behavioural and emotional symptoms can make it challenging for someone with ADHD to complete everyday tasks and routines. ADHD often presents with other co-occurring mental health disorders such as [[wikipedia:Anxiety_disorder|anxiety]], [[wikipedia:Major_depressive_disorder|depression]] and [[wikipedia:Addiction|addiction]] (Wilens & Spencer, 2010). The diagnosis of ADHD involves the individual meeting criteria outlined in the [[wikipedia:DSM-5|DSM-5]] (see Figure 2) (ADHD Australia, 2019). {| class="wikitable" |+Figure 3. List of ADHD symptoms{{fact}} !Behavioural !Physiological !Psychological |- | Impaired emotion regulation Impaired impulse control Hyperactivity Inattention Mood swings Impaired focus and concentration Poor time management and task prioritisation |Impaired brain regions * Frontal cortex * Limbic system * Basal ganglia * Reticular activating system Neurochemical imbalances * Norepinephrine * Dopamine |Impaired executive function Emotionally reactive Problem-solving challenges Poor sensory relegation and processing Impaired memory and recall Interrupted sleep |} == How does ADHD impact motivation? == This section explores the specific neurological differences in brain chemistry and biology observed in the ADHD brain that are believed to impact motivation. {{fact}} === Brain chemistry === One of the main reasons people with an ADHD diagnosis can experience a lack of motivation is an imbalance in their [[wikipedia:Neurochemistry|brain chemistry]] stemming from deficits in [[wikipedia:Dopamine|dopamine]] and [[wikipedia:Norepinephrine#Sympathetic_nervous_system|norepinephrine]] production and processing. === Dopamine === [[File:Dopamin - Dopamine.svg|thumb|Figure 3. The chemical structure of dopamine]] [[wikipedia:Dopamine|Dopamine]] (see Figure 3) is a neurotransmitter that is released during or in anticipation of pleasurable experiences. It helps people to identify events and actions that can lead to a reward or pleasurable experience and motivate them to take action to experience these results. The pleasurable experience from a dopamine release helps the brain make a connection between the action and pleasure, making the individual likely to repeat the behaviour to experience the pleasure again, i.e. like the brain saying "doing that felt good... do it again!" (Arias-Carrión & Pŏppel, 2007). For people with ADHD, their brains have lower levels of dopamine than a neurotypical brain. This deficit of dopamine causes motivation challenges regarding the completion of tasks and goals as those with ADHD can identify that completing a task will lead to a reward later, but their brain doesn't release dopamine in anticipation of that reward, meaning the individual doesn't experience the boost in motivation to complete the task for the reward.{{fact}} When someone with ADHD does receive a reward, the associated dopamine release and motivation boost is weaker and shorter than it is for the neurotypical brain. This weaker reaction makes it harder for the brain to connect the action to the pleasurable reward, making the individual less likely to repeat the behaviour (Levy & Swanson, 2001). {{Robelbox|theme={{{theme|13}}}|title=Case study example 1}} <div style="{{Robelbox/pad}}"> Frank is wanting to apply for university and knows he needs to fill out an application. In normal circumstances, a person would be able to link the filling out of the application with the anticipation of the pleasurable feeling that will come with graduating and the dopamine from the anticipation of the reward would motivate the action of filling out the form. In Frank's case, he knows that filling out the application will lead to graduating but his brain doesn't 'light up' in anticipation of that reward, meaning he doesn't experience the dopamine fuelled motivation boost to help him fill out the form. </div> {{Robelbox/close}} === Norepinephrine === [[File:Scout Girl in Concentration.jpg|left|thumb|Figure 4. Norepinephrine helps control attention ]] [[wikipedia:Norepinephrine#Sympathetic_nervous_system|Norepinephrine]] is a neurotransmitter that provides a trigger to 'get the brain going', and is responsible for controlling attention, alertness and arousal (see figure 4). Norepinephrine assists with motivation as it can control or increase the brain and body's responses, perception and processing of stimuli and changes in situations by adjusting the input and processing of the [[wikipedia:Prefrontal_cortex|prefrontal cortex]]. This controls the brain's levels of attentiveness and alertness towards specific stimuli and initiates actions towards the stimuli, such as listening to instructions and then taking action to follow the instruction to complete a task (Sara, 2015). Research surrounding the connection between norepinephrine, ADHD and motivation is still developing, but has shown evidence of lower levels of norepinephrine in the ADHD brain resulting from altered production and processing and suggests that norepinephrine deficits could be responsible for a lack of motivation to complete tasks experienced by people with ADHD{{fact}}. Evidence suggests a disconnect between the recognition of a task, the planning of task completion and the actual motivation to take action to complete the task as there are insufficient levels of norepinephrine to motivate and 'get the brain going' to work to achieve a pleasurable experience or reward (Faraone et al., 2014). {{Robelbox|theme={{{theme|13}}}|title=Case study example 2}} <div style="{{Robelbox/pad}}"> Frank's struggle with wanting to clean and maintain his room or fill out his university application but being unable to get motivated to do either task could be explained by an imbalance of norepinephrine, leading him to not have the means to physically be motivated to complete these tasks. </div> {{Robelbox/close}} === Neurobiology === [[File:Prefrontal cortex.png|thumb|Figure 5. Location of the frontal cortex in the human brain|243x243px]] The ADHD brain experiences impaired functioning in four main areas. It affects their functions and the role they play in motivation are discussed below.{{fact}} ==== Frontal Cortex ==== The frontal cortex (see figure 5) plays a key role in motivation, more specifically in areas including high-level functioning, [[wikipedia:Executive_functions|executive function]], attention and organisation (Kouneiher et al., 2009). Studies have identified weaker connections, functions and structure of the frontal cortex, resulting in impaired attention delegation and weakened links between internal goals and behaviour in the ADHD brain. Deficits in norepinephrine and dopamine are believed to contribute to the impaired function of this brain region (Arnsten, 2009) (F. T. Arnsten, 2009). ==== Limbic System ==== The [[wikipedia:Limbic_system|limbic system]] regulates emotion and attention and is connected to the [[wikipedia:Mesolimbic_pathway|reward pathway]]. Research has found size differences {{ic|between?}} in structures in the limbic system (Plessen et al., 2006). Evidence suggests that a link between ADHD and differences in the limbic system could cause restlessness and inattentiveness, resulting in impaired motivation. ==== Basal Ganglia ==== The [[wikipedia:Basal_ganglia|basal ganglia]] (BG) assists with inter-brain communication and controls the giving or withdrawal of energy to or from actions, depending on if the action is deemed 'worth it' or not by the individual{{fact}}. Evidence shows size and volume differences in the BG in the ADHD brain and suggests this could cause impaired brain processing capacity, leading the individual to feel overwhelmed, resulting in 'zoning out' and inattentiveness (Qiu et al., 2009) (Lauren, 2013). Research suggests these differences could cause over or underactive energy delegation towards tasks, causing [[wikipedia:Hyperfocus|hyper-fixation]] on a specific task resulting in struggles with effective task prioritisation and motivation challenges for completing tasks (Lauren, 2013). ==== Reticular activating system ==== The [[wikipedia:Reticular_formation|reticular activating system]] (RAS) acts as an 'on and off switch' for the brain, {{gr}} it controls what messages enter and leave the brain and the arousal and regulation of thoughts, behaviour and emotions, which directly influences concentration and focus. Individuals with ADHD have an impaired RAS, resulting in impaired focus and behaviour control, {{gr}} this can result in 'over or under-arousal'{{fact}}. This causes challenges with sustaining focus on a task due to the state of arousal becoming too high to focus on a task, resulting in hyperactivity, or the arousal is not high enough to produce enough energy to motivate the individual to complete a task. == Theoretical underpinnings about ADHD and motivation == This section explores theories underpinning the connection between ADHD and motivation. ==== Executive function ==== Concerning motivation, [[wikipedia:Executive_functions|executive function]] refers to a person's ability to control concentration, attention and identification of rewards to then direct and adapt behaviour towards achieving that reward. These skills allow the organisation of thoughts and actions so the individual can effectively prioritise and perform actions to achieve goals and rewards.{{fact}} Individuals with ADHD struggle to exercise their executive function (this is called [[wikipedia:Executive_dysfunction|executive dysfunction]]), meaning they can struggle and get overwhelmed by the ordering and planning of the 'what', 'why', 'when' and 'how' in regards to tasks, goals and actions and then struggle to direct energy and emotion effectively to fuel their motivation and use this energy to work towards an action to achieve a goal.{{fact}} ==== Self-determination theory ==== [[wikipedia:Self-determination_theory]] (SDT) provides a framework for motivation suggesting that to achieve psychological growth, an individual needs to feel like they have control over their behaviours and goals, the individual needs to acquire skills and knowledge to enable them to gain a sense of independence and to help them achieve their goals, and the individual needs to feel connected to other people (Pesantez, 2021). Historically, ADHD therapy has predominantly focused on positive reinforcement strategies to enhance motivation, often relying heavily on extrinsic factors such as rewards and recognition. However, emerging research indicates that nurturing intrinsic motivation can be equally, if not more, effective in the treatment of ADHD. By integrating SDT into ADHD interventions, the focus can transition towards cultivating an internal drive for motivation. This method emphasises fostering a sense of autonomy, competence, and relatedness, aiming to instigate profound and sustainable transformations for individuals managing ADHD. It not only accommodates their unique symptoms but also empowers them to develop robust coping strategies and adaptive behaviours, enabling them to adeptly navigate the complexities of motivation challenges in their daily lives (Morsink et al., 2021). ==== Reward pathway and the pleasure-pain principle ==== The [[wikipedia:Mesolimbic_pathway|reward pathway]] is an avenue in the brain that allows for connections to be made between behaviour and events that bring a pleasurable feeling or experience and then increased the likelihood of the behaviour being repeated. Due to impaired and imbalances, the reward pathway can be impaired, making it challenging for people with ADHD to form connections between pleasurable experiences and behaviours or events, meaning they are less likely to repeat actions that lead to a pleasurable experience or reward (Volkow et al., 2009). The pleasure-pain principle suggests that humans decide their actions based on the avoidance of pain or the pursuit of pleasure. People decide what they consider to be pain or pleasure based on their beliefs and values. Due to an impaired reward pathway, individuals with ADHD can struggle to identify which actions and events will or will not bring pleasure or pain.{{fact}} == Solutions and treatment == [[File:VariousPills.jpg|thumb|260x260px|''Figure 6.'' Various pills as an example of stimulant, SSRI or SNRI medication]] This section explores different treatments and strategies used to treat or address motivation challenges in ADHD. The most successful treatment cases of ADHD are treatment plans that have used an integrated approach and used multiple treatments together, according to the client's unique circumstances (Johnson & Safranek, 2005) (Pelham & Gnagy, 1999). === Medication === Table 1 summarises information on ADHD medication, what the medications do and the types of medication in each category (see example in Figure 6). {| class="wikitable" |+Table 1. Medications for ADHD ! !Stimulant !Non-stimulant |- |What does it do? |[[wikipedia:Stimulant|Stimulant]] medications for ADHD target specific cells in the brain that produce dopamine and norepinephrine and stimulate the cells to increase the production of both neurotransmitters to create and maintain healthy levels of these neurotransmitters. These effects work to lessen or alleviate the effects that insufficient levels of both neurotransmitters have on motivation. |The most common non-stimulant drugs that are used to treat ADHD fit into the categories of either [[wikipedia:Selective_serotonin_reuptake_inhibitor|SSRIs]] or [[wikipedia:Serotonin–norepinephrine_reuptake_inhibitor|SNRIs]] - these medications can be used to treat symptoms of co-occurring psychological disorders (e.g. anxiety or depression) and help balance out dopamine and norepinephrine levels in the ADHD. |- |Common types of medication |Stimulant medications can be short-acting, intermediate-acting or long-acting according. Short-acting stimulants: * Methylphenidate, also known as Ritalin * Dextroamphetamine, also known as Dexedrine, ProCentra and Zenzedi * Amphetamine/dextroamphetamine also known as Adderall * Dexmethylphenidate, also known as Focalin Intermediate-acting stimulants: * Methylphenidate, also known as Ritalin SR, Metadate ER, Methylin ER * Amphetamine sulfate, also known as Evekeo Long-acting stimulants: * Amphetamine, also known as Dyanavel XR and Adzenys XR-ODT * Dexmethylphenidate, also known as Focalin XR * Dextroamphetamine, also known as Adderall XR * Lisdexamfetamine, also known as Vyvanse * Methylphenidate, also known as Concerta, Ritalin LA and Quillivant XR |Common non-stimulant ADHD medications: * Atomoxetine, also known as Strattera * Viloxazine, also known as Qelbree * Clonidine, also known as Catapres and Kapvay * Guanfacine, also known as Tenex, Intuniv * Buproprion, also known as Wellbutrin * Qelbree * Tricyclic antidepressants * Monoamine oxidase inhibitors (MAOIs) such as phenelzine and tranylcypromine |- |How these medications help with motivation | colspan="2" |These medication treatments, by balancing out the levels of dopamine and/or norepinephrine neurotransmitters in the brain, can lessen or alleviate the effects and impacts on motivation caused by deficiencies of these neurotransmitters. |} (Bhandari, 2021), (''Non-Stimulant ADHD Medication Overview'', 2019) === Therapies and training === The therapies and training discussed below are used to treat motivation challenges in ADHD. * [[wikipedia:Behavior_modification|Behaviour modification]] Behaviour modification for ADHD uses rewards to reinforce positive behaviours that are challenging for the individual, such as completing a task or doing 20 minutes of an assignment. By reinforcing these actions, the individual is more likely to repeat the behaviour. Reinforcements are usually in response to small behaviours or progress, in shorter intervals, to account for difficulties with attention and the reward pathway. The more these actions are reinforced, the simpler and easier it is for the individual to make a connection between the action and the reward and pleasurable experience, which helps the individual get motivated for these actions and to be more likely to repeat them. * [[wikipedia:Cognitive_therapy|Cognitive therapy]] This therapeutic approach focuses on thought patterns. The individual and therapist role-play situations and associated thoughts and then practice using different reactions and thoughts to the given situation. For example, this could look like, {{gr}} being given a task for school, catching the thought processes associated with 'leaving it till later' or 'being overwhelmed and not knowing where to start' and instead thinking and acting on breaking the task into smaller bits and staggering the completion of each little task. Other thoughts and actions can involve using behaviour modification techniques of using rewards and positive reinforcement. * Education and training [[File:Wiki Education attends Dr. Amin Azzam's spring 2018 UCSF class 24.jpg|thumb|272x272px|Figure 8. Some treatments involve education and training workshops for ADHD]] Education and training focus on educating the client and possibly people in their support network, about the AHDH and strategies to overcome and cope with motivation challenges through therapy sessions, workshops (see figure 8), group therapy sessions and many more avenues. When the client or parents of a child with ADHD are equipped with this knowledge, they can notice thoughts, behaviours etc associated with motivation challenges and then use their knowledge of possible coping strategies to find a solution that works for them to help motivate them in different scenarios. * Strategies People with ADHD often can be overwhelmed by tasks and experience challenges with their executive function which can inhibit motivation for short-term and long-term goals. These challenges can be addressed through strategic goal setting and achievement, this could involve ideas such as breaking tasks down into smaller tasks, setting time limits and working for shorter periods, using rewards to reinforce the completion of small tasks or goals.{{fact}} {{Robelbox|theme={{{theme|13}}}|title=Case study example 3}} <div style="{{Robelbox/pad}}"> An example of an ADHD treatment for Frank could be the use of stimulant medication, coupled with education for him and his mum, allowing them both to be equipped to create change and ensure Frank is supported in working on his motivation challenges. Frank could attend cognitive therapy to identify his thought patterns and when they occur. He could then implement behaviour modification strategies such as breaking his assignment into small parts and rewarding the completion of each part. </div> {{Robelbox/close}} === Quiz === <quiz display="simple"> {Education and training treatment is only beneficial to the client with ADHD: |type="()"} - True + False {Education and training treatment is beneficial for the client with ADHD and people in their support network: |type="()"} + True - False </quiz> == Conclusion == This chapter provides an understanding of what ADHD is, how it affects motivation and explores solutions and treatments to help reduce motivation challenges for those with ADHD. ADHD is a common psychological disorder with symptoms that are linked to motivation challenges (ADHD Australia, 2019). These challenges are caused by neurological impairments and deficits in the human brain that occur when the individual has ADHD. The research surrounding treatments and knowledge of the disorder has grown extensively in recent years, leading to the creation and improvement of interventions and treatment and helping to combat the stigmas and stereotypes associated with ADHD (Retz et al., 2021). The use of stimulant and non-stimulant medication are some of the main treatment approaches and are often used in a combined approach with therapy, education and training to equip people with ADHD with strategies and tools to overcome or lessen their challenges with motivation. ADHD research and applications of this research still have room to develop, however, the current knowledge available provides the public with opportunities to seek treatment to be able to overcome motivation challenges or be equipped to support people in their life with ADHD in their treatment journey as they explore and overcome their struggles with motivation. The more information about ADHD and motivation is shared, the more support becomes accessible for those living with or supporting someone with ADHD and the more negative stigmas and stereotypes associated with ADHD and motivation challenges can be challenged and reduced. == See also == [[Motivation and emotion/Book/2021/Dopamine and motivational drive|Dopamine and motivational drive]] (Book chapter, 2021) [[Motivation and emotion/Book/2017/Dopamine and drug addiction|Dopamine and drug addiction]] (Book chapter, 2017) [[Motivation and emotion/Book/2017/Norepinephrine and motivation|Norepinephrine and motivation]] (Book chapter, 2017) [[Motivation and emotion/Book/2017/Impulsivity|Impulsivity]] (Book chapter, 2017) [[Motivation and emotion/Book/2013/Intrinsic motivation|Intrinsic motivation]] (Book chapter, 2013) [[Motivation and emotion/Book/2013/Extrinsic motivation|Extrinsic motivation]] (Book chapter, 2013) [https://www.youtube.com/watch?v=4JqswmelblQ What is ADHD? | National Geographic] (YouTube) == References == {{Hanging indent|1= ADHD Australia. (2019). What is Attention Deficit Hyperactivity Disorder (ADHD)? ADHD Australia. https://www.adhdaustralia.org.au/about-adhd/what-is-attention-deficit-hyperactivity-disorder-adhd/ Arias-Carrión, O., & Pŏppel, E. (2007). Dopamine, learning, and reward-seeking behaviour. Acta Neurobiologiae Experimentalis, 67(4), 481–488. https://pubmed.ncbi.nlm.nih.gov/18320725/ Arnsten, A. F. T. (2009). ADHD and the Prefrontal Cortex. The Journal of Pediatrics, 154(5), I-S43. https://doi.org/10.1016/j.jpeds.2009.01.018 F. T. Arnsten, A. (2009). The Emerging Neurobiology of Attention Deficit Hyperactivity Disorder: The Key Role of the Prefrontal Association Cortex [Review of The Emerging Neurobiology of Attention Deficit Hyperactivity Disorder: The Key Role of the Prefrontal Association Cortex]. The Journal of Paediatrics, 5. National library of medicine. https://doi.org/10.1016/j.jpeds.2009.01.018 Faraone, S. V., Bonvicini, C., & Scassellati, C. (2014). Biomarkers in the Diagnosis of ADHD – Promising Directions. Current Psychiatry Reports, 16(11). https://doi.org/10.1007/s11920-014-0497-1 Johnson, L. A., & Safranek, S. (2005). What is the most effective treatment for ADHD in children? Mospace.umsystem.edu. https://mospace.umsystem.edu/xmlui/handle/10355/3377 Kouneiher, F., Charron, S., & Koechlin, E. (2009). Motivation and cognitive control in the human prefrontal cortex. Nature Neuroscience, 12(7), 939–945. https://doi.org/10.1038/nn.2321 Levy, F., & Swanson, J. M. (2001). Timing, Space and ADHD: The Dopamine Theory Revisited. Australian & New Zealand Journal of Psychiatry, 35(4), 504–511. https://doi.org/10.1046/j.1440-1614.2001.00923.x McGoey, K. E., Eckert, T. L., & DuPaul, G. J. (2002). Early intervention for preschool-age children with ADHD: a literature review. In www.ncbi.nlm.nih.gov. Centre for Reviews and Dissemination (UK). https://www.ncbi.nlm.nih.gov/books/NBK69230/ Morsink, S., Van der Oord, S., Antrop, I., Danckaerts, M., & Scheres, A. (2021). Studying Motivation in ADHD: The Role of Internal Motives and the Relevance of Self Determination Theory. Journal of Attention Disorders, 108705472110509. https://doi.org/10.1177/10870547211050948 Pelham, W. E., & Gnagy, E. M. (1999). Psychosocial and combined treatments for ADHD. Mental Retardation and Developmental Disabilities Research Reviews, 5(3), 225–236. https://doi.org/10.1002/(SICI)1098-2779(1999)5:3<225::AID-MRDD9>3.0.CO;2-E Plessen, K. J., Bansal, R., Zhu, H., Whiteman, R., Amat, J., Quackenbush, G. A., Martin, L., Durkin, K., Blair, C., Royal, J., Hugdahl, K., & Peterson, B. S. (2006). Hippocampus and Amygdala Morphology in Attention-Deficit/Hyperactivity Disorder. Archives of General Psychiatry, 63(7), 795–807. https://doi.org/10.1001/archpsyc.63.7.795 Qiu, A., Crocetti, D., Adler, M., Mahone, E. M., Denckla, M. B., Miller, M. I., & Mostofsky, S. H. (2009). Basal Ganglia Volume and Shape in Children With Attention Deficit Hyperactivity Disorder. American Journal of Psychiatry, 166(1), 74–82. https://doi.org/10.1176/appi.ajp.2008.08030426 Retz, W., Ginsberg, Y., Turner, D., Barra, S., Retz-Junginger, P., Larsson, H., & Asherson, P. (2021). Attention-Deficit/Hyperactivity Disorder (ADHD), antisociality and delinquent behaviour over the lifespan. Neuroscience & Biobehavioral Reviews, 120, 236–248. https://doi.org/10.1016/j.neubiorev.2020.11.025 Sara, S. J. (2015). Locus Coeruleus in time with the making of memories. Current Opinion in Neurobiology, 35, 87–94. https://doi.org/10.1016/j.conb.2015.07.004 Sara, Susan J., & Bouret, S. (2012). Orienting and Reorienting: The Locus Coeruleus Mediates Cognition through Arousal. Neuron, 76(1), 130–141. https://doi.org/10.1016/j.neuron.2012.09.011 Volkow, N. D., Wang, G.-J., Kollins, S. H., Wigal, T. L., Newcorn, J. H., Telang, F., Fowler, J. S., Zhu, W., Logan, J., Ma, Y., Pradhan, K., Wong, C., & Swanson, J. M. (2009). Evaluating Dopamine Reward Pathway in ADHD. JAMA, 302(10), 1084. https://doi.org/10.1001/jama.2009.1308 Wilens, T. E., & Spencer, T. J. (2010). Understanding Attention-Deficit/Hyperactivity Disorder from Childhood to Adulthood. Postgraduate Medicine, 122(5), 97–109. https://doi.org/10.3810/pgm.2010.09.2206 }} == External links == [https://www.drlaurendeville.com/articles/anxiety-adhd-basal-ganglia/ Anxiety, ADHD, and the Basal Ganglia] (Lauren, 2013) [https://www.youtube.com/watch?v=s_P6sNFjLzI How Dopamine Affects Learning and Motivation in ADHD Brains] (How to ADHD, 2021) [https://www.medicalnewstoday.com/articles/clonidine-for-adhd Is clonidine an effective treatment for ADHD?] (''Medical News Today'', 2021) [https://www.additudemag.com/non-stimulant-adhd-medication/ Non-Stimulant ADHD Medication Overview] (Janice Rodden, 2022) [https://www.webmd.com/add-adhd/adhd-stimulant-therapy Stimulant Medications for ADHD] (Bhandari, 2021) [https://www.additudemag.com/self-determination-theory-adhd-motivation-research-news/ Study: Self-Determination Theory Could Guide Research on ADHD and Motivation] (Nathaly Pesantez, 2022) [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Attention]] [[Category:Motivation and emotion/Book/ADHD]] 7e4f86p0qxp27lhhfytazskqikioxoo Draft:How to write a song 118 299310 2813009 2554378 2026-06-06T00:33:20Z Kirby - Electrotechnics 3074947 changed to 'category:Music composition' instead of category:Music 2813009 wikitext text/x-wiki {{Draft}} '''[[w:Songwriter|Songwriting]]''' is the process by which someone creates either lyrics for a song and/or the music itself. Songwriting can be very intimidating for new songwriters, and many struggle to make it past the 3 minute mark. The purpose of this guide is to help new songwriters through that block. Keep in mind, there is no "correct" way to write a song. == Before beginning == Think about the following: * Are you in a {{W|Musical ensemble|band}}, or are you a {{W|Solo (music)|solo artist}}? * What {{W|Musical genre|genre}} are you? * What instrumentation do you want? * How complex do you want the song to be? After thinking about those, think about what you want the song to be about? Most songs are generally about: * Personal experiences (being poor, being heartbroken) * Emotions * {{W|Protest song|Protesting}} * A fictional story == Starting out == === Structure === With an idea of what the song is going to sound like, start out thinking of a short, catchy melody. This will be called the {{W|refrain}}, or chorus. It is the line or set of lines that repeat and seldom change throughout the song. You may also optionally write an opening, or an {{W|Introduction (music)|introduction}} (intro for short) that usually resembles the chorus. Then, think of a longer melody. This will be called the {{W|Verse (poetry)|verse}}. The verses, while they repeat through the song, almost always change throughout the song. They contain a few lines that progress the story of the song forward. Then, feel free to experiment in the {{W|Bridge (music)|bridge}}. The bridge is an optional portion of the song typically after the last verse and before the last chorus, but this is not always the case. The bridge does not typically sound like the rest of the song. It may change key (such as in {{W|Red Hot Chili Peppers}}' {{W|Otherside}}, from {{W|A minor}} to {{W|E Minor}}), change vocalists and instruments, and may even include a {{W|Solo (music)|solo}}. There is no one song structure, but the most common are: * [[w:Verse-chorus_form|Intro]] * [[w:Verse-chorus_form|Verse]] * [[w:Verse-chorus_form|Chorus]] * [[w:Verse-chorus_form|Verse]] * [[w:Verse-chorus_form|Chorus]] * [[w:Verse-chorus_form|Bridge]] * [[w:Verse-chorus_form|Chorus]] and * Intro * Chorus * Verse * Chorus * Bridge * Chorus [[Category:Music composition]] hwr0k9mpc5z6quvca54evw4trqh0co4 Film Music 0 301640 2812954 2810314 2026-06-05T21:41:49Z Kirby - Electrotechnics 3074947 changed to 'category:Music in Film' instead of category:Music 2812954 wikitext text/x-wiki [[Category:Film]] [[Category:Cinema Aesthetics]] [[Category:Music in Film]] '''Film Music''' refers to any music that is featured in a film, either as a background accompaniment and/or in-world, such as music a character listens to. Film music can be separated into two main categories: '''''soundtrack''''' and '''''film score.''''' == Soundtrack == A film soundtrack is the selection of music that is used in a given film, typically pre-existing music that is licensed. Generally, music for a film's soundtrack is selected by a music supervisor in consultation with a film's director and producer. Often these songs are also released as an album. Sometimes a film’s soundtrack will feature original songs - songs produced specifically for the film. An example of this is Celine Dion’s "My Heart Will Go On", produced for and featured in ''[[Titanic (1997)]]''. === Aesthetics === [[File:Baz Luhrmann Cannes 2013.jpg|thumb|Baz Luhrmann at the Cannes Premiere of ''The Great Gatsby'' in 2013.]] The choice of music featured in a film’s soundtrack has wide implications for the aesthetics of a film. For example, period pieces typically utilize music from the period being portrayed. This creates a greater sense of historical immersion for the audience and can conjure feelings of nostalgia. [[Wes Anderson]]’s ''[[Fantastic Mr. Fox|Fantastic Mr. Fox (2012)]]'' does this, matching its 1960’s/70’s setting with a soundtrack full of music from that time period. However, another technique is purposely using music anachronistic to a film’s setting. Baz Luhrmann chose to feature modern pop music in his remake of ''The Great Gatsby (2013)'', a film set during the 1920’s. In an interview with Rolling Stone, he explained that: ''“The question for me in approaching Gatsby was how to elicit from our audience the same level of excitement and pop cultural immediacy toward the world that Fitzgerald did for his audience? And in our age, the energy of jazz is caught in the energy of hip-hop.”<ref>RJ Cubarrubia, “Jay-Z Producing Music for ‘Great Gatsby’”, ''Rolling Stone,'' 2013. https://www.rollingstone.com/music/music-news/jay-z-producing-music-for-great-gatsby-181443/</ref>'' Some filmmakers have become known for their preference for using soundtracks rather than film scores, a good example being [[Sofia coppola|Sofia Coppola]]. Filmmakers who work in this mold will often plan scenes around the use of a particular song. These moments are referred to as "needle-drops." Another common technique is the use of pre-existing classical/opera pieces in a film’s soundtrack. These pieces function similarly to how a film’s score does. Stanley Kubrick chose to use a selection of classical pieces rather than a film score for much of the run-time of ''The Shining (1980).'' == Film Score == A film score is the collection of music composed and produced specifically for a film, typically by a composer in consultation with a film’s director and producers. === Production === Film scores vary by genre, but most commonly are recorded live by a symphony orchestra, such as James Horner’s original score for ''[[Titanic (1997)]].'' Some film scores heavily feature electronic music, synths, or other forms of non-traditional orchestration. The score for [[Kelly Reichardt]]’s ''First Cow (2019),'' composed by William Tyler, features electric guitar, toy piano, banjos, and mandolins<ref>Nadine Smith, “Music from First Cow”, ''Pitchfork,'' 2020. https://pitchfork.com/reviews/albums/william-tyler-music-from-first-cow/</ref>. Film scores, similarly to film soundtracks, are often released as albums. It’s common for film score albums to feature altered versions of film score cues to improve their listenability<ref>Michael W. Harris, “Hollywood Film Scores.” ''Notes,'' vol. 74, no. 3, Mar. 2018, pp. 497–502. EBSCOhost, search.ebscohost.com/login.aspx?direct=true&AuthType=ip,shib&db=edsjsr&AN=edsjsr.26397539&site=eds-live&scope=site.</ref>. [[File:Hans Zimmer 2010.jpg|thumb|Hans Zimmer in 2010.]] In the traditional film scoring process, a composer would write all the music for a film. John Williams is known for this approach. However, a new model is becoming the dominant one in Hollywood film scoring. In this one, the film’s composer is more of a team leader, dictating the “tonal palette” for a film, Hans Zimmer refers to this as “sketching”, and assigning cues to additional composers or studio assistants to compose for specific scenes<ref>Mark Rozzo, ““The Minions Do the Actual Writing”: The Ugly Truth of How Movie Scores Are Made”, ''Vanity Fair,'' 2022. https://www.vanityfair.com/hollywood/2022/02/the-ugly-truth-of-how-movie-scores-are-made</ref>. Under this model, the actual moment to moment scoring of a film is done by these additional composers, who are often uncredited. === Aesthetics === One of the most common techniques in film scoring is known as ''leitmotif'' - the use of a recurrent theme. Repeated use of a theme can create an association between it and a person, object, place, or idea. A composer can then develop or change the theme to reflect the changes occurring within the story to that person, object, place, or idea. An example of this is Howard Shore’s score for ''The Lord of the Rings trilogy,'' which makes use of dozens of leitmotifs, all associated with various people (ex. Aragorn), objects (ex. the Ring), places (ex. the Shire), and ideas (ex. nature)<ref>Doug Adams, "The Music of The Lord of the Rings Films", ''Alfred Music,'' 2010.</ref>. Another common technique is the use of ''[[Texture in film|texture]]''. Composers can make use of non-traditional instrumentation to add a unique texture to a film’s score. This can create/add to a film’s atmosphere. It can also be used in a similar fashion to leitmotif, communicating change to people, objects, places, and ideas through the texture of the music. An example of this is Hans Zimmer’s score for ''[[Interstellar|Interstellar (2014)]],'' which heavily features organ music. Zimmer used the organ’s association with religion to musically express the idea of the vastness of space and time<ref>J. Bryan Lowder, "How Interstellar’s Stunning Score Was Made", ''Slate,'' 2014. https://slate.com/culture/2014/11/making-interstellars-score-hans-zimmer-s-soundtrack-explored-in-exclusive-featurette-video.html</ref>. == Diegetic vs Non-Diegetic Film Music == Film music can be diegetic or non-diegetic. Diegetic means existing within the film’s world, i.e. the characters in the film can hear the music. Non-diegetic film music is music only the audience can hear. Film music can cross over these boundaries, starting off as either diegetic or non-diegetic and transitioning into the other within a scene. This is commonly done at the beginning/end of montage sequences or to transition from one scene to the next. == Terminology == * Cue - an individual piece of music in a film score. * Leitmotif - a recurring theme that represents a person, object, place, and/or idea. * Needle drop - when a piece of pre-existing music is used in a scene. * Theme - a musical phrase. == References == <references /> m053xdqn9ozkntmjy8th66qm228rzzw Category:Hip hop 14 301989 2812972 2588805 2026-06-05T22:05:51Z Kirby - Electrotechnics 3074947 changed to 'category:Music genres and styles' instead of category:Music 2812972 wikitext text/x-wiki [[Category:Music genres and styles]] qhimz008596xnk5444wagy28xmwcppl Music appreciation and history/Why we need essays about music 0 302008 2812981 2591584 2026-06-05T22:18:55Z Kirby - Electrotechnics 3074947 changed to 'category:Music appreciation' instead of category:Music 2812981 wikitext text/x-wiki I'm not sure why we need essays about music. Personally, I need music, but can't explain why. '''But I have lots to say about why we need essays.''' I am a retired physics professor who also taught a lot of courses for people with little or no interest in science. A typical example would be Introductory Astronomy. I tried to make the course relevant to students by hinting at either the history of the subject, or why an understanding of science is essential to everybody in a democratic society. At all levels of everything we do, communication is essential. Scientific knowledge is developed and verified through scientific publications. These publications are an essential part of what we call the [[w:Scientific method|''scientific method'']]. And democracies make decisions based on knowledge that is created by scientific publications and communicated to the public through other publications (called [[w:Secondary source|''secondary sources'']].) When I look around the world, almost without exception, the only places I would want to live are democracies. I can't say whether the link between ''livability'' and ''democracy'' is [[w:Correlation does not imply causation|causal]]: Perhaps only wealthy countries can afford democracy, or perhaps ''democracy'' itself is a source of wealth. Either way, supporting democracy is a safe bet. And democracies require an educated public, and that means we need to learn to communicate. When I look at social media, political dialogue, and even much of what appears in the news, I do not see much communication. Instead I see a desert of cliches and meaningless attempts at rhetoric. So that brings us to the idea that you should learn how to write a thoughtful essay. I wrote a thoughtful essay, not on music, but on the need to write essays. It rambles a bit, but here it is: *[[Socialism/Is it a thing of the left?]] [[Category:Music appreciation]] e0da7pta6tww0of8j0k1r6r53n12o74 Motivation and emotion/Book/2024/ADHD and emotional regulation 0 306473 2812835 2657992 2026-06-05T12:06:48Z Jtneill 10242 Adjust category 2812835 wikitext text/x-wiki {{title|ADHD and emotional regulation:<br>How does ADHD affect emotional regulation and how can this be managed?}} {{MECR3|1=https://yourlinkgoeshere.com}} <div align=center>Replace the link in the box above once the multimedia presentation has been published.</div> __TOC__ ==Overview== {{RoundBoxTop}} [[File:Large bonfire.jpg|thumb|160px|''Figure 1.'' Fire. ]] '''Imagine''' ... something horrible has happened. A ferocious bushfire has torn through your property, leaving pile of rubble where your home once stood. Your family are alive and well, however, everything you own has been destroyed. This disaster rattles you. You're angry and heartbroken. Attempting to process these emotions, you seek ways to cope. Will you lash out at family members, binge drink alcohol, maybe run away? Or will you reach out to loved ones for support, reappraise the situation, and practice mindfulness? {{RoundBoxBottom}} Emotional regulation is like a muscle. It's a set of strategies we build and exercise throughout the lifespan to cope with the love, loss, joy, and fear associated with being human. Emotional regulation is an important area of study as well-developed emotional regulation skills can lead to happiness and health, whereas deficits can lead to maladaptive coping, poor mental health outcomes and antisocial behaviour. People with ADHD are vulnerable to such deficits. Increased emotional sensitivity and executive function deficits can make emotions feel bigger, and strategies more difficult to access. This can lead to increased reliance on maladaptive coping strategies and poorer predicted outcomes. However, there are treatments and strategies available to assist those with ADHD. {{RoundBoxTop|theme=3}} '''Focus questions:''' * What is emotional regulation and why is it important? * What is the relationship between ADHD and emotional regulation? * What emotional regulation strategies are recommended for those with ADHD? {{RoundBoxBottom}} == Emotional regulation and its importance == Emotional regulation refers to the processes involved in monitoring, evaluating and modifying one's emotions (Thompson, 1994). === Stages of emotional regulation === [[File:Process Model of Emotional Regulation.png|none|thumb|613x613px|Figure 2. Stages of emotional regulation according to Gross (1998)]] One of the most prominent theories of emotional regulation is the process model, first published by Gross (1998; see Figure 2). The stages of the process model of emotional regulation (Gross, 1998) are listed in Table 1. {| class="wikitable" |+'''Table 1.''' Process model of emotional regulation |'''Situation Selection''' |Making a decision to approach or avoiding certain situations or stimuli. E.g. avoiding someone who often belittles you. |- |'''Situation Modification''' |Efforts made to alter the emotional impart of a stimuli or situation. E.g. if someone is acting in a way that irritates you, you could ask them to stop. |- |'''Attentional Deployment''' |Focus either given to or removed from an emotionally arousing stimuli. E.g. you may avoid thinking about something that makes you stressed, or you could ruminate on the situation and become increasingly upset. |- |'''Cognitive Change''' |Also referred to as 'appraisal'. This includes initial judgement as to whether the stimuli is deemed relevant (does this matter to me?), assumed implication (will it pose a threat?), and coping potential (can I handle it?). |- |'''Response Modulation''' |Often referred to as 'strategies'. The response stage of emotional regulation refers to behaviours a person utilizes once they feel the emotion. For example, if feeling stressed you may utilise mindfulness or drink a glass of wine to 'calm down'. |} <quiz display="simple"> { In Gross' process model of emotional regulation, attentional deployment refers to choosing whether or not you enter a given situation: |type="()"} - True + False {Emotional regulation involves monitoring of emotions: |type="()"} + True - False </quiz> === Capacity for emotional regulation === Emotional regulation skills are developed across the lifespan (Cole, 2014). Extrinsic influences: Parents help an infant regulate their emotions by caring for them. Once in childhood, positive reinforcement and modelling become more relevant (Thompson, 1991), (Morris et al., 2007). === Consequences of emotional dysregulation === Emotion regulation deficits have been consistently linked to psychopathology in cross-sectional studies (McLaughlin et al., 2011). Excessive attention can lead to rumination and exacerbation of negative emotions (Wadlinger & Isaacowitz, 2011). Poor emotional regulation has been linked with negative health/social outcomes such as substance use disorders (Stellern et al., 2023), self harm (Slee et al, 2008) and increased aggression (Roberton et al., 2012). == Relationship between ADHD and emotional regulation == When first delving into the topic and searching for "ADHD emotional regulation", you will be confronted with an immediate reframing. Rather than articles related to "ADHD and emotional regulation", you'll instead find a plethora of information regarding ADHD and emotional ''dysregulation''. This is the first hint that there might be an interaction here, and it may not be positive. People with ADHD often exhibit symptoms of emotional dysregulation such as a hot temper, emotional lability, and over-reactivity (Retz, et al.,2012). A literature review undertaken by Shaw et al., (2014) concluded that 34–70% of adults with ADHD experience deficits in emotional regulation. Attention-Deficit/Hyperactivity Disorder (ADHD) is a neurodevelopmental disorder characterised by inattention and/or hyperactivity-impulsivity that interferes with functioning or development (American Psychiatric Association, 2013). It is estimated that prevalence of ADHD is approximately 2.5% worldwide (Simon et al., 2009). {| class="wikitable" |+ADHD Definition Breakdown |'''Inattention''' |Includes symptoms such as disorganisation and difficulty with focus. |- |'''Hyperactivity''' |Includes symptoms such as restlessness and excessive motor activity. |- |'''Impulsivity''' |Actions or decisions made quickly without forethought. |} === Bottom-up processing === Bottom-up processing refers to the generation of emotion based on the presentation of an external stimulus. This type of processing is often associated with activity from the amygdala (McRae et al., 2012). For example, you may be overwhelmed by an instant wave of fear when you see a snake slither across the floor. People with ADHD are prone to generating more intense emotions. A meta-analysis undertaken by by Graziano & Garcia (2016) found that children with ADHD had higher emotional reactivity, meaning the stimulus threshold required to illicit an emotional response was lower in children with ADHD than in their neurotypical peers. Discuss whether this has been shown in adults. === Top-down processing === Top-down processing refers to the regulation of emotion based on a more conscious appraisal of the stimuli. This process is related to executive function. Top-down processing is sometimes considered 'higher order' thinking as it recruits a wide range of brain regions, including the inferior frontal gyrus, medial frontal cortex, anterior cingulate cortex, supplementary motor area, and right temporoparietal junction. These brain regions are known to be involved in self regulation, action/response inhibition, and emotion processing (Langner et al., 2018). People with ADHD are known to have deficits in these executive neural networks (Christiansen et al., 2019). === Maladaptive emotional regulation strategies === People with ADHD are more likely to utilise maladaptive emotional regulation strategies than neurotypical peers (Barra et al., 2021). Avoidance is particularly prominent in the research (Torrente et al., 2014). Need to find other emotional regulation research featuring maladaptive strategies. Is there anything about drugs/alcohol? ADHD has high comorbidity with habit-forming illnesses such as alcohol dependency and substance addiction (Ohlmeier et al., 2008), (Baker et al., 2012). However, neither study makes a link directly to emotional regulation, meaning this increased drug/alcohol use could be related to other features of ADHD. == Emotional regulation interventions for people with ADHD == === Psychopharmacological treatments === Surman & Walsh (2022) undertook a systemic review and analysis of clinical trials investigating the affect of stimulant medication on emotional regulation in participants with ADHD. Methylphenidate (commonly known as Ritalin), Lisdexamfetamine (Vyvanse), and Atomoxetine (Strattera) have been found to improve emotional regulation. However, it was not concluded how or why that was the case. Future studies are required to determine whether emotional effects are due to change in the range or pattern of emotions individuals experience (bottom-up processing), or their ability to regulate expression of those emotions (top-down processing). === Mindfulness === Mindfulness is an active and focused awareness of one's surroundings and inner thoughts. Mindfulness has historically been used to treat anxiety and depression symptoms. A pilot trial undertaken by (Mitchell, et al., 2017) found mindfulness practice assisted with emotional dysfunction in ADHD. Cairncross & Miller (2020) undertook meta-analysis into mindfulness based therapies for ADHD and found that MBT was more effective for adults than children. === Reappraisal === Idk maybe there's a better one to add here. will reconsider. ==Conclusion== Overall, available literature cements the relationship between ADHD and emotional regulation. Many sources recommend that emotional dysregulation should be added to the main diagnostic criteria of ADHD. This would perhaps lead to further studies into efficacy of emotional regulation interventions for those with ADHD. Will flesh out conclusion further once content is finalised. ==See also== * [[Motivation and emotion/Book/2014/Appraisal and emotion|Appraisal and emotion]] (Book chapter, 2014) * [[wikipedia:Attention deficit hyperactivity disorder|Attention deficit hyperactivity disorder]] (Wikipedia) * [[wikipedia:Emotional_dysregulation|Emotional dysregulation]] (Wikipedia) * [[Motivation and emotion/Book/2021/Developmental changes in emotion regulation|Developmental changes in emotion regulation]] (Book chapter, 2021) ==References== {{Hanging indent|1= American Psychiatric Association. (2013). ''Diagnostic and statistical manual of mental disorders'' (5th ed.). https://doi.org/10.1176/appi.books.9780890425596 Baker, L., Prevatt, F., & Proctor, B. (2012). Drug and Alcohol Use in College Students With and Without ADHD. ''Journal of Attention Disorders'', ''16''(3), 255–263. <nowiki>https://doi.org/10.1177/1087054711416314</nowiki> Barkley, R.A. (2015). ''Attention Deficit Hyperactivity Disorder: A Handbook for Diagnosis and Treatment'', 4th ed. New York: Guilford Publications. Barra, S., Grub, A., Roesler, M., Retz-Junginger, P., Philipp, F., & Retz, W. (2021). The role of stress coping strategies for life impairments in ADHD. ''Journal of Neural Transmission'', ''128''(7), 981–992. <nowiki>https://doi.org/10.1007/s00702-021-02311-5</nowiki> Cairncross, M., & Miller, C. J. (2020). The Effectiveness of Mindfulness-Based Therapies for ADHD: A Meta-Analytic Review. ''Journal of Attention Disorders'', ''24''(5), 627–643. <nowiki>https://doi.org/10.1177/1087054715625301</nowiki> Christiansen, H., Hirsch, O., Albrecht, B., & Chavanon, M.-L. (2019). Attention-Deficit/Hyperactivity Disorder (ADHD) and Emotion Regulation Over the Life Span. ''Current Psychiatry Reports'', ''21''(3), 17–17. <nowiki>https://doi.org/10.1007/s11920-019-1003-6</nowiki> Cole, P. M. (2014). Moving ahead in the study of the development of emotion regulation. ''International Journal of Behavioral Development'', ''38''(2), 203–207. <nowiki>https://doi.org/10.1177/0165025414522170</nowiki> Graziano, P. A., & Garcia, A. (2016). Attention-deficit hyperactivity disorder and children’s emotion dysregulation: A meta-analysis. ''Clinical Psychology Review'', ''46'', 106–123. <nowiki>https://doi.org/10.1016/j.cpr.2016.04.011</nowiki> Gross, J. J. (1998). The Emerging Field of Emotion Regulation: An Integrative Review. ''Review of General Psychology'', ''2''(3), 271-299. https://doi.org/10.1037/1089-2680.2.3.271 Langner, R., Leiberg, S., Hoffstaedter, F., & Eickhoff, S. B. (2018). Towards a human self-regulation system: Common and distinct neural signatures of emotional and behavioural control. ''Neuroscience and Biobehavioral Reviews'', ''90'', 400–410. <nowiki>https://doi.org/10.1016/j.neubiorev.2018.04.022</nowiki> McLaughlin, K. A., Hatzenbuehler, M. L., Mennin, D. S., & Nolen-Hoeksema, S. (2011). Emotion dysregulation and adolescent psychopathology: A prospective study. ''Behaviour Research and Therapy'', ''49''(9), 544–554. <nowiki>https://doi.org/10.1016/j.brat.2011.06.003</nowiki> McRae, K., Misra, S., Prasad, A. K., Pereira, S. C., & Gross, J. J. (2012). Bottom-up and top-down emotion generation: implications for emotion regulation. ''Social Cognitive and Affective Neuroscience'', ''7''(3), 253–262. <nowiki>https://doi.org/10.1093/scan/nsq103</nowiki> Mitchell, J. T., McIntyre, E. M., English, J. S., Dennis, M. F., Beckham, J. C., & Kollins, S. H. (2017). A Pilot Trial of Mindfulness Meditation Training for ADHD in Adulthood: Impact on Core Symptoms, Executive Functioning, and Emotion Dysregulation. ''Journal of attention disorders'', ''21''(13), 1105–1120. <nowiki>https://doi.org/10.1177/1087054713513328</nowiki> Morris, A. S., Silk, J. S., Steinberg, L., Myers, S. S., & Robinson, L. R. (2007). The Role of the Family Context in the Development of Emotion Regulation. ''Social Development (Oxford, England)'', ''16''(2), 361–388. <nowiki>https://doi.org/10.1111/j.1467-9507.2007.00389.x</nowiki> Ohlmeier, M. D., Peters, K., Wildt, B. T. T., Zedler, M., Ziegenbein, M., Wiese, B., Emrich, H. M., & Schneider, U. (2008). Comorbidity of alcohol and substance dependence with attention-deficit/hyperactivity disorder (ADHD). ''Alcohol and Alcoholism (Oxford)'', ''43''(3), 300–304. <nowiki>https://doi.org/10.1093/alcalc/agn014</nowiki> Retz, W., Stieglitz, R.-D., Corbisiero, S., Retz-Junginger, P., & Rösler, M. (2012). Emotional dysregulation in adult ADHD: what is the empirical evidence? ''Expert Review of Neurotherapeutics'', ''12''(10), 1241–1251. <nowiki>https://doi.org/10.1586/ern.12.109</nowiki> Roberton, T., Daffern, M., & Bucks, R. S. (2012). Emotion regulation and aggression. ''Aggression and Violent Behavior'', ''17''(1), 72–82. <nowiki>https://doi.org/10.1016/j.avb.2011.09.006</nowiki> Shaw, P., Stringaris, A., Nigg, J., & Leibenluft, E. (2014). Emotion dysregulation in attention deficit hyperactivity disorder. ''The American journal of psychiatry'', ''171''(3), 276–293. https://doi.org/10.1176/appi.ajp.2013.13070966 Simon, V., Czobor, P., Bálint, S., Mészáros, A., & Bitter, I. (2009). Prevalence and correlates of adult attention-deficit hyperactivity disorder: meta-analysis. ''The British journal of psychiatry : the journal of mental science'', ''194''(3), 204–211. https://doi.org/10.1192/bjp.bp.107.048827 Stellern, J., Xiao, K. B., Grennell, E., Sanches, M., Gowin, J. L., & Sloan, M. E. (2023). Emotion regulation in substance use disorders: a systematic review and meta‐analysis. ''Addiction (Abingdon, England)'', ''118''(1), 30–47. <nowiki>https://doi.org/10.1111/add.16001</nowiki> Surman, C. B. H., & Walsh, D. M. (2022). Do Treatments for Adult ADHD Improve Emotional Behavior? A Systematic Review and Analysis. ''Journal of Attention Disorders'', ''26''(14), 1822–1832. <nowiki>https://doi.org/10.1177/10870547221110926</nowiki> Thompson, R. A. (1991). Emotional Regulation and Emotional Development. ''Educational Psychology Review'', ''3''(4), 269–307. <nowiki>https://doi.org/10.1007/BF01319934</nowiki> Wadlinger, H. A., & Isaacowitz, D. M. (2011). Fixing our focus: training attention to regulate emotion. ''Personality and social psychology review: an official journal of the Society for Personality and Social Psychology'', ''15''(1), 75–102. https://doi.org/10.1177/1088868310365565 }} ==External links== * [https://news.mit.edu/2024/new-strategy-cope-emotional-stress-0708 A new strategy to cope with emotional stress] (MIT News) * [https://www.apa.org/monitor/2024/04/adhd-managing-emotion-dysregulation Emotional dysregulation is part of ADHD. See how psychologists are helping] (American Psychological Association) * [https://www.psychologytoday.com/au/basics/emotion-regulation Emotion Regulation] (Psychology Today) [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/ADHD]] [[Category:Motivation and emotion/Book/Emotion regulation]] 9dx5ee47rkya0vt3wbtnn0j3m1g32gt Universal Bibliography/Music 0 321521 2812963 2812114 2026-06-05T21:52:52Z Kirby - Electrotechnics 3074947 changed to 'category:Music resources' instead of category:Music 2812963 wikitext text/x-wiki {{Bibliography}} See [[s:Category:Music]] and [[w:Category:Music books]] This part of the [[Universal Bibliography]] is a bibliography of music. Bibliography *[[w:Bibliography of Music Literature|Bibliography of Music Literature]] *Green (ed). Foundations in Music Bibliography. 1993. [https://books.google.co.uk/books?id=rADdpZN9UhAC&pg=PR3#v=onepage&q&f=false] *Krummel. The Literature of Music Bibliography: An Account of the Writings on the History of Music Printing & Publishing. 2nd Ed: 1992. [https://books.google.com/books?id=3AZsiITI-IEC] *Bibliography of Music Bibliographies. 1967. [https://books.google.co.uk/books?id=d6YJAQAAMAAJ] *Bayne. A Guide to Library Research in Music. 2008. [https://books.google.co.uk/books?id=ExGbDqu9gPAC&pg=PP1#v=onepage&q&f=false] *A Selected Bibliography of Music Librarianship [https://books.google.co.uk/books?id=X5AeOl4O-osC] *Bradley. American Music Librarianship: A Research and Information Guide. [https://books.google.co.uk/books?id=VabcAAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Music Reference and Research Materials. 3rd Ed: 1974: [https://books.google.com/books?id=5Y1IAAAAMAAJ] *Agruss. Guide to Reference Books on Music. 1948. [https://books.google.co.uk/books?id=wX06AAAAIAAJ] *Haggerty. A Guide to Popular Music Reference Books: An Annotated Bibliography. 1995. [https://books.google.co.uk/books?id=2OnEEAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Coover. A Bibliography of Music Dictionaries. 1952: [https://books.google.co.uk/books?id=NH06AAAAIAAJ]. Music Lexicography. 2nd Ed: 1958. Including a Study of Lacunae in Music Lexicography and a Bibliography of Music Dictionaries. 3rd Ed: 1971: [https://books.google.co.uk/books?id=jKMJAQAAMAAJ]. *A Bibliography of Books on Music and Collections of Music. 1948. [https://books.google.co.uk/books?id=vfvpnwWWlZwC] *Deakin. Musical Bibliography: A Catalogue of the Musical Works. 1892. [https://books.google.co.uk/books?id=-UgQAAAAYAAJ&pg=PP7#v=onepage&q&f=false] (England 15th to 18th century) *Matthew. The Literature of Music. 1896. [https://books.google.co.uk/books?id=fTQ6AAAAMAAJ&pg=PR3#v=onepage&q&f=false]. Reviews: [https://books.google.co.uk/books?id=bjdVAAAAYAAJ&pg=RA1-PA56#v=onepage&q&f=false] [https://books.google.co.uk/books?id=dzcZAAAAYAAJ&pg=PA22#v=onepage&q&f=false] [https://books.google.co.uk/books?id=R0gcAQAAMAAJ&pg=PA470#v=onepage&q&f=false] [https://books.google.co.uk/books?id=qK5OAQAAMAAJ&pg=PA55#v=onepage&q&f=false] [https://books.google.co.uk/books?id=1chZAAAAYAAJ&pg=PA155#v=onepage&q&f=false] [https://books.google.co.uk/books?id=ezszAQAAMAAJ] [https://books.google.co.uk/books?id=5h61TMyTmOMC] [https://books.google.co.uk/books?id=8k8wAQAAIAAJ] [https://books.google.co.uk/books?id=i8W8LKTuc0AC]. Author: [https://books.google.co.uk/books?id=awIQAAAAYAAJ&pg=PA275#v=onepage&q&f=false]. *Hoek. Analyses of Nineteenth- and Twentieth-Century Music, 1940-2000. 2007. [https://books.google.co.uk/books?id=CRG4AQAAQBAJ&pg=PP1#v=onepage&q&f=false] *RILM Abstracts of Music Literature. [https://books.google.co.uk/books?id=HxjjAAAAMAAJ] *Elliker. The Periodical Literature of Music: Trends from 1952 to 1987. 1996. [https://books.google.co.uk/books?id=T5ifAAAAMAAJ] *Forkel. Allgemeine Litteratur der Musik. 1792. [https://books.google.co.uk/books?id=VTRDAAAAcAAJ&pg=PR1#v=onepage&q&f=false] Review: [https://books.google.co.uk/books?id=3N8sAAAAYAAJ&pg=PA33#v=onepage&q&f=false] History and bibliography *Matthew. A Handbook of Musical History and Bibliography. 1898. [https://books.google.co.uk/books?id=V1g5AAAAIAAJ&pg=PR3#v=onepage&q&f=false] Review: [https://books.google.co.uk/books?id=P1lDAQAAMAAJ&pg=PA229#v=onepage&q&f=false] *Boyden. The History and Literature of Music: 1750 to the Present. 1959. [https://books.google.co.uk/books?id=XcAZAQAAIAAJ] *Brown. An Introduction to the History and Literature of Music in Western Culture. 2nd Ed: 2011. [https://books.google.co.uk/books?id=aKpGAAAACAAJ] Chronology, annuals, year books, years *Eisler. World Chronology of Music History. *Lowe. A Chronological Cyclopædia of Musicians and Musical Events. 1896. *Tokyo Ongaku Gakko. Kinsei Hogaku Nempyo. [Chronology of Japanese Music in Recent Ages.] Rokugatsu-Kan. Volume 1. 1912. Volume 2. 1914. Volume 3. 1927. [https://books.google.co.uk/books?id=drMQAQAAMAAJ] *Cossar. This Day in Music. 2005. 2010. *Glassman. The Year in Music. Columbia House. *[[w:Herman Klein|Hermann Klein]]. Musical Notes. Annual Critical Record of Important Musical Events. *[[w:Joseph Bennett (critic)|Bennett]]. The Musical Year. *Hinrichsen's Musical Year Book *The Musical Year Book of the United States **The Boston Musical Year Book *Billboard. Overview. 1982: [https://books.google.co.uk/books?id=YyQEAAAAMBAJ&pg=PT53#v=onepage&q&f=false]. *Billboard. The Year in Music. 1994: [https://books.google.co.uk/books?id=ZAgEAAAAMBAJ&pg=PA62#v=onepage&q&f=false]. 2003: [https://books.google.co.uk/books?id=bA8EAAAAMBAJ&pg=PA47#v=onepage&q&f=false]. **The Year in Music and Video. 1985: [https://books.google.co.uk/books?id=uyQEAAAAMBAJ&pg=PT50#v=onepage&q&f=false]. 1986: [https://books.google.co.uk/books?id=tiQEAAAAMBAJ&pg=PA49#v=onepage&q&f=false]. *Jackson. 1965: The Most Revolutionary Year in Music. *Porter. A Musical Season: 1972-1973. **Music of Three Seasons: 1974-1977 **Music of Three More Seasons 1977-1980 **Musical Events: A Chronicle, 1980-1983. *[https://news.1242.com/article/tag/大人のmusic-calendar 【大人のMusic Calendar】]. Nippon Broadcasting System. [Articles from 2016 are included in [https://news.1242.com/article/author/toritani/page/42 NEWS ONLINE 編集部の記事一覧].] *[http://music-calendar.jp Music Calendar] History *"Recorded Sound: The First Century: 1877-1977". Billboard. 21 May 1977. pp [https://books.google.co.uk/books?id=XCMEAAAAMBAJ&pg=PT39#v=onepage&q&f=false RS-1] to RS-117. Encyclopedias See also [[w:List of encyclopedias by branch of knowledge/Music]] and [[w:Bibliography of encyclopedias#Music and dance]] *Encyclopedia of Music in the 20th Century [https://books.google.co.uk/books?id=m8W2AgAAQBAJ&pg=PP1#v=onepage&q&f=false] *Moore. Complete Encyclopædia of Music. 1852. [https://books.google.co.uk/books?id=-QBFAQAAMAAJ&pg=PA1#v=onepage&q&f=false] Dictionaries *Apel. "Dictionaries of music". Harvard Dictionary of Music. 1969. pp [https://books.google.co.uk/books?id=TMdf1SioFk4C&pg=PA232#v=onepage&q&f=false 232] to 234. United Kingdom: *Billboard. Spotlight on the United Kingdom. 1978: [https://books.google.co.uk/books?id=TSQEAAAAMBAJ&pg=PT78#v=onepage&q&f=false]. 1979: [https://books.google.co.uk/books?id=MCUEAAAAMBAJ&pg=PT100#v=onepage&q&f=false]. Australia: *Billboard. Spotlight on Australia/New Zealand. 1982: [https://books.google.co.uk/books?id=GCQEAAAAMBAJ&pg=PT54#v=onepage&q&f=false]. 1985: [https://books.google.co.uk/books?id=hiQEAAAAMBAJ&pg=PT29#v=onepage&q&f=false]. 1986: [https://books.google.co.uk/books?id=UCQEAAAAMBAJ&pg=PA60#v=onepage&q&f=false]. **Live Talent of Australia: [https://books.google.co.uk/books?id=YyQEAAAAMBAJ&pg=PT94#v=onepage&q&f=false] New Zealand: *Harvey. A Bibliography of Writings about New Zealand Music Published to the End of 1983. 1985. [https://books.google.co.uk/books?id=B1ROA_sP-xsC&pg=PP1#v=onepage&q&f=false] *The Complete New Zealand Music Charts, 1966-2006: Singles, Albums, DVDs, Compilations. 2007. [https://books.google.co.uk/books?id=wyU5AQAAIAAJ] *Billboard. New Zealand. 2002: [https://books.google.co.uk/books?id=Rg0EAAAAMBAJ&pg=PA37#v=onepage&q&f=false] Canada: *Billboard. Spotlight on Canada. 1981: [https://books.google.co.uk/books?id=DSQEAAAAMBAJ&pg=PT50#v=onepage&q&f=false]. Scandanavia: *Billboard. Spotlight on Scandanavia. 1981: [https://books.google.co.uk/books?id=GCUEAAAAMBAJ&pg=PT86#v=onepage&q&f=false]. France: *Billboard. Spotlight on France. 1971: [https://books.google.co.uk/books?id=-wgEAAAAMBAJ&pg=PA35#v=onepage&q&f=false]. 1972: [https://books.google.co.uk/books?id=REUEAAAAMBAJ&pg=PA35#v=onepage&q&f=false]. 1982: [https://books.google.co.uk/books?id=AyQEAAAAMBAJ&pg=PT66#v=onepage&q&f=false]. 1986: [https://books.google.co.uk/books?id=ICUEAAAAMBAJ&pg=PA41#v=onepage&q&f=false] Germany: *Billboard. Spotlight on West Germany. 1971: [https://books.google.co.uk/books?id=zQgEAAAAMBAJ&pg=PA45#v=onepage&q&f=false]. 1985: [https://books.google.co.uk/books?id=-iMEAAAAMBAJ&pg=PT12#v=onepage&q&f=false]. **Spotlight on West Germany, Austria and Switzerland. 1986: [https://books.google.co.uk/books?id=CSUEAAAAMBAJ&pg=RA1-PA35#v=onepage&q&f=false] Italy: *Billboard. Spotlight on Italy. 1981: [https://books.google.co.uk/books?id=8iQEAAAAMBAJ&pg=PT3#v=onepage&q&f=false]. 1985: [https://books.google.co.uk/books?id=3yQEAAAAMBAJ&pg=PT36#v=onepage&q&f=false]. 1986: [https://books.google.co.uk/books?id=2SQEAAAAMBAJ&pg=PA38-IA1#v=onepage&q&f=false]. 1994: [https://books.google.co.uk/books?id=XQgEAAAAMBAJ&pg=PA67#v=onepage&q&f=false]. Spain: *Billboard. Spotlight on Spain. 1971: [https://books.google.co.uk/books?id=5Q8EAAAAMBAJ&pg=PA49#v=onepage&q&f=false] Philipines: *[https://billboardphilippines.com/culture/scenes/lost-history-how-filipino-music-was-documented-in-the-40s-to-2010s/ Lost History: How Filipino Music Was Documented In The ’40s To 2010s]. Billboard Philippines. 18 January 2024. *[[w:en:Billboard Philippines|Billboard Philippines]] Brazil: *Billboard. Spotlight on Brazil. 1996: [https://books.google.co.uk/books?id=NA0EAAAAMBAJ&pg=PA51#v=onepage&q&f=false]. United States *Krummel. Bibliographical Handbook of American Music. 1987. [https://books.google.co.uk/books?id=G4wcnkvFZl4C&pg=PP1#v=onepage&q&f=false] *Krummel. Resources of American Music History: A Directory of Source Materials from Colonial Times to World War II. 1981. [https://books.google.co.uk/books?id=bJcYAAAAIAAJ] Soviet *Aschmann. Current Soviet Music Bibliography. 1976. [https://books.google.co.uk/books?id=2i7jAAAAMAAJ] Decline of pop music: *[https://www.smithsonianmag.com/smart-news/science-proves-pop-music-has-actually-gotten-worse-8173368/ Science Proves: Pop Music Has Actually Gotten Worse]. [[w:Smithsonian (magazine)|Smithsonian]]. 27 July 2012. *[https://faroutmagazine.co.uk/new-study-discovers-pop-music-has-suffered-significant-decline-in-one-area/ New study discovers pop music has suffered “significant decline” in one area]. [[w:Far Out (website)|Far Out]]. 5 July 2024. *[https://www.globalnews.ca/news/9001083/why-older-music-more-popular-than-new-music/amp/ There is something very, very wrong with today’s music. It just may not be very good.] [[w:Global News|Global News]]. 24 July 2022. *[https://www.bbc.co.uk/music/articles/fb84bf19-29c9-4ed3-b6b6-953e8a083334 Has pop music lost its fun?]. BBC. 12 January 2018. *[https://www.spectator.co.uk/article/its-official-modern-music-is-bad/ It’s official: modern music is bad]. The Spectator. 13 February 2024. Homogeneity of pop music: *[https://www.theguardian.com/music/2012/jul/27/pop-music-sounds-same-survey-reveals Pop music these days: it all sounds the same, survey reveals]. The Guardian. 27 July 2012. *[https://www.nbcnews.com/id/wbna48356108 Pop Music All Sounds the Same Nowadays]. NBC News. 27 July 2012. *[https://www.independent.co.uk/voices/comment/why-does-today-s-pop-music-sound-the-same-because-the-same-people-make-it-8368714.html Why does today's pop music sound the same? Because the same people make it]. The Independent. 29 November 2012. *[https://www.reuters.com/article/lifestyle/science/pop-music-too-loud-and-all-sounds-the-same-official-idUSBRE86P0R9/ Pop music too loud and all sounds the same: official]. Reuters. 26 July 2012. *[https://theconversation.com/from-art-form-to-asset-our-study-found-popular-songs-are-becoming-more-generic-266097 From art form to asset: our study found popular songs are becoming more generic]. The Conversation. 3 October 2025. Conferences: *International Music Industry Conference. 1971: [https://books.google.co.uk/books?id=tggEAAAAMBAJ&pg=PA29#v=onepage&q&f=false] Laserdisc/Karaoke/CES *Billboard. Karaoke. 1992: [https://books.google.co.uk/books?id=jg8EAAAAMBAJ&pg=PA41-IA1#v=onepage&q&f=false] **CES and Karaoke. 1994. [https://books.google.co.uk/books?id=UggEAAAAMBAJ&pg=PA77#v=onepage&q&f=false] **Laserdisc. 1995. [https://books.google.co.uk/books?id=7AsEAAAAMBAJ&pg=PA67#v=onepage&q&f=false] **Laserdisc/Karaoke. 1996: [https://books.google.co.uk/books?id=iQ8EAAAAMBAJ&pg=PA59#v=onepage&q&f=false] Classical music *Billboard spotlights: 1995 [https://books.google.co.uk/books?id=1g0EAAAAMBAJ&pg=PA39#v=onepage&q&f=false] (9 September 1995) **"Classical Music Recording Market". Billboard. 12 April 1980. pp C-1 to C-12 and p 32. (A Billboard Spotlight). **"Classical Music: Discovering New Dimensions". Billboard. 10 September 1983. pp C-1 to C-18. (A Billboard Spotlight). *"Classical" section, and "Best Selling Classical LPs" chart, in Billboard Jazz *[[w:en:All About Jazz|All About Jazz]] Oldies *"Oldies stations find their place in radio market". Star-News. 13 March 1988. pp 1D & [https://books.google.co.uk/books?id=2OoyAAAAIBAJ&pg=PA16#v=onepage&q&f=false 6D]: "Oldies". *Billboard. 15 April 1972. [https://books.google.co.uk/books?id=a0UEAAAAMBAJ&pg=PT7#v=onepage&q&f=false p 47]. *Billboard. 17 April 1961, [https://books.google.co.uk/books?id=JiIEAAAAMBAJ&pg=PA1#v=onepage&q&f=false p 1]. *Billboard. 4 January 1960, [https://books.google.co.uk/books?id=Ch8EAAAAMBAJ&pg=PA1#v=onepage&q&f=false p 1] Nostalgia See also [[Universal Bibliography/Nostalgia]] *"A Perspective on the Future of Nostalgia". Billboard. 4 May 1974. pp [https://books.google.co.uk/books?id=cgkEAAAAMBAJ&pg=PA37#v=onepage&q&f=false N-1] to N-54 and two more pages. *Carr. Nostalgia, Song and the Quest for Home: Production, Text, Reception. 2025. [https://books.google.co.uk/books?id=xz1jEQAAQBAJ&pg=PP1#v=onepage&q&f=false] Charts *Carroll, " Did Billboard, Cash Box, and Record World Charts Tell the Same Story? Perception and Reality, 1960-1979"(2022) 9 Rock Music Studies [https://www.tandfonline.com/doi/full/10.1080/19401159.2022.2054107 199] Magazines See also [[w:Category:Music magazines]] *Billboard. Google: [https://books.google.co.uk/books/serial/ISSN:00062510?rview=1&lr=&sa=N&start=2770 1942] onwards ==Japanese and Japan== *The Ashgate Research Companion to Japanese Music. 2017. [https://books.google.co.uk/books?id=W2JTgQGc99EC&pg=PP1#v=onepage&q&f=false] [https://books.google.co.uk/books?id=4tINDgAAQBAJ&pg=PA2#v=onepage&q&f=false] *Billboard. Spotlight on Japan. 1970: 19 December 1970 [https://books.google.co.uk/books?id=mSkEAAAAMBAJ&pg=PA37#v=onepage&q&f=false]. 1971: 11 December 1971 [https://books.google.co.uk/books?id=Fg8EAAAAMBAJ&pg=PA39#v=onepage&q&f=false]. 1973: 17 February 1973 [https://books.google.co.uk/books?id=QEUEAAAAMBAJ&pg=PT25#v=onepage&q&f=false]. 1977: 30 April 1977 [https://books.google.co.uk/books?id=USMEAAAAMBAJ&pg=PT46#v=onepage&q&f=false]. 1979: [https://books.google.co.uk/books?id=_iQEAAAAMBAJ&pg=PT48#v=onepage&q&f=false]. 1982:[https://books.google.co.uk/books?id=byQEAAAAMBAJ&pg=PT38#v=onepage&q&f=false]. 1985: [https://books.google.co.uk/books?id=1CQEAAAAMBAJ&pg=PT65#v=onepage&q&f=false]. 1986:[https://books.google.co.uk/books?id=-CMEAAAAMBAJ&pg=RA1-PA79#v=onepage&q&f=false]. 1993: 12 June 1993 [https://books.google.co.uk/books?id=9A8EAAAAMBAJ&pg=PA57#v=onepage&q&f=false]. 1995: 5 August 1995 [https://books.google.co.uk/books?id=xwsEAAAAMBAJ&pg=PA52-IA1#v=onepage&q&f=false]. 1996: 31 August 1996 [https://books.google.co.uk/books?id=vwcEAAAAMBAJ&pg=PA66#v=onepage&q&f=false]. 1997: 30 August 1997 [https://books.google.co.uk/books?id=_gkEAAAAMBAJ&pg=PA61#v=onepage&q&f=false]. 1998: 26 September 1998 [https://books.google.co.uk/books?id=GgoEAAAAMBAJ&pg=PA117#v=onepage&q&f=false]. 2000: 9 September 2000 [https://books.google.co.uk/books?id=aREEAAAAMBAJ&pg=PA65#v=onepage&q&f=false]. 2002: 7 September 2002 [https://books.google.co.uk/books?id=-QwEAAAAMBAJ&pg=PA53#v=onepage&q&f=false]. 2003: 5 July 2003 [https://books.google.co.uk/books?id=3w0EAAAAMBAJ&pg=PA45#v=onepage&q&f=false]. **"Japan in 1974: Business Bristles While Shortages Are Met". Billboard. 23 February 1974. pp J-1 to J-30. (A Billboard Spotlight). **"Made in Japan: A Dynamic Music Industry". Billboard. 1 March 1975. pp J-1 to J-23. (A Billboard Spotlight). **"Japan '76". Billboard. 17 April 1976. pp 36 to 59. (A Billboard Spotlight). **"Japanese Music: The Challenge of Recession". Billboard. 27 May 1978. pp J-1 to J-31. (A Billboard Spotlight). **"Music in Japan: Industry Views 1981 With Quiet Optimism". Billboard. 30 May 1981. pp J-1 to J-18. **"Japan: Where Technology Greets Tradition". (An International Market Profile). Billboard. 21 May 1983. pp J-1 to J-13. Follows p 34. **"Billboard Spotlight on Japan: VCRs and CDs Will Be Pacemakers". Billboard. 26 May 1984. pp J-1 to J-11. Follows p 38. **"Spotlight on Japan". Billboard. 6 June 1987. pp J-1 to J-12. **"Japan '88". Billboard. 9 July 1988. pp J-1 to J-11. (A Billboard International Spotlight). **"Japan". ("Japan '89"/"Spotlight on Japan"). Billboard. 3 June 1989. pp J-1 to J-20. (International Spotlight). **"Japan". ("International Spotlight"/"A Billboard Spotlight"). Billboard. 25 May 1991. pp J-1 to J-26. Follows p 50. Called "Japan '91" on front page. *[[w:The Best Ten|The Best Ten]] (ザ・ベストテン). [Television programme]. [https://www.tbs.co.jp/tbs-ch/special/the_bestten/ Episodes]. *[[w:ja:Music Station|Music Station]]. [Television programme]. Episodes: [https://www.tv-asahi.co.jp/music/contents/m_lineup/0003/index.html episode 1] etc. *Wade. Music in Japan: Experiencing Music, Expressing Culture. 2005. [https://books.google.co.uk/books?id=XXYIAQAAMAAJ] *Malm. Japanese Music & Musical Instruments. 1959. [https://books.google.com/books?id=QkTaAAAAMAAJ] *[[w:Francis Taylor Piggott|Pigott]]. The Music and Musical Instruments of Japan. 1893 [https://books.google.co.uk/books?id=ttKTUwmjzMwC&pg=PR3#v=onepage&q&f=false]. 1909. [https://books.google.co.uk/books?id=MAM5AAAAIAAJ] Bibliography *Tsuge. Japanese Music: An Annotated Bibliography. 1986. [https://books.google.com/books?id=YCsKAQAAMAAJ] *[[w:ja:三井徹|Tōru Mitsui]]. Popyurā Ongaku Kankei Tosho Mokuroku: Ryūkōka, Jazu, Rokku, J-poppu no Hyakunen. (Japanese: ポピュラー音楽関係図書目録: 流行歌、ジャズ、ロック、Jポップの百年). Nichigai Associates. 2009. [https://books.google.co.uk/books?id=dSAxAQAAIAAJ]. Catalogues: [https://search.worldcat.org/title/406243182] [https://cir.nii.ac.jp/crid/1970586434933272116] *[https://ndlsearch.ndl.go.jp/rnavi/avmaterials/post_572 音楽に関する文献を探すには（主題書誌）]. NDL. Dictionaries *[[w:ja:下中弥三郎|Shimonaka Yasaburo]] (ed). Ongaku Jiten. Heibonsha. Review: (1959) 18 Journal of Asian Studies 295 [https://www.cambridge.org/core/journals/journal-of-asian-studies/article/abs/ongaku-jiten-dictionary-of-music-ed-shimonaka-yasaburo-tokyo-heibonsha-195557-12-volumes-900-yen-per-volume/F3067B1CE61B5B2C647091E69CE8C8DD] [https://read.dukeupress.edu/journal-of-asian-studies/article-abstract/18/2/295/322980/Ongaku-jiten-Dictionary-of-Music?redirectedFrom=fulltext] History *Eta Harich-Schneider. A History of Japanese Music. 1973. [https://books.google.com/books?id=3AraAAAAMAAJ] *Koh-ichi Hattori. 123 Years of Japanese Music: The Culture of Japan Through a Look at Its Music. 2004. [https://books.google.com/books?id=znzsAAAAMAAJ] **Koh-ichi Hattori. 36,000 Days of Japanese Music: The Culture of Japan Through A Look At Its Music. Pacific Vision. Pierce, Southfield, Michigan. 1996. ISBN 0965364208. *Shinpan Nihon Ryūkōkashi. (Japanese: 新版日本流行歌史). [[w:ja:社会思想社|Shakaishisosha]]. 1994. Review: [https://books.google.co.uk/books?id=XQdIAAAAMAAJ]. Catalogue: [https://ndlsearch.ndl.go.jp/en/books/R100000002-I000002420287] **新版日本流行歌史: 1960-1994. [https://books.google.com/books?id=_b4pAQAAIAAJ] [https://books.google.co.uk/books?id=nb4pAQAAIAAJ]. **新版日本流行歌史: 1938-1959 **1867-1937 *Mehl. Music and the Making of Modern Japan: Joining the Global Concert. 2024. [https://books.google.co.uk/books?id=P3QMEQAAQBAJ&pg=PA2#v=onepage&q&f=false] Modern, contemporary, today *Johnson. Handbook of Japanese Music in the Modern Era. 2024. [https://books.google.co.uk/books?id=KNP7EAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Matsue. Focus: Music in Contemporary Japan. 2016. [https://books.google.co.uk/books?id=AQgtCgAAQBAJ&pg=PP1#v=onepage&q&f=false] *Music of Japan Today. [https://books.google.co.uk/books?id=YZQYEAAAQBAJ&pg=PP1#v=onepage&q&f=false] Popular music *Mitsui (ed). Made in Japan: Studies in Popular Music. 2014. [https://books.google.co.uk/books?id=YWQKBAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Stevens. Japanese Popular Music: Culture, Authenticity and Power. 2008. [https://books.google.co.uk/books?id=OHMkdcL9DAMC&pg=PP1#v=onepage&q&f=false] *Mitsui. Popular Music in Japan: Transformation Inspired by the West. 2020. [https://books.google.co.uk/books?id=FpbqDwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Nagahara. Tokyo Boogie-Woogie: Japan’s Pop Era and Its Discontents. 2017. [https://books.google.co.uk/books?id=iTxYDgAAQBAJ&pg=PP1#v=onepage&q&f=false] *Patterson. Music and Words: Producing Popular Songs in Modern Japan, 1887–1952. 2019. [https://books.google.co.uk/books?id=P0FvDwAAQBAJ&pg=PP1#v=onepage&q&f=false] *James Stanlaw. "Using English identity markers in Japanese Popular Music". English in East and South Asia. Chapter 14. [https://books.google.co.uk/books?id=88A1EAAAQBAJ&pg=PT109#v=onepage&q&f=false] *"Japanese Popular Music in Singapore". Asian Music. vol 34. No 1: Fall/Winter 2002/2003. p 1. [https://books.google.co.uk/books?id=_D4JAQAAMAAJ] *Steve McClure. Nipponpop. Tuttle Publishing. 1998. ISBN 9780804821070. ISBN 0804821070. [Sometimes called "Nippon Pop"]. Catalogue: [https://search.worldcat.org/title/Nipponpop/oclc/247384040] Review: (1998) [https://books.google.co.uk/books?id=f9egmeZ8YywC 245] The Publishers Weekly 2 From folk to J-pop *[[w:ja:富澤一誠|Issei Tomizawa]]. Ano subarashii kyoku o mō ichido: fōku kara J-poppu made. (Japanese: あの素晴しい曲をもう一度: フォークからJポップまで). [[w:Shinchosha|Shinchosha]]. 2010. [https://books.google.com/books?id=ju9MAQAAIAAJ]. Catalogue: [https://search.worldcat.org/title/501749494]. Commentary on book: [https://www.ytv.co.jp/michiura/time/2010/01/j2010110.html]. Review of the CD: [https://www.cdjournal.com/i/disc/great-agefree-music-forever-and-great-music-are-o/4109110788]. J-pop *Bourdaghs. Sayonara Amerika, Sayonara Nippon: A Geopolitical Prehistory of J-pop. 2012. [https://books.google.co.uk/books?id=K_y88JwibrMC&pg=PP1#v=onepage&q&f=false] *"The Rise of J-Pop in Asia and Its Impact" (2004) Japan Spotlight. vol 23. p 24. [https://books.google.co.uk/books?id=i7C0AAAAIAAJ] *Terence Lancashire. "J-pop's elusive J: Is Japanese popular music Japanese?" (2008) Perfect Beat. vol 9. No 1. p 38. [https://books.google.co.uk/books?id=5No4AQAAIAAJ] *Tetsu Misaki. J-poppu no Nihongo: kashiron. (Japanese: Jポップの日本語: 歌詞論). [[w:ja:彩流社|彩流社 (Sairyusha)]]. 2002. [https://books.google.com/books?id=dsMpAQAAIAAJ] [https://search.worldcat.org/ja/title/J-:/oclc/52005194] *[[w:ja:烏賀陽弘道|Hiromichi Ugaya]]. Jpoppu Towa Nanika: Kyodaikasuru Ongaku Sangyō. (Japanese: Jポップとは何か: 巨大化する音楽産業). 2005. [https://books.google.co.uk/books?id=TLlOAAAAMAAJ] catalogue [https://search.worldcat.org/ja/title/J-:/oclc/676652594] [https://ci.nii.ac.jp/ncid/BA71618018] Japanese rock *Takarajima Special Edition: Encyclopedia of Japanese Rock 1955-1990. Nihon rokku daihyakka: Rokabirī kara bando būmu made. (Japanese: 日本ロック大百科 [年表編] ロカビリーからバンド・ブームまで 1955〜1990). [[w:ja:JICC出版局|JICC Shuppankyoku]]. 1992. ISBN 9784796602907. ISBN 4796602909. Catalogues: [https://ci.nii.ac.jp/ncid/BN07889172] [https://catalogue.nla.gov.au/catalog/2263400]. *Japanese Rock: Standard: 1967-1985. 日本のロック名曲徹底ガイド: 名曲263決定盤846. CDJournal. 2008. ISBN 9784861710469. ISBN 4861710464. [https://www.cdjournal.com/Company/products/mook.php?mno=20081002]. Catalogue: [https://ci.nii.ac.jp/ncid/BA8932668X?l=en]. *Kojima Satoshi (Japanese: 小島智). 検証・80年代日本のロック. アルファベータブックス. 2024. ISBN 9784865981179. ISBN 4865981179. [https://books.google.com/books?id=0gbl0AEACAAJ]. Review: [https://mainichi.jp/articles/20241026/ddm/015/070/005000c]. Jazz *[[w:ja:スイングジャーナル|Swing Journal]] (1947 to 2010) Commentary: [https://www.allaboutjazz.com/news/swing-journal-long-standing-jazz-magazine-to-be-suspended-in-june/] Japanese fusion: *THE DIG presents 日本のフュージョン. Shinko Music Mook. Released 19 April 2013. Commentary: [https://www.cdjournal.com/news/casiopea/50967]. No II. Released 23 October 2014. Commentary: [https://www.cdjournal.com/news/takanaka-masayoshi/62225] Classical *[[w:ja:ぶらあぼ|Bravo]] (Japanese: ぶらあぼ) ebravo.jp *[[w:ja:音楽芸術 (雑誌)|Ongaku Geijutsu]] (Japanese: 音楽芸術) Magazines For Japanese music magazines, see [[w:ja:日本の音楽雑誌]]. *Music Periodicals in Japan — A Comprehensive List (1988) 35 Fontes Artis Musicae 116 [https://www.jstor.org/stable/23507222] [https://books.google.com/books?id=qHYWAAAAIAAJ] **Kishimoto, "Additional Corrections and Alphabetical Title Index" (1989) 36 Fontes Artis Musicae 38 [https://www.jstor.org/stable/23507313] [https://books.google.co.uk/books?id=7XYWAAAAIAAJ] *Special Bibliography: A Bibliography of Japanese Magazines and Music (1959) 3 Ethnomusicology 76 [https://www.jstor.org/stable/924290] *A Historical Survey of Music Periodicals in Japan: 1881—1920 (1989) 36 Fontes Artis Musicae 44 [https://www.jstor.org/stable/23507314] *[[w:ja:篠原章|Akira Shinohara]]. 日本ロック雑誌クロニクル. [[w:en:Ohta Publishing|Ohta Publishing]]. 2005. [https://books.google.co.uk/books?id=L8opAQAAIAAJ] *[[w:Oricon|Oricon]] (オリコン) **[https://web.archive.org/web/19970412131857/http://www.999.com/Oricon/index.html Oricon Music Site]. Commentary: [https://internet.watch.impress.co.jp/www/article/980309/oms.htm]. *[[w:Billboard Japan|Billboard Japan]] (ビルボード・ジャパン) **Music Labo (ミュージック・ラボ) (1970 to 1994) *Music Research (ミュージック・リサーチ) ["Weekly Music Magazine"]. Catalogue: [https://web.archive.org/web/20260319070908/https://ndlsearch.ndl.go.jp/books/R100000002-I000000039804]. *Rolling Stone Japan *新譜ジャーナル (Shinpu Journal). Catalogue: [https://ndlsearch.ndl.go.jp/books/R100000002-I000000012315]. Began 1968 [https://books.google.co.uk/books?id=L8opAQAAIAAJ], later called シンプジャーナル **シンプジャーナル *Myūjikku mansurī [ミュージック・マンスリー]  [https://ci.nii.ac.jp/ncid/AN00396190] *カセットライフ. (Cassette Life). [[w:ja:シンコーミュージック・エンタテイメント|Shinko Music Entertainment]] *[[w:ja:CDジャーナル|CDJournal]] *[[w:ja:Rockin'on Japan|Rockin'on Japan]]. (ロッキング・オン・ジャパン). (1986 onwards) *[[w:ja:Rooftop|Rooftop]] (1976 onwards) *[[w:ja:FOOL'S MATE|Fool's Mate]] Columns in periodicals *"Japanese Newsnotes". Billboard. (eg 17 April 1961, [https://books.google.co.uk/books?id=JiIEAAAAMBAJ&pg=PA13#v=onepage&q&f=false p 13].) Websites *[[w:ja:ナタリー (ニュースサイト)|Natalie]] (ナタリー) *[[w:ja:BARKS|Barks]] *OKMusic. Commentary: [https://xtech.nikkei.com/it/article/NEWS/20120626/405442/]. Charts For Japanese music charts, see [[w:ja:日本の音楽チャート]] Chart books *Oricon Chart Book (Japanese: オリコンチャート・ブック) **1987 to 1998 Oricon Chart Book. All Albums. [https://books.google.co.uk/books?id=KvEoNwAACAAJ] **Album Chart Book Complete Edition 1970〜2005. Catalogue:[https://www.tosyokan.pref.shizuoka.jp/licsxp-opac/WOpacMsgNewListToTifTilDetailAction.do?tilcod=1000610247212] *澤山博之. ミュージック・ライフ 東京で1番売れていたレコード 1958~1966. Shinko Music Entertainment. 2019. [Charts published in Music Life from 1958 onwards]. Commentary: [https://mikiki.tokyo.jp/articles/-/20952 Mikiki] Number ones *Oricon No.1 Hits 500. Clubhouse (Japanese: クラブハウス). 1994. 1998. **[https://books.google.com/books?id=GlsnNwAACAAJ vol 1 (1968~1985)]. ISBN 9784906496129. **[https://books.google.com/books?id=icInNwAACAAJ vol 2 (1986~1994)]. ISBN 9784906496136. Awards Japan Record Awards *輝く!日本レコード大賞 公式データブック: 放送60回記念: TBS公認. Shinko Music Entertainment. ISBN 9784401647019. [https://books.google.co.uk/books?id=JcDqvwEACAAJ] [https://ci.nii.ac.jp/ncid/BB2773137X] Traditional, Hogaku *Malm. Traditional Japanese Music and Musical Instruments. [https://books.google.co.uk/books?id=Yn3VQbqywCsC&pg=PP1#v=onepage&q&f=false] *Miyuki Yoshikami. Japan's Musical Tradition: Hogaku from Prehistory to the Present. 2020. [https://books.google.co.uk/books?id=X3XTDwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Hughes. Traditional Folk Song in Modern Japan: Sources, Sentiment and Society. 2008. [https://books.google.co.uk/books?id=yfV5DwAAQBAJ&pg=PR1#v=onepage&q&f=false] Koto: *Tokyo Academy of Music. Collection of Japanese Koto Music. 1888. [https://books.google.co.uk/books?id=RncQAAAAYAAJ&pg=PP13#v=onepage&q&f=false][https://babel.hathitrust.org/cgi/pt?id=hvd.32044040839565&seq=1] Exam guides: For the 音楽CD検定 exam on music CDs: *音楽CD検定公式ガイドブック. 2007. [[w:ja:音楽出版社 (企業)|Ongaku Shuppansha Co Ltd]] (音楽出版社). [https://books.google.co.uk/books?id=sbjdeDJMkQcC&pg=PP1#v=onepage&q&f=false vol 1]. [https://books.google.co.uk/books?id=AoFgIowII48C&pg=PP1#v=onepage&q&f=false vol 2]. Commentary: [https://www.cdjournal.com/i/news/-/15303] [https://www.oricon.co.jp/news/46065/full/] [https://allabout.co.jp/gm/gc/57723/] [https://www.oricon.co.jp/news/45388/full/]. Children's music *Elizabeth May. The Influence of the Meiji Period on Japanese Children's Music. University of California Press. 1963. [https://books.google.co.uk/books?id=54cHAQAAMAAJ] **Japanese Children's Music Before and After Contact with the West. University of California at Los Angeles. 1958. (doctoral dissertation). DJs *Masahiro Yasuda, "How Japanese DJs cut across Market Boundaries" (1999) [https://books.google.co.uk/books?id=H5QJAQAAMAAJ 4] Perfect Beat 45 [[Category:Music resources]] tiyogprrjvx42k6lcdovnnsuubgo9wk Motivation and emotion/Book/2026 0 323153 2812838 2812804 2026-06-05T12:19:16Z Jtneill 10242 /* Motivation */ +[[/Charismatic leadership and follower motivation/]] - How does charismatic leadership inspire commitment, effort, and persistence among followers? 2812838 wikitext text/x-wiki {{/Banner}} ==Motivation== # [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour? {{ME-By|User Name}} # [[/Charismatic leadership and follower motivation/]] - How does charismatic leadership inspire commitment, effort, and persistence among followers? {{ME-By|User Name}} # [[/Citizen science motivation/]] - What motivates participation in citizen science projects? {{ME-By|User Name}} # [[/Competence and intrinsic motivation in SDT/]] - How does the need for competence influence intrinsic motivation according to self-determination theory? {{ME-By|User Name}} # [[/Cost-benefit motivation and effort regulation/]] - How is effort dynamically adjusted I based on changing cost-benefit analysis during goal pursuit? {{ME-By|User Name}} # [[/Deliberative vs implemental mindset/]] - What are the motivational and cognitive differences between these mindsets? {{ME-By|User Name}} # [[/Dopamine and reward prediction/]] - How does dopamine affect the anticipation of rewards and subsequent emotional responses? {{ME-By|U3228742}} # [[/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}} # [[/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}} # [[/Fogg behaviour model/]] - How can the FBM be applied to understanding and changing behaviour? {{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}} # [[/Goal striving dynamics/]] - What is the role of pushing and coasting in goal striving? {{ME-By|User Name}} # [[/Hypothalamus and homeostatic motivation/]] - How do hypothalamic circuits regulate hunger, thirst, and other survival-related motivations? {{ME-By|User Name}} # [[/Mental contrasting and motivation/]] - How does mental contrasting affect motivation? {{ME-By|User Name}} # [[/Mesolimbic pathway development and adolescent risk-taking/]] - How does maturation of reward circuits influence teenage sensation-seeking and impulsive behaviours? {{ME-By|User Name}} # [[/Metacognitive monitoring and productivity/]] - How does metacognitive monitoring influence goal attainment and productivity? {{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}} # [[/Perfectionism and procrastination/]] - What is the role of perfectionism in procrastination and what can be done about it? {{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 and leadership/]] - How does power motivation influence leadership styles and effectiveness? {{ME-By|User Name}} # [[/Prevention vs promotion mindset/]] - What are the motivational differences between focusing on safety versus growth? {{ME-By|User Name}} # [[/Relatedness and intrinsic motivation in SDT/]] - How does the need for relatedness influence intrinsic motivation according to self-determination theory? {{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 social media use/]] - How do basic psychological needs explain patterns of social media engagement? {{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}} # [[/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}} # [[/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}} # [[/Value congruence and motivation/]] - How does alignment between personal and situational values influence motivation? {{ME-By|User Name}} # [[/Windfall gain effect/]] - How do unexpected financial gains 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== # [[/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}} # [[/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/]] - How does feedback about physiological states influence emotion? {{ME-By|User Name}} # [[/Breathing exercises and relaxation/]] - How can breathing exercises promote relaxation? {{ME-By|User Name}} # [[/Cognitive hardiness in the workplace/]] - How does cognitive hardiness protect against occupational stress and burnout? {{ME-By|User Name}} # [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? {{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 versus strategy/]] – How do ability and strategy differ in shaping effective emotion regulation? {{ME-By|User Name}} # [[/Emotional expressivity/]] – What is emotional expressivity, why does it matter, and how can it be developed? {{ME-By|User Name}} # [[/Empathy fatigue and emotional exhaustion/]] - How does sustained empathic engagement contribute to emotional exhaustion in caregiving and helping roles? {{ME-By|User Name}} # [[/Enjoyment and learning/]] - How does enjoyment influence learning? {{ME-By|User Name}} # [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? {{ME-By|User Name}} # [[/Introjection and guilt-based motivation/]] - What role does shame and guilt play in introjected forms of behavioural regulation? {{ME-By|User Name}} # [[/Laughing gas (nitrous oxide) and emotion/]] - How does nitrous oxide influence emotion? {{ME-By|User Name}} # [[/Noise and emotion/]] - What are the typical emotional responses to different types of noise? {{ME-By|User Name}} # [[/Opponent process theory and emotion/]] - What role do opposing affective states play in emotional experience? {{ME-By|User Name}} # [[/Positive emotion dysregulation/]] – What is positive emotion dysregulation, what are its consequences, and what can be done about it? {{ME-By|User Name}} # [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters? {{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 trust/]] - How does responsiveness influence the development and maintenance of trust? {{ME-By|User Name}} # [[/Secondary trauma in healthcare workers/]] - What are the emotional consequences of secondary trauma in healthcare settings? {{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}} # [[/Wayfinding and emotion/]] - What are the affective aspects of wayfinding? {{ME-By|User Name}} ==Motivation and emotion== # [[/Reinforcement sensitivity theory/]] – What is reinforcement sensitivity theory and how does it explain motivation and emotion? {{ME-By|User Name}} # [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}} # [[/Warm-glow giving, motivation, and emotion/]] - What are the motivational and emotional aspects of warm-glow giving? {{ME-By|User Name}} # [[/Wisdom, motivation, and emotion/]] - What are the motivational-emotional aspects of wisdom? {{ME-By|User Name}} [[Category:Motivation and emotion/Book]] 7ivq5iu4mqf4h3tjg2t0xylpqytcdo6 User:Harold Foppele 2 324382 2813024 2804858 2026-06-06T04:33:47Z PieWriter 3039865 Replaced content with "{{blocked user}}" 2813024 wikitext text/x-wiki {{blocked user}} 9nsq3566y9qqur57bxjyd46o6xcoegz Just sustainability transitions: a living review 0 326060 2812861 2811777 2026-06-05T15:28:24Z Jeanne Noiraud 1366702 /* Wikidata and the semantic web */ 2812861 wikitext text/x-wiki == Contributors == {| class="wikitable" |+ !Name !Affiliation !ORCID !Contribution |- |Adélie Ranville |IAE de Grenoble, CERAG lab (https://ror.org/0509qp208) |https://orcid.org/0000-0002-3993-6135 |Research design, database search, article screening, knowledge modelling |- |Amélie Pereira | | |Meta-data enrichement |- | | | | |} == Introduction == === Definition of living review === The concept of living systematic reviews is recent (2014), so the definition has been regularly reworked<ref name="Why1">{{Cite Q |Q40040379 }}</ref>. Living systematic reviews complement the older concept of [[literature review]]. Its objective is the same : obtain an accurate overview of the state of scientific knowledge on a subject<ref name="Why1" /><ref name="Why4">{{Cite journal |last=Akl |first=Elie A. |last2=Meerpohl |first2=Joerg J. |last3=Elliott |first3=Julian |last4=Kahale |first4=Lara A. |last5=Schünemann |first5=Holger J. |last6=Agoritsas |first6=Thomas |last7=Hilton |first7=John |last8=Perron |first8=Caroline |last9=Akl |first9=Elie |last10=Hodder |first10=Rebecca |last11=Pestridge |first11=Charlotte |last12=Albrecht |first12=Lauren |last13=Horsley |first13=Tanya |last14=Platt |first14=Joanne |last15=Armstrong |first15=Rebecca |date=2017-11 |title=Living systematic reviews: 4. Living guideline recommendations |url=https://www.wikidata.org/wiki/Q50084143 |journal=Journal of Clinical Epidemiology |language=en |volume=91 |pages=47–53 |doi=10.1016/j.jclinepi.2017.08.009}}</ref><ref name=":6">{{Citation|title=Living Systematic Reviews|url=https://doi.org/10.1007/978-1-0716-1566-9_7|publisher=Springer US|work=Meta-Research: Methods and Protocols|date=2022|access-date=2026-01-16|place=New York, NY|isbn=978-1-0716-1566-9|pages=121–134|doi=10.1007/978-1-0716-1566-9_7|language=en|first=Mark|last=Simmonds|first2=Julian H.|last2=Elliott|first3=Anneliese|last3=Synnot|first4=Tari|last4=Turner|editor-first=Evangelos|editor-last=Evangelou|editor2-first=Areti Angeliki|editor2-last=Veroniki}}</ref>. A traditional review may be obsolete by the time it is published, as new studies have emerged between the submission of the manuscript and its publication<ref name="Why1"/><ref name="Why4" /><ref name=":6" />. Living systematic reviews exists to address this common problem<ref name="Why1" /><ref name="Why4" /><ref name=":6" /><ref name=":2">https://blogs.lse.ac.uk/impactofsocialsciences/2019/05/14/the-death-of-the-literature-review-and-the-rise-of-the-dynamic-knowledge-map/</ref>. It is therefore particularly useful in rapidly evolving fields of research<ref name="Why1" /><ref name=":6" />, such as just transition. [[wikidata:Q33002955|Knowledge graphs]], a structured representation of knowledge in the form of a graph, linked together by relationships that encode explicit meanings between these entities, are very suitable for conducting living systematic reviews<ref name=":2" /><ref name="Fotopoulou">{{Cite journal|first1=Eleni |last1=Fotopoulou|first2=Ioanna|last2=Mandilara|first3=Anastasios|last3=Zafeiropoulos|first4=Chrysi|last4=Laspidou|first5=Giannis |last5=Adamos|first6=Phoebe|last6=Koundouri|first7=Symeon|last7=Papavassiliou|title=SustainGraph: A knowledge graph for tracking the progress and the interlinking among the sustainable development goals’ targets|journal=Frontiers in environmental science, Frontiers|volume=10|date=2022-10-26|issn=2296-665X|doi=10.3389/FENVS.2022.1003599|url=https://www.wikidata.org/wiki/Q117837999}}.</ref>. Advances in AI could render certain older methodological types of living systematic reviews obsoletes<ref>{{Cite journal|last=Krlev|first=Gorgi|last2=Hannigan|first2=Tim|last3=Spicer|first3=André|date=2025-01|title=What Makes a Good Review Article? Empirical Evidence From Management and Organization Research|url=https://journals.aom.org/doi/abs/10.5465/annals.2021.0051|journal=Academy of Management Annals|volume=19|issue=1|pages=376–403|doi=10.5465/annals.2021.0051|issn=1941-6520}}</ref>, as IA are useful to extract, filter and classify datas<ref>{{Cite web|url=https://arxiv.org/abs/2504.20276v1|title=Enhancing Systematic Reviews with Large Language Models: Using GPT-4 and Kimi|last=Kaptur|first=Dandan Chen|last2=Huang|first2=Yue|date=2025-04-28|website=arXiv.org|language=en|doi=10.48550/arXiv.2504.20276|access-date=2026-01-21|last3=Ji|first3=Xuejun Ryan|last4=Guo|first4=Yanhui|last5=Kaptur|first5=Bradley}}</ref><ref>{{Cite web|url=https://arxiv.org/abs/2504.20276v1|title=Enhancing Systematic Reviews with Large Language Models: Using GPT-4 and Kimi|last=Kaptur|first=Dandan Chen|last2=Huang|first2=Yue|date=2025-04-28|website=arXiv.org|language=en|doi=10.48550/arXiv.2504.20276|access-date=2026-01-21|last3=Ji|first3=Xuejun Ryan|last4=Guo|first4=Yanhui|last5=Kaptur|first5=Bradley}}</ref>. [[Large language models]] (LLM) are "on the rise" (2025), but "not yet ready for use"<ref>{{Cite journal |last=Lieberum |first=Judith-Lisa |last2=Toews |first2=Markus |last3=Metzendorf |first3=Maria-Inti |last4=Heilmeyer |first4=Felix |last5=Siemens |first5=Waldemar |last6=Haverkamp |first6=Christian |last7=Böhringer |first7=Daniel |last8=Meerpohl |first8=Joerg J. |last9=Eisele-Metzger |first9=Angelika |date=2025-05 |title=Large language models for conducting systematic reviews: on the rise, but not yet ready for use—a scoping review |url=https://www.wikidata.org/wiki/Q134545593|journal=Journal of Clinical Epidemiology |language=en |volume=181 |pages=111746 |doi=10.1016/j.jclinepi.2025.111746}}</ref>. The living review method relevant for just transition because it includes topic such as energy democracy which necessitate transdisciplinarity and consolidation of fragmented literature<ref>{{Cite journal|last=Droubi|first=Sufyan|last2=Heffron|first2=Raphael|last3=McCauley|first3=Darren|date=2022-04-01|title=A critical review of energy democracy: A failure to deliver justice?|url=https://www.wikidata.org/wiki/Q137901182|journal=Energy Research & Social Science|volume=86|doi=10.1016/J.ERSS.2021.102444}}</ref>. === Definitions of just transition : === * «a fair and equitable process of moving towards a post-carbon society’. »<ref name=":0">{{Cite journal|last=McCauley|first=Darren|last2=Heffron|first2=Raphael|date=2018-08-01|title=Just transition: Integrating climate, energy and environmental justice|url=https://www.wikidata.org/wiki/Q129947262|journal=Energy Policy|language=English|volume=119|pages=1–7|doi=10.1016/J.ENPOL.2018.04.014}}</ref>. The concept of just transition originated from global trade unions in the 1980s to promote green jobs creation as a key element of sustainability transitions<ref name=":0" />. However, scholars have broadened the use of this term to develop frameworks for analysing issues of fairness in these transitions<ref name=":0" />. The concept of just transition can be used to bridge various bodies of scholarship : climate justice, environmental justiceand energy justice<ref name=":3">{{Cite journal|last=Wang|first=Xinxin|last2=Lo|first2=Kevin|date=2021-12-01|title=Just transition: A conceptual review|url=https://www.wikidata.org/wiki/Q137209041|journal=Energy Research & Social Science|volume=82|pages=102291|doi=10.1016/J.ERSS.2021.102291}}</ref><ref name=":1">{{Cite book|url=https://www.wikidata.org/wiki/Q134545572|title=What is the “Just Transition”?|last=Heffron|first=Raphael J.|date=2021-01-01|pages=9–19|language=English}}</ref> and take into account various aspects of justice including distributional justice, procedural justice, restorative justice, recognition justice<ref name=":0" /><ref name=":3" /><ref name=":1" /><ref name=":4">{{Cite journal|last=Jenkins|first=Kirsten|last2=McCauley|first2=Darren|last3=Heffron|first3=Raphael|last4=Stephan|first4=Hannes|last5=Rehner|first5=Robert|date=2016-01-01|title=Energy justice: A conceptual review|url=https://www.wikidata.org/wiki/Q137210566|journal=Energy Research & Social Science|volume=11|pages=174–182|doi=10.1016/J.ERSS.2015.10.004}}</ref>. === Definition of Procedural justice === Procedural justice is about the fairness of decision-making processes related to transitions<ref name=":4" /> such as the inclusion of those impacted by these decisions<ref name=":5">{{Cite journal|last=Stark|first=Anthony|last2=Gale|first2=Fred|last3=Murphy-Gregory|first3=Hannah|date=2023-05-05|title=Just Transitions’ Meanings: A Systematic Review|url=https://www.wikidata.org/wiki/Q137210229|journal=Society and Natural Resources|volume=36|issue=10|pages=1277–1297|doi=10.1080/08941920.2023.2207166}}</ref>. Procedural justice can include issues of community and citizen participation in decision making, their political representation their consultation or the integration of their knowledge, with a focus on neglected population (indigenous people, women, gender and ethnic minorities<ref>{{Cite journal|last=Jenkins|first=Kirsten|last2=McCauley|first2=Darren|last3=Heffron|first3=Raphael|last4=Stephan|first4=Hannes|last5=Rehner|first5=Robert|date=2016-01-01|title=Energy justice: A conceptual review|url=https://www.wikidata.org/wiki/Q137210566|journal=Energy Research & Social Science|volume=11|pages=174–182|doi=10.1016/J.ERSS.2015.10.004}}</ref>. For example, the participation of affected communities in decisions related to the construction of new infrastructures<ref name=":0" />. == Methodology == === Wikidata and the semantic web ===<!-- Add introduction to what wikidata is and how the triplet works in a pedagogical manner Example of good description here : https://pubs.rsc.org/en/content/articlelanding/2025/np/d4np00008k#fig1 --> "A knowledge graph is a structured representation of knowledge that captures information in a machine-readable format.<ref>{{Cite journal|last=Hogan|first=Aidan|last2=Blomqvist|first2=Eva|last3=Cochez|first3=Michael|last4=D’amato|first4=Claudia|last5=Melo|first5=Gerard De|last6=Gutierrez|first6=Claudio|last7=Kirrane|first7=Sabrina|last8=Gayo|first8=José Emilio Labra|last9=Navigli|first9=Roberto|date=2022-05-31|title=Knowledge Graphs|url=https://dl.acm.org/doi/10.1145/3447772|journal=ACM Computing Surveys|language=en|volume=54|issue=4|pages=1–37|doi=10.1145/3447772|issn=0360-0300}}</ref> A knowledge graph consists of a graph or network of interconnected data points, where each data point represents a piece of information or a concept, and the relationships between them are explicitly defined. Knowledge graphs organize and store data in a format that facilitates information retrieval, data analysis, and reasoning."<ref>{{Cite journal|last=Meijer|first=David|last2=Beniddir|first2=Mehdi A.|last3=Coley|first3=Connor W.|last4=Mejri|first4=Yassine M.|last5=Öztürk|first5=Meltem|last6=Hooft|first6=Justin J. J. van der|last7=Medema|first7=Marnix H.|last8=Skiredj|first8=Adam|date=2025-04-16|title=Empowering natural product science with AI: leveraging multimodal data and knowledge graphs|url=https://pubs.rsc.org/en/content/articlelanding/2025/np/d4np00008k|journal=Natural Product Reports|language=en|volume=42|issue=4|pages=654–662|doi=10.1039/D4NP00008K|issn=1460-4752}}</ref> == Building a corpus and enriching bibliographic metadata == === Database search === We conducted preliminary searches in various databases including Web of science, Go Triple, Dimensions and OpenAlex. Web of Science was the database offering the most relevant restults and included the possibility to filter results to display only litterature reviews. Articles metadata were exported (in .ris format) and then imported into the reference manager software Zotero. {| class="wikitable" |+ !Keywords search !Database !Search date !Filters !Number of results |- |(((TS=(procedural justice OR procedural fairness OR democracy OR participation OR participatory)) AND TS=(sustainability OR energy OR climate)) AND TS=(transition OR transitions)) AND TS=(review OR reviews) |Web of Science (all databases, all dates) |December 2025 |Document type: Review Article |362 |} === Article screening === Articles abstract were then screened and we selected only articles which were litterature reviews focusing on concepts related to procedural justice as their main topics. We excluded article which were * Not related to sustainability transition (e.g. sustainable shift in..., hard science papers...) * Not literature reviews (e.g. review of policies, initiatives, cases, review notes, book review...) * Not related to procedural justice but to participation into markets, participation in eco-friendly behaviors or included justice consideration only in “future research” suggestions * Discussing participatory research methodologies (e.g. participatory modelling) without approaching it as an issue of justice, power or democracy * Discussing procedural justice concepts as key variables or key results without it being the main focus of the paper === Importing selected articles into Wikidata === To import the selected articles meta-data into Wikidata, we first ran [https://gist.github.com/zuphilip/aa9f59271fcb0807fb20c7d0110d26e4 a script] to check if any article was already present in the database. Next we used [https://gist.github.com/zuphilip/90acdc3eac4109830db1b3ab855fcb24 another script] that checks the ISSN of the publication in Wikidata and add P-Q-pairs in the extra field of Zotero. Then we exported the articles data using the "export to Wikidata QuickStatements" function of Zotero and use the QuickStatements tool to add them to Wikidata. Next we used the [[wikidata:Wikidata:Zotero/Cita|Cita]] (V1.0.0-beta.17) Zotero add-on to add articles QID in Zotero. At this point we identified that duplicates had been created in Wikidata (possibly because the initial [https://gist.github.com/zuphilip/aa9f59271fcb0807fb20c7d0110d26e4 script] did not work that well because of the recent [[wikidata:Wikidata:SPARQL_query_service/WDQS_graph_split|Graph Split]] on Wikidata). We merged duplicates on wikidata using the [[wikidata:Help:Merge|"Merge" gadget]] on Wikidata. We checked manually for duplicated statments in those items. === Article classification through meta-data enrichement ===<!-- Add : What is meta-data enrichement --> Existing review try to classify existing articles according to various criteria such as industry focus, academic discipline, geography of research sites (countries), stakeholder focus (community, consumer, worker...), type of study (case study, theory development) or methodology (quantitative, qualitative, mixt).<ref name=":5" /> We selected the most relevant properties in Wikidata to reflect these classifications : {{Wikidata entity link|P921}} to describe what the article is about, {{Wikidata entity link|P8363}} to describe its main methodology/research design and {{Wikidata entity link|P6153}} to describe its geographical focus. ==== Main subjects ==== We first read the articles abstracts and listed relevant topics and their Wikidata ID in a shared spreadsheet. These topics were : {| class="wikitable" |+ !Qid !Main topic !Description |- |[[d:Q42377797|Q42377797]] |acceptability |characteristic of a thing being subject to acceptance for some purpose |- |[[d:Q2798912|Q2798912]] |accountability |concept of responsibility in ethics, governance and decision-making |- |[[d:Q421953|Q421953]] |actor–network theory |theory within social science |- |[[d:Q84459973|Q84459973]] |affordability | |- |[[d:Q185836|Q185836]] |age of a person |time elapsed since a person was born |- |[[d:Q4764988|Q4764988]] |animal studies |field in which animals are studied in a variety of cross-disciplinary ways |- |[[d:Q4338318|Q4338318]] |awareness |state or ability to perceive, to feel, or to be conscious of events, objects, or sensory patterns |- |[[d:Q4930066|Q4930066]] |blue carbon |carbon captured by the world's coastal ocean ecosystems |- |[[d:Q430460|Q430460]] |capability approach |economic theory |- |[[d:Q7569|Q7569]] |child |human between birth and puberty |- |[[d:Q4116870|Q4116870]] |civic engagement |individual or group activity addressing issues of public concern |- |[[d:Q125928|Q125928]] |climate change |human-caused changes to climate on Earth |- |[[d:Q260607|Q260607]] |climate change adaptation |process of adjustment to actual or expected climate change and its effects, seeking to moderate or avoid harm or exploit beneficial opportunities |- |[[d:Q1291678|Q1291678]] |climate justice |term linking the climate crisis with environmental and social justice |- |[[d:Q2270945|Q2270945]] |co-creation |product or service design process in which input from consumers plays a central role |- |[[d:Q16972712|Q16972712]] |co-design |approach to design attempting to actively involve all stakeholders |- |[[d:Q16324410|Q16324410]] |coproduction |product or service design process in which input from consumers plays a central role |- |[[d:Q11024|Q11024]] |communication |act of conveying intended meaning |- |[[d:Q177634|Q177634]] |community |social unit of human organisms who share common values |- |[[d:Q5154673|Q5154673]] |community choice aggregation |alternative energy supply system |- |[[d:Q113514984|Q113514984]] |community energy |delivery of community-led renewable energy, energy demand reduction and energy supply projects |- |[[d:Q65807646|Q65807646]] |community participation |The taking part by members of a community in decisionmaking processes related to the development of their community |- |[[d:Q188843|Q188843]] |cosmopolitanism |ideology that all human beings belong to a single community, based on a shared morality |- |[[d:Q11693783|Q11693783]] |decarbonization |change of economy, especially of energy industries, towards lower carbon dioxide emissions |- |[[d:Q284289|Q284289]] |deliberative democracy |form of democracy focusing on consensus |- |[[d:Q7174|Q7174]] |democracy |form of government |- |[[d:Q552284|Q552284]] |distributive justice |concept of the socially just allocation of goods |- |[[d:Q1230584|Q1230584]] |diversity |concept in sociology and political studies |- |[[d:Q1049066|Q1049066]] |ecological economics |research field on the interdependence of human economies and natural ecosystems |- |[[d:Q8134|Q8134]] |economics |social science that studies the production, distribution, and consumption of goods and services |- |[[d:Q868575|Q868575]] |empowerment |providing increased autonomy |- |[[d:Q295865|Q295865]] |ecosystem service |benefits created by nature, forests and environmental systems |- |[[d:Q138359220|Q138359220]] |energy citizenship |involvement of citizens in energy-related decisions |- |[https://www.wikidata.org/w/index.php?title=Q131444737&redirect=no Q131444737] |community energy |[redirection] |- |[[d:Q16869822|Q16869822]] |energy consumption |amount of energy or power used |- |[[d:Q1358789|Q1358789]] |senior |elderly person |- |[[d:Q14944319|Q14944319]] |energy democracy |concept in environmental justice movement |- |[[d:Q192704|Q192704]] |energy efficiency |ratio between the useful energy output and the input of a machine |- |[[d:Q24965464|Q24965464]] |energy modeling |process of building computer models of energy systems in order to analyze them |- |[[d:Q1805337|Q1805337]] |energy policy |policy addressing energy issues |- |[[d:Q1341244|Q1341244]] |energy poverty |lack of access to modern energy services |- |[[d:Q3406659|Q3406659]] |energy production |conversion of energy from a primary source into a form useful to humans |- |[[d:Q117091181|Q117091181]] |energy justice |subconcept of economic equality |- |[[d:Q3456219|Q3456219]] |energy renovation |building works aimed at reducing energy consumption and decarbonising the energy sources used |- |[[d:Q2700433|Q2700433]] |energy security |national security considerations of energy availability |- |[[d:Q837718|Q837718]] |energy storage |capture of energy produced at one time for use at a later time |- |[[d:Q795757|Q795757]] |energy transition |long-term structural change towards sustainable energy systems |- |[[d:Q1479527|Q1479527]] |environmental justice |system of fairness |- |[[d:Q771773|Q771773]] |fairness |concept in sociology and generally the interaction of society |- |[[d:Q56395513|Q56395513]] |farming system |method of agricultural production defined by its physical practices and economic characteristics |- |[[d:Q5465532|Q5465532]] |food system |all processes and infrastructure involved in feeding a population |- |[[d:Q4421|Q4421]] |forest |dense collection of trees covering a relatively large area |- |[[d:Q48277|Q48277]] |gender |social concept which distinguish the different gender categories |- |[[d:Q1553864|Q1553864]] |governance |all of the processes of governing, whether undertaken by a government, market or network, whether over a family, tribe, formal or informal organization or territory and whether through the laws, norms, power or language of an organized society |- |[[d:Q8458|Q8458]] |human rights |inalienable fundamental rights to which a person is inherently entitled |- |[[d:Q11376059|Q11376059]] |human rights violation |act or omission which contravene the principles of human rights |- |[[d:Q103817|Q103817]] |indigenous people |first inhabitants of an area and their descendants |- |[[d:Q113561794|Q113561794]] |indigenous science |indigenous knowledge applied to the scientific method |- |[[d:Q770480|Q770480]] |injustice |quality relating to unfairness or undeserved outcomes |- |[[d:Q17142211|Q17142211]] |interactional justice |the perceived appropriateness of interpersonal treatment |- |[[d:Q1516555|Q1516555]] |intersectionnality |theoretical framework of multidimensional oppression |- |[[d:Q6316391|Q6316391]] |just transition |Framework developed by the trade union movement to encompass wide range of social interventions needed to secure decent work opportunities and a greener economy. |- |[[d:Q366139|Q366139]] |legitimation |the process of making something acceptable and normative to a group |- |[[d:Q3027857|Q3027857]] |living lab |user-centered, open innovation ecosystem integrating research and innovation in real life communities |- |[[d:Q59679511|Q59679511]] |low income |home with little money |- |[[d:Q43619|Q43619]] |natural environment |all living and non-living things occurring naturally on Earth or some region thereof |- |[[d:Q127514833|Q127514833]] |nature-positive |global goal to halt and reverse nature loss by 2030 |- |[[d:Q13023682|Q13023682]] |non-human |organism not in the genus Homo |- |[[d:Q728646|Q728646]] |partnership |arrangement in which parties agree to cooperate to advance their mutual interests |- |[[d:Q3907287|Q3907287]] |policy making |the act of developing policy |- |[[d:Q9357091|Q9357091]] |political theory |class of theory |- |[[d:Q265425|Q265425]] |postcolonialism |academic discipline |- |[[d:Q25107|Q25107]] |power |ability to influence the behavior of others |- |[[d:Q442100|Q442100]] |procedural justice |fairness in the processes that resolve disputes and allocate resources |- |[[d:Q7249406|Q7249406]] |project governance |management framework |- |[[d:Q7257735|Q7257735]] |public engagement |Policy-making practice |- |[[d:Q541936|Q541936]] |public participation |participation of citizens in various policy decisions and planning processes |- |[[d:Q6142016|Q6142016]] |recognition justice |social philosophy theory |- |[[d:Q10509953|Q10509953]] |renewable electricity |electricity from renweable sources |- |[[d:Q12705|Q12705]] |renewable energy |energy collected from renewable resources |- |[[d:Q56510941|Q56510941]] |renewable energy policy | |- |[[d:Q1165392|Q1165392]] |restorative justice |approach to justice where victims and perpetrators mediate a restitution agreement |- |[[d:Q4414036|Q4414036]] |rural population |inhabitants of rural areas or of small towns classified as rural |- |[[d:Q17152351|Q17152351]] |smart system |adaptive intelligent systems |- |[[d:Q187588|Q187588]] |social class |group of people categorized in a hierarchy based on socioeconomic factors |- |[[d:Q264892|Q264892]] |social justice |concept that discrimination recognized in society should be remedied |- |[[d:Q34749|Q34749]] |social science |academic disciplines concerned with society and the relationships between individuals in society |- |[[d:Q2930198|Q2930198]] |stakeholder participation |involvement of groups or individuals affected by the actions of an entity |- |[[d:Q125359881|Q125359881]] |sustainability transition | |- |[[d:Q219416|Q219416]] |sustainability |ability of human civilization to coexist with the biosphere in a steady state |- |[[d:Q131201|Q131201]] |sustainable development |mode of human development that meets current demands without compromising the needs of future generations |- |[[d:Q7649586|Q7649586]] |Sustainable Development Goals |set of United Nations-defined global development goals and climate change |- |[[d:Q69883|Q69883]] |urban planning |technical and political process concerned with the use of land and design of the urban environment |- |[[d:Q920600|Q920600]] |urban renewal |program of land redevelopment in cities, often where there is urban decay |- |[[d:Q3376054|Q3376054]] |vulnerable population |group of persons whose range of options is severely limited, are subjected to coercion, or who may be compromised in their ability to give informed consent |- |[[d:Q107389921|Q107389921]] |water-management | |- |[[d:Q7981051|Q7981051]] |well-being |measure of how well life is to someone or a group with factors such as health, happiness and satisfaction |- |[[d:Q467|Q467]] |woman |female adult human |- |[[d:Q188867|Q188867]] |future studies |study of possible, probable, and preferable social, technological and political futures |- |[[d:Q1038171|Q1038171]] |participatory design |active involvement of all stakeholders in the design process |} <!-- include all below items using the wikidata link template --> Then, for each article, we inferred what the {{Wikidata entity link|P921}} was from the abstracts and author provided keywords. ==== Study types ==== Our review included only litterature reviews. We first read abstracts to identify all the [https://angryloki.github.io/wikidata-graph-builder/?item=Q2412849&property=P279&mode=reverse different types of litterature reviews] present in the corpus and created wikidata items which did not exist, for example {{Wikidata entity link|Q137209848}} and {{Wikidata entity link|Q137174203}}. We improved these method items using the methodological references cited in the reviewed papers. The types of reviews were : {| class="wikitable" |+ !Qid !Study type !Description |- |[[d:Q603441|Q603441]] |bibliometrics |statistical analysis of written publications, such as books or articles |- |[[d:Q472342|Q472342]] |scientometrics |study of measuring and analysing science, technology and innovation |- |[[d:Q815382|Q815382]] |meta-analysis |statistical method that summarizes data from multiple sources |- |[[d:Q1504425|Q1504425]] |systematic review |publication type, study that gathers, analyzes, and communicates the results of research and information on a topic |- |[[d:Q2412849|Q2412849]] |literature review |process of information search and text of a review article (Q7318358), which includes the current knowledge including substantive findings, as well as theoretical and methodological contributions to a particular topic |- |[[d:Q6822263|Q6822263]] |meta-regression |statistical tool used in meta-analyses |- |[[d:Q7301211|Q7301211]] |realist evaluation |[...] |- |[[d:Q17007303|Q17007303]] |combinatorial meta-analysis |[...] |- |[[d:Q70470634|Q70470634]] |network meta-analysis |meta-analysis of randomized trials in which estimates of comparative treatment effects are visualized and interpreted from a network of interventions |- |[[d:Q101116078|Q101116078]] |scoping review |search for concepts by mapping the language and data which surrounds those concepts and adjusting the search method iteratively to synthesize evidence and assess the scope of an area of inquiry |- |[[d:Q110665014|Q110665014]] |narrative review |type of literature review, without structured method of retrieval and analysis |- |[[d:Q137174203|Q137174203]] |conceptual review |academic research aiming to review existing concepts and definitions in the litterature |- |[[d:Q137174450|Q137174450]] |critical review |type of literature review analysing strenghts, major contributions, mistakes and neglected issues in an academic field of research |- |[[d:Q137209848|Q137209848]] |integrative literature review |type of literature review |- |[[d:Q110665014|Q137211242]] |narrative review |type of literature review, without structured method of retrieval and analysis |}<!-- include all below items using the wikidata link template --> [Include list and description of types of litterature reviews] Then, we added the {{Wikidata entity link|P8363}} of each articles based on the abstract and method sections. In case of doubt, we compared our interpretation. ==== Research site ==== When an article had a specific geographical focus, we used the property {{Wikidata entity link|P6153}} to describe it. For example, the article "{{Wikidata entity link|Q137901202}}" focused on {{Wikidata entity link|Q132959}}. ==== Results ==== The table listing all the papers in the sample can be visualized [https://tabernacle.toolforge.org/?#/tab/manual/Q137211155%0A%0A%0A%0A%0A%0A%0AQ114306483%0A%0A%0A%0A%0AQ137901181%0A%0A%0A%0AQ137901182%0A%0A%0A%0A%0A%0A%0A%0AQ137901183%0A%0A%0AQ114306476%0A%0A%0A%0A%0AQ137901184%0A%0A%0A%0A%0AQ137901185%0A%0A%0A%0A%0A%0AQ137901186%0A%0A%0A%0A%0A%0A%0AQ137901187%0A%0A%0A%0A%0A%0A%0AQ137901188%0A%0A%0A%0A%0AQ137210566%0A%0A%0A%0A%0AQ114306511%0A%0A%0A%0A%0A%0AQ137901191%0A%0A%0A%0A%0AQ137901192%0A%0A%0A%0A%0AQ137901193%0A%0A%0A%0A%0AQ135979013%0A%0A%0A%0A%0A%0A%0A%0AQ137901195%0A%0A%0A%0A%0A%0AQ137901196%0A%0A%0A%0A%0A%0A%0AQ137901197%0A%0A%0A%0A%0AQ136447761%0A%0A%0A%0AQ137901199%0A%0A%0A%0A%0A%0A%0AQ129652515%0A%0A%0A%0A%0A%0A%0AQ137901201%0A%0A%0A%0A%0A%0AQ137901202%0A%0A%0A%0A%0AQ137901203%0A%0A%0A%0AQ137901204%0A%0A%0A%0A%0A%0A%0A%0AQ137901205%0A%0A%0A%0A%0AQ137901206%0A%0A%0A%0A%0A%0A%0A%0A%0AQ137901207%0A%0A%0A%0A%0AQ129203992%0A%0A%0A%0A%0A%0A%0AQ114197507%0A%0A%0A%0AQ137901161%0A%0A%0A%0A%0A%0A%0A%0AQ137901209%0A%0A%0A%0A%0A%0AQ137901210%0A%0A%0A%0A%0A%0AQ137901211%0A%0A%0A%0A%0AQ11420462%0A%0AQ137901213%0A%0A%0A%0A%0A%0A%0A%0A%0A%0A%0A%0A%0A%0A%0AQ104887325%0A%0A%0A%0A%0A%0AQ137901162%0A%0A%0AQ137901163%0A%0A%0A%0A%0AQ137901164%0A%0A%0A%0A%0A%0AQ137901215%0A%0A%0A%0A%0AQ137901216%0A%0A%0A%0A%0A%0A%0A%0A%0AQ137901217%0A%0A%0A%0A%0AQ115448818%0A%0A%0A%0A%0AQ137901218%0A%0A%0A%0AQ137901219%0A%0A%0A%0A%0AQ137901220%0A%0A%0A%0A%0A%0AQ137901221%0A%0A%0A%0A%0A%0AQ137901222%0A%0A%0A%0A%0AQ137901223%0A%0A%0AQ137901224%0A%0A%0A%0AQ137901225%0A%0A%0A%0A%0A%0A%0AQ137901226%0A%0A%0A%0AQ137901227%0A%0A%0AQ137901182/Len%3BP921%3BP6153%3BP8363%3BP50 here]. == Modelling knowledge == Concept maps can be a powerful literature review tool<ref>{{Cite journal|last=Lewis|first=John Kennedy|date=2016|title=Using ATLAS.ti to Facilitate Data Analysis for a Systematic Review of Leadership Competencies in the Completion of a Doctoral Dissertation|url=https://www.ssrn.com/abstract=2850726|journal=SSRN Electronic Journal|language=en|doi=10.2139/ssrn.2850726|issn=1556-5068}}</ref> allowing to synthetize theoretical statements about relationship between concepts<ref>{{Cite journal|last=Panniers|first=Teresa L|last2=Feuerbach|first2=Renee Daiuta|last3=Soeken|first3=Karen L|date=2003-08-01|title=Methods in informatics: using data derived from a systematic review of health care texts to develop a concept map for use in the neonatal intensive care setting|url=https://www.sciencedirect.com/science/article/pii/S1532046403000911|journal=Journal of Biomedical Informatics|series=Building Nursing Knowledge through Informatics: From Concept Representation to Data Mining|volume=36|issue=4|pages=232–239|doi=10.1016/j.jbi.2003.09.010|issn=1532-0464}}</ref>. In the present study, we explored how concept map can be used to model the knowledge present in the paper we selected. [define knowledge modelling] ==== Wikidata ontology ==== Wikidata "supports multiple coexisting classification" and allow multiple ontological frameworks to coexist.<ref name=":8">{{Cite web|url=https://arxiv.org/abs/2512.12260v1|title=A Multi-Axial Mindset for Ontology Design Lessons from Wikidata's Polyhierarchical Structure|last=Doğan|first=Ege Atacan|last2=Patel-Schneider|first2=Peter F.|date=2025-12-13|website=arXiv.org|language=en|access-date=2026-05-26}}</ref> It also supports epistemic pluralism : different worldviews can be represented in wikidata, even though scientific knowledge is preferred.<ref name=":8" /> See more on membership properties : https://www.wikidata.org/wiki/Help:Basic_membership_properties See the discussion on cause modelling : https://www.wikidata.org/wiki/Help:Modeling_causes/en ==== Conceptual modelling ==== We first reflected on what kind of wikidata properties could be used to represent concepts and theories in wikidata. Capturing the content of a concept is not straightforward and there are various approaches coming from psychology and philosophy on the matter<ref>{{Cite book|title=The Origin of Concepts|last=Carey|first=Susan|date=2011|publisher=Oxford University Press USA - OSO|isbn=978-0-19-536763-8|series=Oxford Series in Cognitive Development Ser|location=Cary}}</ref> we summarize these approaches below and examine which wikidata properties exist to represent them. * Definition: the content of a concept can be formed by its decomposition into other concepts. Many Wikidata properties can be relevant to model definitions, for example: {{Wikidata entity link|P1269}}, {{Wikidata entity link|P361}}/{{Wikidata entity link|P527}}, {{Wikidata entity link|P2670}}, {{Wikidata entity link|P1552}}/{{Wikidata entity link|P6477}}, {{Wikidata entity link|P3712}}... * Categorization: the content of a concept is formed by its illustration by an exemplar (a [[wikipedia:Prototype_theory|prototype]]) that best represent the concept. Apart from the inclusion of images to illustrate an item, Wikidata structure do not highlight exemplars. However, properties signifying relations of categorizations are among the most used with {{Wikidata entity link|P31}} and {{Wikidata entity link|P279}}. * Theory: the content of a concept is formed by its role in providing explanation of the world. Wikidata includes several properties to describe causal relationships: {{Wikidata entity link|P828}}/{{Wikidata entity link|P1542}}, {{Wikidata entity link|P1537}}/{{Wikidata entity link|P1479}}. * Essence: the content of a concept is "something" deep explaning the entity's existence and its properties. We can use concepts before knowing what they mean, and this is what allows us to revise our knowledge about it. The idea of essence is well represented by the QID of Wikidata entities: it is independent of language and definitions and we can create it before really knowing what all its properties will be. * Origin: the content of the concept is determined causally by social and historial factors (e.g. someone inventing the concept and introducing its use in a language community). This can be represented by the property {{Wikidata entity link|P3938}}. ==== Thematic networks ==== [[File:Thematic network example.jpg|thumb|547x547px|Structure of a thematic network (Source: Attride-Stirling 2001)]] A thematic network is “simply a way of organizing a thematic analysis of qualitative data”<ref name=":7">{{Cite journal|last=Attride-Stirling|first=Jennifer|date=2001-12|title=Thematic networks: an analytic tool for qualitative research|url=https://journals.sagepub.com/doi/10.1177/146879410100100307|journal=Qualitative Research|language=en|volume=1|issue=3|pages=385–405|doi=10.1177/146879410100100307|issn=1468-7941}}</ref>. It is compatible with classical coding strategies such as [[grounded theory]]<ref>{{Cite journal|last=Corbin|first=Juliet|last2=Strauss|first2=Anselm|date=1990-12-01|title=Grounded Theory Research: Procedures, Canons and Evaluative Criteria|url=https://www.degruyter.com/document/doi/10.1515/zfsoz-1990-0602/html|journal=Zeitschrift für Soziologie|language=en|volume=19|issue=6|pages=418–427|doi=10.1515/zfsoz-1990-0602|issn=2366-0325}}</ref>. Thematic networks can be used to visualise the data structure after identifying themes and help structure and interpret the data<ref name=":7" />. The principle is to assemble basic themes into more general themes. Qualitative researchers usually use {{Wikidata entity link|Q4550939}} and qualitative coding (e.g. grounded theory) to identify themes and sub-themes. However, the nature of the relationship between these various themes and sub-themes is often not specified. * ==== Causal networks ==== The use of diagrams to represent causal relationship exist in various research practices. In statistics, researchers sometime present models with boxes and arrows representing correlations and/or causations<ref>{{Cite book|url=https://mirror.vcu.edu/pub/mx/doc/mxmang10.pdf|title=Statistical Modeling|last=Neale|first=Michael C.|last2=Boker|first2=Steven M.|last3=Xie|first3=Gary|last4=Maes|first4=Hermine H.|publisher=Richmond, VA: Department of Psychiatry|year=1999|location=Virginia Commonwealth University}}</ref>. In qualitative research, building grounded theory models is about "[accounting] for not only all the major emergent concepts, themes, and dimensions, but also for their dynamic interrelationships. Speaking in classic boxes-and-arrows terms, this process amounts to assembling the constellation of boxes with a special focus on the arrows."<ref>{{Cite journal|last=Gioia|first=Dennis A.|last2=Corley|first2=Kevin G.|last3=Hamilton|first3=Aimee L.|date=2013-01|title=Seeking Qualitative Rigor in Inductive Research: Notes on the Gioia Methodology|url=https://journals.sagepub.com/doi/10.1177/1094428112452151|journal=Organizational Research Methods|language=en|volume=16|issue=1|pages=15–31|doi=10.1177/1094428112452151|issn=1094-4281}}</ref> Researchers relying on system theory also use causal loop diagram where boxes represent variables and arrows represent causal influence (positive or negative), causal relationship can "feedback" (two variables can influence each other)<ref>{{Cite book|url=https://link.springer.com/10.1007/978-3-031-01919-7_4|title=Causal Loop Diagrams|last=Barbrook-Johnson|first=Pete|last2=Penn|first2=Alexandra S.|date=2022|publisher=Springer International Publishing|isbn=978-3-031-01833-6|location=Cham|pages=47–59|language=en|doi=10.1007/978-3-031-01919-7_4}}</ref>. Wikidata includes several properties to describe causal relationships: * {{Wikidata entity link|P828}} * {{Wikidata entity link|P1542}} * {{Wikidata entity link|P1537}} * {{Wikidata entity link|P1479}} : it is difficult to identify single causes for social phenomenons, many factors having an effect on the subject item will likely be contributing factors ==== Modelling concepts ==== To model concepts related to just transition. We read the selected papers and used them as source to build a knowledge graph in wikidata. For example, the paper {{Wikidata entity link|Q137901182}} mention "Energy democracy is both an ideal and a process"<ref>{{Cite journal|last=Droubi|first=Sufyan|last2=Heffron|first2=Raphael|last3=McCauley|first3=Darren|date=2022-04-01|title=A critical review of energy democracy: A failure to deliver justice?|url=https://www.wikidata.org/wiki/Q137901182|journal=Energy Research & Social Science|volume=86|pages=4|doi=10.1016/J.ERSS.2021.102444}}</ref>, we thus entered the wikidata statement {{Wikidata entity link|Q14944319}} is an {{Wikidata entity link|P31}} {{Wikidata entity link|Q840396}}, using the paper as source. Ontology challenges: *{{Wikidata entity link|P31}}: concepts may have a dual nature because they designate at the same time an idea and the entity that this idea represent. Energy democracy is a concept, an ideal, a process and an outcome. *'''Process versus outcome :''' For material processes, the distinction between process and outcome is rather simple. For example, in Wikidata, {{Wikidata entity link|Q11629}} (practice of applying paint) is different from {{Wikidata entity link|Q3305213}} (visual artwork), and this distinction is based on the criterion "{{Wikidata entity link|Q127270577}}". However, this distinction is less straightforward for social processes that do not have an end. Such processes are ongoing and outcomes cannot be separated as clearly. * '''Ideal versus reality :''' Concepts do not have goals in themselves, but the reality they represent can have goals. To distinguish goals from the process to reach it, we used {{Wikidata entity link|P3712}} to describe ideals and {{Wikidata entity link|P2670}} to describe processes. * '''Phenomenon versus theory :''' Wikidata current items are not really suited to model "meta-research" statements. For example, modelling the idea tha the literature on energy democracy is fragmented would require creating an item representing the energy democracy literature, not just energy democracy in general. Similarly, it can be difficult to model the chronological evolution of the definition of an idea (although it could be technically possible). It is hard to represent in Wikidata affirmations related to missing knowlege, propositions of untested hypothesis, critique of existing research or research agenda recommandations * '''Origin of discourses versus origin of practices :''' To distinguish the causes of the concepts/discourses and the causes of the phenomenon itself, we used {{Wikidata entity link|P3938}} to indicate the origins of the concept or the movements promoting it. Other challenges * Wikidata does not seem to be the best tool to model quantitative statements, for example, the paper {{Wikidata entity link|Q137901196}} states that "9.8% of the final energy consumed in developing countries comes from modern renewable energy sources". Including energy data in Wikidata require using or creating specific properties (e.g. {{Wikidata entity link|P6826}}) * When concepts are not precisely defined, statements cannot be modelled correctly. For example, in the sentence "management of social affairs by voluntary and self-governing associations is deemed to ensure that both citizen choice and public welfare are best served"<ref>{{Cite journal|last=Veelen|first=Bregje van|last2=Horst|first2=Dan van der|date=2018-12-01|title=What is energy democracy? Connecting social science energy research and political theory|url=https://www.wikidata.org/wiki/Q129652515|journal=Energy Research & Social Science|language=English|volume=46|pages=19–28|doi=10.1016/J.ERSS.2018.06.010}}</ref>, "choice" could refer to {{Wikidata entity link|Q111986453}}, {{Wikidata entity link|Q1331926}}, or {{Wikidata entity link|Q12888920}} as "choice" can refer to the availability of different options, or the decision process to chose among them. Advantages : * Link toward unique identifiers for concepts, but also laws (e.g. {{Wikidata entity link|Q139764294}}) == Data visualisation == === Filter statements === * Visualize only statements using a specitic source. Example : https://w.wiki/PFqH * Visualize only items which are part to the present project (require that all items of the project include the statement {{Wikidata entity link|P6104}} {{Wikidata entity link|Q134545539}}). === Mapping a concept === Scholia request "topic in context" === Mapping sources consensus === Visualise graphs and use the number of references to determine edge thickness/weight. == Writing == To cite articles we used the [[Template:Cite Q|Cite Q template.]] Each reference is an item in Wikidata and the template retrieve the necessary data to generate the citation references below. == Data == {| class="wikitable sortable" ! QID !! Year !! DOI !! Title |- | [[d:Q137901191|Q137901191]] || 2025 || [https://doi.org/10.1002/GEO2.70040 10.1002/GEO2.70040] || Place-Based Sustainability Transformations for Just Futures: A Systematic Review |- | [[d:Q137901187|Q137901187]] || 2025 || [https://doi.org/10.1002/WCC.932 10.1002/WCC.932] || Public Communication of Climate and Justice: A Scoping Review |- | [[d:Q135979013|Q135979013]] || 2025 || [https://doi.org/10.1007/S13280-025-02202-Z 10.1007/S13280-025-02202-Z] || Participatory approaches to climate adaptation, resilience, and mitigation: A systematic review |- | [[d:Q137901223|Q137901223]] || 2022 || [https://doi.org/10.1007/S13412-021-00726-W 10.1007/S13412-021-00726-W] || A review of stakeholder participation studies in renewable electricity and water: does the resource context matter? |- | [[d:Q137901184|Q137901184]] || 2021 || [https://doi.org/10.1007/S40518-021-00184-6 10.1007/S40518-021-00184-6] || Energy Storage as an Equity Asset. |- | [[d:Q114204627|Q114204627]] || 2021 || [https://doi.org/10.1007/S43621-021-00024-Z 10.1007/S43621-021-00024-Z] || Can public awareness, knowledge and engagement improve climate change adaptation policies? |- | [[d:Q137901209|Q137901209]] || 2026 || [https://doi.org/10.1016/J.AGSY.2025.104512 10.1016/J.AGSY.2025.104512] || Designing with non-humans for agricultural systems transformation: An interdisciplinary review and framework for reflection |- | [[d:Q137901201|Q137901201]] || 2025 || [https://doi.org/10.1016/J.COPSYC.2024.101987 10.1016/J.COPSYC.2024.101987] || Individual and community catalysts for Renewable Energy Communities (RECs) development |- | [[d:Q114197507|Q114197507]] || 2022 || [https://doi.org/10.1016/J.CRM.2022.100438 10.1016/J.CRM.2022.100438] || Advancements of sustainable development goals in co-production for climate change adaptation research |- | [[d:Q129203992|Q129203992]] || 2024 || [https://doi.org/10.1016/J.EGYR.2024.01.040 10.1016/J.EGYR.2024.01.040] || Empowering energy citizenship: Exploring dimensions and drivers in citizen engagement during the energy transition |- | [[d:Q137901216|Q137901216]] || 2026 || [https://doi.org/10.1016/J.EIAR.2025.108187 10.1016/J.EIAR.2025.108187] || From participation to partnership: A systematic review of public engagement in sustainable urban planning |- | [[d:Q137210566|Q137210566]] || 2016 || [https://doi.org/10.1016/J.ERSS.2015.10.004 10.1016/J.ERSS.2015.10.004] || Energy justice: A conceptual review |- | [[d:Q115448818|Q115448818]] || 2016 || [https://doi.org/10.1016/J.ERSS.2016.04.001 10.1016/J.ERSS.2016.04.001] || Stakeholder involvement in sustainability science—A critical view |- | [[d:Q129652515|Q129652515]] || 2018 || [https://doi.org/10.1016/J.ERSS.2018.06.010 10.1016/J.ERSS.2018.06.010] || What is energy democracy? Connecting social science energy research and political theory |- | [[d:Q137901196|Q137901196]] || 2020 || [https://doi.org/10.1016/J.ERSS.2020.101716 10.1016/J.ERSS.2020.101716] || Of renewable energy, energy democracy, and sustainable development: A roadmap to accelerate the energy transition in developing countries |- | [[d:Q136447761|Q136447761]] || 2020 || [https://doi.org/10.1016/J.ERSS.2020.101768 10.1016/J.ERSS.2020.101768] || Energy democracy as a process, an outcome and a goal: A conceptual review |- | [[d:Q137901204|Q137901204]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101834 10.1016/J.ERSS.2020.101834] || Identities, innovation, and governance: A systematic review of co-creation in wind energy transitions |- | [[d:Q137901183|Q137901183]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101837 10.1016/J.ERSS.2020.101837] || Renewable energy for whom? A global systematic review of the environmental justice implications of renewable energy technologies |- | [[d:Q137901207|Q137901207]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101871 10.1016/J.ERSS.2020.101871] || Rethinking community empowerment in the energy transformation: A critical review of the definitions, drivers and outcomes |- | [[d:Q137901215|Q137901215]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101876 10.1016/J.ERSS.2020.101876] || Co-production in the wind energy sector: A systematic literature review of public engagement beyond invited stakeholder participation |- | [[d:Q114306511|Q114306511]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101907 10.1016/J.ERSS.2020.101907] || From consultation toward co-production in science and policy: A critical systematic review of participatory climate and energy initiatives |- | [[d:Q137901221|Q137901221]] || 2021 || [https://doi.org/10.1016/J.ERSS.2021.102257 10.1016/J.ERSS.2021.102257] || The challenges of engaging island communities: Lessons on renewable energy from a review of 17 case studies |- | [[d:Q137901218|Q137901218]] || 2022 || [https://doi.org/10.1016/J.ERSS.2021.102333 10.1016/J.ERSS.2021.102333] || The (in)justices of smart local energy systems: A systematic review, integrated framework, and future research agenda |- | [[d:Q137901182|Q137901182]] || 2022 || [https://doi.org/10.1016/J.ERSS.2021.102444 10.1016/J.ERSS.2021.102444] || A critical review of energy democracy: A failure to deliver justice? |- | [[d:Q114306483|Q114306483]] || 2022 || [https://doi.org/10.1016/J.ERSS.2021.102482 10.1016/J.ERSS.2021.102482] || The role of energy democracy and energy citizenship for participatory energy transitions: A comprehensive review |- | [[d:Q114306476|Q114306476]] || 2022 || [https://doi.org/10.1016/J.ERSS.2022.102714 10.1016/J.ERSS.2022.102714] || What about citizens? A literature review of citizen engagement in sustainability transitions research |- | [[d:Q137901193|Q137901193]] || 2022 || [https://doi.org/10.1016/J.ERSS.2022.102862 10.1016/J.ERSS.2022.102862] || When energy justice is contested: A systematic review of a decade of research on Sweden?s conflicted energy landscape |- | [[d:Q137901219|Q137901219]] || 2023 || [https://doi.org/10.1016/J.ERSS.2022.102913 10.1016/J.ERSS.2022.102913] || Can we optimise for justice? Reviewing the inclusion of energy justice in energy system optimisation models |- | [[d:Q137901186|Q137901186]] || 2023 || [https://doi.org/10.1016/J.ERSS.2023.103010 10.1016/J.ERSS.2023.103010] || Analysing intersections of justice with energy transitions in India- A systematic literature review |- | [[d:Q137901181|Q137901181]] || 2023 || [https://doi.org/10.1016/J.ERSS.2023.103053 10.1016/J.ERSS.2023.103053] || Fostering justice through engagement: A literature review of public engagement in energy transitions |- | [[d:Q137211155|Q137211155]] || 2023 || [https://doi.org/10.1016/J.ERSS.2023.103213 10.1016/J.ERSS.2023.103213] || A fairway to fairness: Toward a richer conceptualization of fairness perceptions for just energy transitions |- | [[d:Q137901217|Q137901217]] || 2023 || [https://doi.org/10.1016/J.ERSS.2023.103221 10.1016/J.ERSS.2023.103221] || Powering just energy transitions: A review of the justice implications of community choice aggregation |- | [[d:Q137901199|Q137901199]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104016 10.1016/J.ERSS.2025.104016] || Making energy renovations equitable: A literature review of decision-making criteria for a just energy transition in residential buildings |- | [[d:Q137901188|Q137901188]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104036 10.1016/J.ERSS.2025.104036] || Community energy justice: A review of origins, convergence, and a research agenda |- | [[d:Q137901211|Q137901211]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104067 10.1016/J.ERSS.2025.104067] || Psychological and social factors driving citizen involvement in renewable energy communities: A systematic review |- | [[d:Q137901192|Q137901192]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104149 10.1016/J.ERSS.2025.104149] || Assessing social impacts and Energy Justice along green hydrogen supply chains: a capability-based framework |- | [[d:Q137901195|Q137901195]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104422 10.1016/J.ERSS.2025.104422] || Out of place, scale and time? Navigating injustices across mission arenas of the German Energiewende |- | [[d:Q137901185|Q137901185]] || 2024 || [https://doi.org/10.1016/J.ESD.2024.101546 10.1016/J.ESD.2024.101546] || Characterizing 'injustices' in clean energy transitions in Africa |- | [[d:Q137901226|Q137901226]] || 2024 || [https://doi.org/10.1016/J.JCLEPRO.2024.143470 10.1016/J.JCLEPRO.2024.143470] || Energy justice and sustainable urban renewal: A systematic review of low-income old town communities |- | [[d:Q137901222|Q137901222]] || 2024 || [https://doi.org/10.1016/J.JENVMAN.2024.120804 10.1016/J.JENVMAN.2024.120804] || Forest, climate, and policy literature lacks acknowledgement of environmental justice, diversity, equity, and inclusion |- | [[d:Q115441381|Q115441381]] || 2021 || [https://doi.org/10.1016/J.RSER.2021.111504 10.1016/J.RSER.2021.111504] || Participatory methods in energy system modelling and planning – A review |- | [[d:Q137901205|Q137901205]] || 2025 || [https://doi.org/10.1016/J.RSER.2025.115892 10.1016/J.RSER.2025.115892] || A systematic review of the intersection between energy justice and human rights |- | [[d:Q137901225|Q137901225]] || 2024 || [https://doi.org/10.1017/SUS.2024.24 10.1017/SUS.2024.24] || Blue carbon as just transition? A structured literature review |- | [[d:Q137901220|Q137901220]] || 2025 || [https://doi.org/10.1017/SUS.2025.2 10.1017/SUS.2025.2] || Toward an intersectional equity approach in social-ecological transformations |- | [[d:Q137901203|Q137901203]] || 2024 || [https://doi.org/10.1080/14693062.2023.2256697 10.1080/14693062.2023.2256697] || Exploring the democracy-climate nexus: a review of correlations between democracy and climate policy performance |- | [[d:Q137901164|Q137901164]] || 2022 || [https://doi.org/10.1111/GEC3.12662 10.1111/GEC3.12662] || Creating fairer futures for sustainability transitions |- | [[d:Q137901227|Q137901227]] || 2025 || [https://doi.org/10.1139/ER-2024-0018 10.1139/ER-2024-0018] || Community engagement in nature-positive food systems programming and research in East and Southern Africa: a review |- | [[d:Q119955266|Q119955266]] || 2019 || [https://doi.org/10.1146/ANNUREV-ENVIRON-101718-033103 10.1146/ANNUREV-ENVIRON-101718-033103] || Co-Producing Sustainability: Reordering the Governance of Science, Policy, and Practice |- | [[d:Q137901206|Q137901206]] || 2023 || [https://doi.org/10.1146/ANNUREV-ENVIRON-112621-063400 10.1146/ANNUREV-ENVIRON-112621-063400] || Metrics for Decision-Making in Energy Justice |- | [[d:Q137901213|Q137901213]] || 2022 || [https://doi.org/10.1186/S13705-021-00330-4 10.1186/S13705-021-00330-4] || Mapping emergent public engagement in societal transitions: a scoping review |- | [[d:Q137901163|Q137901163]] || 2025 || [https://doi.org/10.17573/CEPAR.2025.2.09 10.17573/CEPAR.2025.2.09] || From Co-Creation to Circular Cities: Exploring Living Labs in EU Governance Frameworks - A Literature Review |- | [[d:Q137901197|Q137901197]] || 2024 || [https://doi.org/10.3390/EN17143512 10.3390/EN17143512] || A Systematic Review on the Path to Inclusive and Sustainable Energy Transitions |- | [[d:Q104887325|Q104887325]] || 2019 || [https://doi.org/10.3390/SU11041023 10.3390/SU11041023] || Deliberation and the Promise of a Deeply Democratic Sustainability Transition |- | [[d:Q137901202|Q137901202]] || 2021 || [https://doi.org/10.3390/SU13042128 10.3390/SU13042128] || A Review of Energy Communities in Sub-Saharan Africa as a Transition Pathway to Energy Democracy |- | [[d:Q137901210|Q137901210]] || 2023 || [https://doi.org/10.3390/SU15032441 10.3390/SU15032441] || Sustainable Project Governance: Scientometric Analysis and Emerging Trends |- | [[d:Q137901224|Q137901224]] || 2024 || [https://doi.org/10.3390/SU16198700 10.3390/SU16198700] || Empowering Communities to Act for a Change: A Review of the Community Empowerment Programs towards Sustainability and Resilience |} == References == {{References}} errckh6ccmlcteapjhfydfg2nq7gqwz 2812862 2812861 2026-06-05T15:33:51Z Jeanne Noiraud 1366702 /* Writing */ 2812862 wikitext text/x-wiki == Contributors == {| class="wikitable" |+ !Name !Affiliation !ORCID !Contribution |- |Adélie Ranville |IAE de Grenoble, CERAG lab (https://ror.org/0509qp208) |https://orcid.org/0000-0002-3993-6135 |Research design, database search, article screening, knowledge modelling |- |Amélie Pereira | | |Meta-data enrichement |- | | | | |} == Introduction == === Definition of living review === The concept of living systematic reviews is recent (2014), so the definition has been regularly reworked<ref name="Why1">{{Cite Q |Q40040379 }}</ref>. Living systematic reviews complement the older concept of [[literature review]]. Its objective is the same : obtain an accurate overview of the state of scientific knowledge on a subject<ref name="Why1" /><ref name="Why4">{{Cite journal |last=Akl |first=Elie A. |last2=Meerpohl |first2=Joerg J. |last3=Elliott |first3=Julian |last4=Kahale |first4=Lara A. |last5=Schünemann |first5=Holger J. |last6=Agoritsas |first6=Thomas |last7=Hilton |first7=John |last8=Perron |first8=Caroline |last9=Akl |first9=Elie |last10=Hodder |first10=Rebecca |last11=Pestridge |first11=Charlotte |last12=Albrecht |first12=Lauren |last13=Horsley |first13=Tanya |last14=Platt |first14=Joanne |last15=Armstrong |first15=Rebecca |date=2017-11 |title=Living systematic reviews: 4. Living guideline recommendations |url=https://www.wikidata.org/wiki/Q50084143 |journal=Journal of Clinical Epidemiology |language=en |volume=91 |pages=47–53 |doi=10.1016/j.jclinepi.2017.08.009}}</ref><ref name=":6">{{Citation|title=Living Systematic Reviews|url=https://doi.org/10.1007/978-1-0716-1566-9_7|publisher=Springer US|work=Meta-Research: Methods and Protocols|date=2022|access-date=2026-01-16|place=New York, NY|isbn=978-1-0716-1566-9|pages=121–134|doi=10.1007/978-1-0716-1566-9_7|language=en|first=Mark|last=Simmonds|first2=Julian H.|last2=Elliott|first3=Anneliese|last3=Synnot|first4=Tari|last4=Turner|editor-first=Evangelos|editor-last=Evangelou|editor2-first=Areti Angeliki|editor2-last=Veroniki}}</ref>. A traditional review may be obsolete by the time it is published, as new studies have emerged between the submission of the manuscript and its publication<ref name="Why1"/><ref name="Why4" /><ref name=":6" />. Living systematic reviews exists to address this common problem<ref name="Why1" /><ref name="Why4" /><ref name=":6" /><ref name=":2">https://blogs.lse.ac.uk/impactofsocialsciences/2019/05/14/the-death-of-the-literature-review-and-the-rise-of-the-dynamic-knowledge-map/</ref>. It is therefore particularly useful in rapidly evolving fields of research<ref name="Why1" /><ref name=":6" />, such as just transition. [[wikidata:Q33002955|Knowledge graphs]], a structured representation of knowledge in the form of a graph, linked together by relationships that encode explicit meanings between these entities, are very suitable for conducting living systematic reviews<ref name=":2" /><ref name="Fotopoulou">{{Cite journal|first1=Eleni |last1=Fotopoulou|first2=Ioanna|last2=Mandilara|first3=Anastasios|last3=Zafeiropoulos|first4=Chrysi|last4=Laspidou|first5=Giannis |last5=Adamos|first6=Phoebe|last6=Koundouri|first7=Symeon|last7=Papavassiliou|title=SustainGraph: A knowledge graph for tracking the progress and the interlinking among the sustainable development goals’ targets|journal=Frontiers in environmental science, Frontiers|volume=10|date=2022-10-26|issn=2296-665X|doi=10.3389/FENVS.2022.1003599|url=https://www.wikidata.org/wiki/Q117837999}}.</ref>. Advances in AI could render certain older methodological types of living systematic reviews obsoletes<ref>{{Cite journal|last=Krlev|first=Gorgi|last2=Hannigan|first2=Tim|last3=Spicer|first3=André|date=2025-01|title=What Makes a Good Review Article? Empirical Evidence From Management and Organization Research|url=https://journals.aom.org/doi/abs/10.5465/annals.2021.0051|journal=Academy of Management Annals|volume=19|issue=1|pages=376–403|doi=10.5465/annals.2021.0051|issn=1941-6520}}</ref>, as IA are useful to extract, filter and classify datas<ref>{{Cite web|url=https://arxiv.org/abs/2504.20276v1|title=Enhancing Systematic Reviews with Large Language Models: Using GPT-4 and Kimi|last=Kaptur|first=Dandan Chen|last2=Huang|first2=Yue|date=2025-04-28|website=arXiv.org|language=en|doi=10.48550/arXiv.2504.20276|access-date=2026-01-21|last3=Ji|first3=Xuejun Ryan|last4=Guo|first4=Yanhui|last5=Kaptur|first5=Bradley}}</ref><ref>{{Cite web|url=https://arxiv.org/abs/2504.20276v1|title=Enhancing Systematic Reviews with Large Language Models: Using GPT-4 and Kimi|last=Kaptur|first=Dandan Chen|last2=Huang|first2=Yue|date=2025-04-28|website=arXiv.org|language=en|doi=10.48550/arXiv.2504.20276|access-date=2026-01-21|last3=Ji|first3=Xuejun Ryan|last4=Guo|first4=Yanhui|last5=Kaptur|first5=Bradley}}</ref>. [[Large language models]] (LLM) are "on the rise" (2025), but "not yet ready for use"<ref>{{Cite journal |last=Lieberum |first=Judith-Lisa |last2=Toews |first2=Markus |last3=Metzendorf |first3=Maria-Inti |last4=Heilmeyer |first4=Felix |last5=Siemens |first5=Waldemar |last6=Haverkamp |first6=Christian |last7=Böhringer |first7=Daniel |last8=Meerpohl |first8=Joerg J. |last9=Eisele-Metzger |first9=Angelika |date=2025-05 |title=Large language models for conducting systematic reviews: on the rise, but not yet ready for use—a scoping review |url=https://www.wikidata.org/wiki/Q134545593|journal=Journal of Clinical Epidemiology |language=en |volume=181 |pages=111746 |doi=10.1016/j.jclinepi.2025.111746}}</ref>. The living review method relevant for just transition because it includes topic such as energy democracy which necessitate transdisciplinarity and consolidation of fragmented literature<ref>{{Cite journal|last=Droubi|first=Sufyan|last2=Heffron|first2=Raphael|last3=McCauley|first3=Darren|date=2022-04-01|title=A critical review of energy democracy: A failure to deliver justice?|url=https://www.wikidata.org/wiki/Q137901182|journal=Energy Research & Social Science|volume=86|doi=10.1016/J.ERSS.2021.102444}}</ref>. === Definitions of just transition : === * «a fair and equitable process of moving towards a post-carbon society’. »<ref name=":0">{{Cite journal|last=McCauley|first=Darren|last2=Heffron|first2=Raphael|date=2018-08-01|title=Just transition: Integrating climate, energy and environmental justice|url=https://www.wikidata.org/wiki/Q129947262|journal=Energy Policy|language=English|volume=119|pages=1–7|doi=10.1016/J.ENPOL.2018.04.014}}</ref>. The concept of just transition originated from global trade unions in the 1980s to promote green jobs creation as a key element of sustainability transitions<ref name=":0" />. However, scholars have broadened the use of this term to develop frameworks for analysing issues of fairness in these transitions<ref name=":0" />. The concept of just transition can be used to bridge various bodies of scholarship : climate justice, environmental justiceand energy justice<ref name=":3">{{Cite journal|last=Wang|first=Xinxin|last2=Lo|first2=Kevin|date=2021-12-01|title=Just transition: A conceptual review|url=https://www.wikidata.org/wiki/Q137209041|journal=Energy Research & Social Science|volume=82|pages=102291|doi=10.1016/J.ERSS.2021.102291}}</ref><ref name=":1">{{Cite book|url=https://www.wikidata.org/wiki/Q134545572|title=What is the “Just Transition”?|last=Heffron|first=Raphael J.|date=2021-01-01|pages=9–19|language=English}}</ref> and take into account various aspects of justice including distributional justice, procedural justice, restorative justice, recognition justice<ref name=":0" /><ref name=":3" /><ref name=":1" /><ref name=":4">{{Cite journal|last=Jenkins|first=Kirsten|last2=McCauley|first2=Darren|last3=Heffron|first3=Raphael|last4=Stephan|first4=Hannes|last5=Rehner|first5=Robert|date=2016-01-01|title=Energy justice: A conceptual review|url=https://www.wikidata.org/wiki/Q137210566|journal=Energy Research & Social Science|volume=11|pages=174–182|doi=10.1016/J.ERSS.2015.10.004}}</ref>. === Definition of Procedural justice === Procedural justice is about the fairness of decision-making processes related to transitions<ref name=":4" /> such as the inclusion of those impacted by these decisions<ref name=":5">{{Cite journal|last=Stark|first=Anthony|last2=Gale|first2=Fred|last3=Murphy-Gregory|first3=Hannah|date=2023-05-05|title=Just Transitions’ Meanings: A Systematic Review|url=https://www.wikidata.org/wiki/Q137210229|journal=Society and Natural Resources|volume=36|issue=10|pages=1277–1297|doi=10.1080/08941920.2023.2207166}}</ref>. Procedural justice can include issues of community and citizen participation in decision making, their political representation their consultation or the integration of their knowledge, with a focus on neglected population (indigenous people, women, gender and ethnic minorities<ref>{{Cite journal|last=Jenkins|first=Kirsten|last2=McCauley|first2=Darren|last3=Heffron|first3=Raphael|last4=Stephan|first4=Hannes|last5=Rehner|first5=Robert|date=2016-01-01|title=Energy justice: A conceptual review|url=https://www.wikidata.org/wiki/Q137210566|journal=Energy Research & Social Science|volume=11|pages=174–182|doi=10.1016/J.ERSS.2015.10.004}}</ref>. For example, the participation of affected communities in decisions related to the construction of new infrastructures<ref name=":0" />. == Methodology == === Wikidata and the semantic web ===<!-- Add introduction to what wikidata is and how the triplet works in a pedagogical manner Example of good description here : https://pubs.rsc.org/en/content/articlelanding/2025/np/d4np00008k#fig1 --> "A knowledge graph is a structured representation of knowledge that captures information in a machine-readable format.<ref name=":9">{{Cite journal|last=Hogan|first=Aidan|last2=Blomqvist|first2=Eva|last3=Cochez|first3=Michael|last4=D’amato|first4=Claudia|last5=Melo|first5=Gerard De|last6=Gutierrez|first6=Claudio|last7=Kirrane|first7=Sabrina|last8=Gayo|first8=José Emilio Labra|last9=Navigli|first9=Roberto|date=2022-05-31|title=Knowledge Graphs|url=https://dl.acm.org/doi/10.1145/3447772|journal=ACM Computing Surveys|language=en|volume=54|issue=4|pages=1–37|doi=10.1145/3447772|issn=0360-0300}}</ref> A knowledge graph consists of a graph or network of interconnected data points, where each data point represents a piece of information or a concept, and the relationships between them are explicitly defined. Knowledge graphs organize and store data in a format that facilitates information retrieval, data analysis, and reasoning."<ref>{{Cite journal|last=Meijer|first=David|last2=Beniddir|first2=Mehdi A.|last3=Coley|first3=Connor W.|last4=Mejri|first4=Yassine M.|last5=Öztürk|first5=Meltem|last6=Hooft|first6=Justin J. J. van der|last7=Medema|first7=Marnix H.|last8=Skiredj|first8=Adam|date=2025-04-16|title=Empowering natural product science with AI: leveraging multimodal data and knowledge graphs|url=https://pubs.rsc.org/en/content/articlelanding/2025/np/d4np00008k|journal=Natural Product Reports|language=en|volume=42|issue=4|pages=654–662|doi=10.1039/D4NP00008K|issn=1460-4752}}</ref> == Building a corpus and enriching bibliographic metadata == === Database search === We conducted preliminary searches in various databases including Web of science, Go Triple, Dimensions and OpenAlex. Web of Science was the database offering the most relevant restults and included the possibility to filter results to display only litterature reviews. Articles metadata were exported (in .ris format) and then imported into the reference manager software Zotero. {| class="wikitable" |+ !Keywords search !Database !Search date !Filters !Number of results |- |(((TS=(procedural justice OR procedural fairness OR democracy OR participation OR participatory)) AND TS=(sustainability OR energy OR climate)) AND TS=(transition OR transitions)) AND TS=(review OR reviews) |Web of Science (all databases, all dates) |December 2025 |Document type: Review Article |362 |} === Article screening === Articles abstract were then screened and we selected only articles which were litterature reviews focusing on concepts related to procedural justice as their main topics. We excluded article which were * Not related to sustainability transition (e.g. sustainable shift in..., hard science papers...) * Not literature reviews (e.g. review of policies, initiatives, cases, review notes, book review...) * Not related to procedural justice but to participation into markets, participation in eco-friendly behaviors or included justice consideration only in “future research” suggestions * Discussing participatory research methodologies (e.g. participatory modelling) without approaching it as an issue of justice, power or democracy * Discussing procedural justice concepts as key variables or key results without it being the main focus of the paper === Importing selected articles into Wikidata === To import the selected articles meta-data into Wikidata, we first ran [https://gist.github.com/zuphilip/aa9f59271fcb0807fb20c7d0110d26e4 a script] to check if any article was already present in the database. Next we used [https://gist.github.com/zuphilip/90acdc3eac4109830db1b3ab855fcb24 another script] that checks the ISSN of the publication in Wikidata and add P-Q-pairs in the extra field of Zotero. Then we exported the articles data using the "export to Wikidata QuickStatements" function of Zotero and use the QuickStatements tool to add them to Wikidata. Next we used the [[wikidata:Wikidata:Zotero/Cita|Cita]] (V1.0.0-beta.17) Zotero add-on to add articles QID in Zotero. At this point we identified that duplicates had been created in Wikidata (possibly because the initial [https://gist.github.com/zuphilip/aa9f59271fcb0807fb20c7d0110d26e4 script] did not work that well because of the recent [[wikidata:Wikidata:SPARQL_query_service/WDQS_graph_split|Graph Split]] on Wikidata). We merged duplicates on wikidata using the [[wikidata:Help:Merge|"Merge" gadget]] on Wikidata. We checked manually for duplicated statments in those items. === Article classification through meta-data enrichement ===<!-- Add : What is meta-data enrichement --> Existing review try to classify existing articles according to various criteria such as industry focus, academic discipline, geography of research sites (countries), stakeholder focus (community, consumer, worker...), type of study (case study, theory development) or methodology (quantitative, qualitative, mixt).<ref name=":5" /> We selected the most relevant properties in Wikidata to reflect these classifications : {{Wikidata entity link|P921}} to describe what the article is about, {{Wikidata entity link|P8363}} to describe its main methodology/research design and {{Wikidata entity link|P6153}} to describe its geographical focus. ==== Main subjects ==== We first read the articles abstracts and listed relevant topics and their Wikidata ID in a shared spreadsheet. These topics were : {| class="wikitable" |+ !Qid !Main topic !Description |- |[[d:Q42377797|Q42377797]] |acceptability |characteristic of a thing being subject to acceptance for some purpose |- |[[d:Q2798912|Q2798912]] |accountability |concept of responsibility in ethics, governance and decision-making |- |[[d:Q421953|Q421953]] |actor–network theory |theory within social science |- |[[d:Q84459973|Q84459973]] |affordability | |- |[[d:Q185836|Q185836]] |age of a person |time elapsed since a person was born |- |[[d:Q4764988|Q4764988]] |animal studies |field in which animals are studied in a variety of cross-disciplinary ways |- |[[d:Q4338318|Q4338318]] |awareness |state or ability to perceive, to feel, or to be conscious of events, objects, or sensory patterns |- |[[d:Q4930066|Q4930066]] |blue carbon |carbon captured by the world's coastal ocean ecosystems |- |[[d:Q430460|Q430460]] |capability approach |economic theory |- |[[d:Q7569|Q7569]] |child |human between birth and puberty |- |[[d:Q4116870|Q4116870]] |civic engagement |individual or group activity addressing issues of public concern |- |[[d:Q125928|Q125928]] |climate change |human-caused changes to climate on Earth |- |[[d:Q260607|Q260607]] |climate change adaptation |process of adjustment to actual or expected climate change and its effects, seeking to moderate or avoid harm or exploit beneficial opportunities |- |[[d:Q1291678|Q1291678]] |climate justice |term linking the climate crisis with environmental and social justice |- |[[d:Q2270945|Q2270945]] |co-creation |product or service design process in which input from consumers plays a central role |- |[[d:Q16972712|Q16972712]] |co-design |approach to design attempting to actively involve all stakeholders |- |[[d:Q16324410|Q16324410]] |coproduction |product or service design process in which input from consumers plays a central role |- |[[d:Q11024|Q11024]] |communication |act of conveying intended meaning |- |[[d:Q177634|Q177634]] |community |social unit of human organisms who share common values |- |[[d:Q5154673|Q5154673]] |community choice aggregation |alternative energy supply system |- |[[d:Q113514984|Q113514984]] |community energy |delivery of community-led renewable energy, energy demand reduction and energy supply projects |- |[[d:Q65807646|Q65807646]] |community participation |The taking part by members of a community in decisionmaking processes related to the development of their community |- |[[d:Q188843|Q188843]] |cosmopolitanism |ideology that all human beings belong to a single community, based on a shared morality |- |[[d:Q11693783|Q11693783]] |decarbonization |change of economy, especially of energy industries, towards lower carbon dioxide emissions |- |[[d:Q284289|Q284289]] |deliberative democracy |form of democracy focusing on consensus |- |[[d:Q7174|Q7174]] |democracy |form of government |- |[[d:Q552284|Q552284]] |distributive justice |concept of the socially just allocation of goods |- |[[d:Q1230584|Q1230584]] |diversity |concept in sociology and political studies |- |[[d:Q1049066|Q1049066]] |ecological economics |research field on the interdependence of human economies and natural ecosystems |- |[[d:Q8134|Q8134]] |economics |social science that studies the production, distribution, and consumption of goods and services |- |[[d:Q868575|Q868575]] |empowerment |providing increased autonomy |- |[[d:Q295865|Q295865]] |ecosystem service |benefits created by nature, forests and environmental systems |- |[[d:Q138359220|Q138359220]] |energy citizenship |involvement of citizens in energy-related decisions |- |[https://www.wikidata.org/w/index.php?title=Q131444737&redirect=no Q131444737] |community energy |[redirection] |- |[[d:Q16869822|Q16869822]] |energy consumption |amount of energy or power used |- |[[d:Q1358789|Q1358789]] |senior |elderly person |- |[[d:Q14944319|Q14944319]] |energy democracy |concept in environmental justice movement |- |[[d:Q192704|Q192704]] |energy efficiency |ratio between the useful energy output and the input of a machine |- |[[d:Q24965464|Q24965464]] |energy modeling |process of building computer models of energy systems in order to analyze them |- |[[d:Q1805337|Q1805337]] |energy policy |policy addressing energy issues |- |[[d:Q1341244|Q1341244]] |energy poverty |lack of access to modern energy services |- |[[d:Q3406659|Q3406659]] |energy production |conversion of energy from a primary source into a form useful to humans |- |[[d:Q117091181|Q117091181]] |energy justice |subconcept of economic equality |- |[[d:Q3456219|Q3456219]] |energy renovation |building works aimed at reducing energy consumption and decarbonising the energy sources used |- |[[d:Q2700433|Q2700433]] |energy security |national security considerations of energy availability |- |[[d:Q837718|Q837718]] |energy storage |capture of energy produced at one time for use at a later time |- |[[d:Q795757|Q795757]] |energy transition |long-term structural change towards sustainable energy systems |- |[[d:Q1479527|Q1479527]] |environmental justice |system of fairness |- |[[d:Q771773|Q771773]] |fairness |concept in sociology and generally the interaction of society |- |[[d:Q56395513|Q56395513]] |farming system |method of agricultural production defined by its physical practices and economic characteristics |- |[[d:Q5465532|Q5465532]] |food system |all processes and infrastructure involved in feeding a population |- |[[d:Q4421|Q4421]] |forest |dense collection of trees covering a relatively large area |- |[[d:Q48277|Q48277]] |gender |social concept which distinguish the different gender categories |- |[[d:Q1553864|Q1553864]] |governance |all of the processes of governing, whether undertaken by a government, market or network, whether over a family, tribe, formal or informal organization or territory and whether through the laws, norms, power or language of an organized society |- |[[d:Q8458|Q8458]] |human rights |inalienable fundamental rights to which a person is inherently entitled |- |[[d:Q11376059|Q11376059]] |human rights violation |act or omission which contravene the principles of human rights |- |[[d:Q103817|Q103817]] |indigenous people |first inhabitants of an area and their descendants |- |[[d:Q113561794|Q113561794]] |indigenous science |indigenous knowledge applied to the scientific method |- |[[d:Q770480|Q770480]] |injustice |quality relating to unfairness or undeserved outcomes |- |[[d:Q17142211|Q17142211]] |interactional justice |the perceived appropriateness of interpersonal treatment |- |[[d:Q1516555|Q1516555]] |intersectionnality |theoretical framework of multidimensional oppression |- |[[d:Q6316391|Q6316391]] |just transition |Framework developed by the trade union movement to encompass wide range of social interventions needed to secure decent work opportunities and a greener economy. |- |[[d:Q366139|Q366139]] |legitimation |the process of making something acceptable and normative to a group |- |[[d:Q3027857|Q3027857]] |living lab |user-centered, open innovation ecosystem integrating research and innovation in real life communities |- |[[d:Q59679511|Q59679511]] |low income |home with little money |- |[[d:Q43619|Q43619]] |natural environment |all living and non-living things occurring naturally on Earth or some region thereof |- |[[d:Q127514833|Q127514833]] |nature-positive |global goal to halt and reverse nature loss by 2030 |- |[[d:Q13023682|Q13023682]] |non-human |organism not in the genus Homo |- |[[d:Q728646|Q728646]] |partnership |arrangement in which parties agree to cooperate to advance their mutual interests |- |[[d:Q3907287|Q3907287]] |policy making |the act of developing policy |- |[[d:Q9357091|Q9357091]] |political theory |class of theory |- |[[d:Q265425|Q265425]] |postcolonialism |academic discipline |- |[[d:Q25107|Q25107]] |power |ability to influence the behavior of others |- |[[d:Q442100|Q442100]] |procedural justice |fairness in the processes that resolve disputes and allocate resources |- |[[d:Q7249406|Q7249406]] |project governance |management framework |- |[[d:Q7257735|Q7257735]] |public engagement |Policy-making practice |- |[[d:Q541936|Q541936]] |public participation |participation of citizens in various policy decisions and planning processes |- |[[d:Q6142016|Q6142016]] |recognition justice |social philosophy theory |- |[[d:Q10509953|Q10509953]] |renewable electricity |electricity from renweable sources |- |[[d:Q12705|Q12705]] |renewable energy |energy collected from renewable resources |- |[[d:Q56510941|Q56510941]] |renewable energy policy | |- |[[d:Q1165392|Q1165392]] |restorative justice |approach to justice where victims and perpetrators mediate a restitution agreement |- |[[d:Q4414036|Q4414036]] |rural population |inhabitants of rural areas or of small towns classified as rural |- |[[d:Q17152351|Q17152351]] |smart system |adaptive intelligent systems |- |[[d:Q187588|Q187588]] |social class |group of people categorized in a hierarchy based on socioeconomic factors |- |[[d:Q264892|Q264892]] |social justice |concept that discrimination recognized in society should be remedied |- |[[d:Q34749|Q34749]] |social science |academic disciplines concerned with society and the relationships between individuals in society |- |[[d:Q2930198|Q2930198]] |stakeholder participation |involvement of groups or individuals affected by the actions of an entity |- |[[d:Q125359881|Q125359881]] |sustainability transition | |- |[[d:Q219416|Q219416]] |sustainability |ability of human civilization to coexist with the biosphere in a steady state |- |[[d:Q131201|Q131201]] |sustainable development |mode of human development that meets current demands without compromising the needs of future generations |- |[[d:Q7649586|Q7649586]] |Sustainable Development Goals |set of United Nations-defined global development goals and climate change |- |[[d:Q69883|Q69883]] |urban planning |technical and political process concerned with the use of land and design of the urban environment |- |[[d:Q920600|Q920600]] |urban renewal |program of land redevelopment in cities, often where there is urban decay |- |[[d:Q3376054|Q3376054]] |vulnerable population |group of persons whose range of options is severely limited, are subjected to coercion, or who may be compromised in their ability to give informed consent |- |[[d:Q107389921|Q107389921]] |water-management | |- |[[d:Q7981051|Q7981051]] |well-being |measure of how well life is to someone or a group with factors such as health, happiness and satisfaction |- |[[d:Q467|Q467]] |woman |female adult human |- |[[d:Q188867|Q188867]] |future studies |study of possible, probable, and preferable social, technological and political futures |- |[[d:Q1038171|Q1038171]] |participatory design |active involvement of all stakeholders in the design process |} <!-- include all below items using the wikidata link template --> Then, for each article, we inferred what the {{Wikidata entity link|P921}} was from the abstracts and author provided keywords. ==== Study types ==== Our review included only litterature reviews. We first read abstracts to identify all the [https://angryloki.github.io/wikidata-graph-builder/?item=Q2412849&property=P279&mode=reverse different types of litterature reviews] present in the corpus and created wikidata items which did not exist, for example {{Wikidata entity link|Q137209848}} and {{Wikidata entity link|Q137174203}}. We improved these method items using the methodological references cited in the reviewed papers. The types of reviews were : {| class="wikitable" |+ !Qid !Study type !Description |- |[[d:Q603441|Q603441]] |bibliometrics |statistical analysis of written publications, such as books or articles |- |[[d:Q472342|Q472342]] |scientometrics |study of measuring and analysing science, technology and innovation |- |[[d:Q815382|Q815382]] |meta-analysis |statistical method that summarizes data from multiple sources |- |[[d:Q1504425|Q1504425]] |systematic review |publication type, study that gathers, analyzes, and communicates the results of research and information on a topic |- |[[d:Q2412849|Q2412849]] |literature review |process of information search and text of a review article (Q7318358), which includes the current knowledge including substantive findings, as well as theoretical and methodological contributions to a particular topic |- |[[d:Q6822263|Q6822263]] |meta-regression |statistical tool used in meta-analyses |- |[[d:Q7301211|Q7301211]] |realist evaluation |[...] |- |[[d:Q17007303|Q17007303]] |combinatorial meta-analysis |[...] |- |[[d:Q70470634|Q70470634]] |network meta-analysis |meta-analysis of randomized trials in which estimates of comparative treatment effects are visualized and interpreted from a network of interventions |- |[[d:Q101116078|Q101116078]] |scoping review |search for concepts by mapping the language and data which surrounds those concepts and adjusting the search method iteratively to synthesize evidence and assess the scope of an area of inquiry |- |[[d:Q110665014|Q110665014]] |narrative review |type of literature review, without structured method of retrieval and analysis |- |[[d:Q137174203|Q137174203]] |conceptual review |academic research aiming to review existing concepts and definitions in the litterature |- |[[d:Q137174450|Q137174450]] |critical review |type of literature review analysing strenghts, major contributions, mistakes and neglected issues in an academic field of research |- |[[d:Q137209848|Q137209848]] |integrative literature review |type of literature review |- |[[d:Q110665014|Q137211242]] |narrative review |type of literature review, without structured method of retrieval and analysis |}<!-- include all below items using the wikidata link template --> [Include list and description of types of litterature reviews] Then, we added the {{Wikidata entity link|P8363}} of each articles based on the abstract and method sections. In case of doubt, we compared our interpretation. ==== Research site ==== When an article had a specific geographical focus, we used the property {{Wikidata entity link|P6153}} to describe it. For example, the article "{{Wikidata entity link|Q137901202}}" focused on {{Wikidata entity link|Q132959}}. ==== Results ==== The table listing all the papers in the sample can be visualized [https://tabernacle.toolforge.org/?#/tab/manual/Q137211155%0A%0A%0A%0A%0A%0A%0AQ114306483%0A%0A%0A%0A%0AQ137901181%0A%0A%0A%0AQ137901182%0A%0A%0A%0A%0A%0A%0A%0AQ137901183%0A%0A%0AQ114306476%0A%0A%0A%0A%0AQ137901184%0A%0A%0A%0A%0AQ137901185%0A%0A%0A%0A%0A%0AQ137901186%0A%0A%0A%0A%0A%0A%0AQ137901187%0A%0A%0A%0A%0A%0A%0AQ137901188%0A%0A%0A%0A%0AQ137210566%0A%0A%0A%0A%0AQ114306511%0A%0A%0A%0A%0A%0AQ137901191%0A%0A%0A%0A%0AQ137901192%0A%0A%0A%0A%0AQ137901193%0A%0A%0A%0A%0AQ135979013%0A%0A%0A%0A%0A%0A%0A%0AQ137901195%0A%0A%0A%0A%0A%0AQ137901196%0A%0A%0A%0A%0A%0A%0AQ137901197%0A%0A%0A%0A%0AQ136447761%0A%0A%0A%0AQ137901199%0A%0A%0A%0A%0A%0A%0AQ129652515%0A%0A%0A%0A%0A%0A%0AQ137901201%0A%0A%0A%0A%0A%0AQ137901202%0A%0A%0A%0A%0AQ137901203%0A%0A%0A%0AQ137901204%0A%0A%0A%0A%0A%0A%0A%0AQ137901205%0A%0A%0A%0A%0AQ137901206%0A%0A%0A%0A%0A%0A%0A%0A%0AQ137901207%0A%0A%0A%0A%0AQ129203992%0A%0A%0A%0A%0A%0A%0AQ114197507%0A%0A%0A%0AQ137901161%0A%0A%0A%0A%0A%0A%0A%0AQ137901209%0A%0A%0A%0A%0A%0AQ137901210%0A%0A%0A%0A%0A%0AQ137901211%0A%0A%0A%0A%0AQ11420462%0A%0AQ137901213%0A%0A%0A%0A%0A%0A%0A%0A%0A%0A%0A%0A%0A%0A%0AQ104887325%0A%0A%0A%0A%0A%0AQ137901162%0A%0A%0AQ137901163%0A%0A%0A%0A%0AQ137901164%0A%0A%0A%0A%0A%0AQ137901215%0A%0A%0A%0A%0AQ137901216%0A%0A%0A%0A%0A%0A%0A%0A%0AQ137901217%0A%0A%0A%0A%0AQ115448818%0A%0A%0A%0A%0AQ137901218%0A%0A%0A%0AQ137901219%0A%0A%0A%0A%0AQ137901220%0A%0A%0A%0A%0A%0AQ137901221%0A%0A%0A%0A%0A%0AQ137901222%0A%0A%0A%0A%0AQ137901223%0A%0A%0AQ137901224%0A%0A%0A%0AQ137901225%0A%0A%0A%0A%0A%0A%0AQ137901226%0A%0A%0A%0AQ137901227%0A%0A%0AQ137901182/Len%3BP921%3BP6153%3BP8363%3BP50 here]. == Modelling knowledge == Concept maps can be a powerful literature review tool<ref>{{Cite journal|last=Lewis|first=John Kennedy|date=2016|title=Using ATLAS.ti to Facilitate Data Analysis for a Systematic Review of Leadership Competencies in the Completion of a Doctoral Dissertation|url=https://www.ssrn.com/abstract=2850726|journal=SSRN Electronic Journal|language=en|doi=10.2139/ssrn.2850726|issn=1556-5068}}</ref> allowing to synthetize theoretical statements about relationship between concepts<ref>{{Cite journal|last=Panniers|first=Teresa L|last2=Feuerbach|first2=Renee Daiuta|last3=Soeken|first3=Karen L|date=2003-08-01|title=Methods in informatics: using data derived from a systematic review of health care texts to develop a concept map for use in the neonatal intensive care setting|url=https://www.sciencedirect.com/science/article/pii/S1532046403000911|journal=Journal of Biomedical Informatics|series=Building Nursing Knowledge through Informatics: From Concept Representation to Data Mining|volume=36|issue=4|pages=232–239|doi=10.1016/j.jbi.2003.09.010|issn=1532-0464}}</ref>. In the present study, we explored how concept map can be used to model the knowledge present in the paper we selected. [define knowledge modelling] ==== Wikidata ontology ==== Wikidata "supports multiple coexisting classification" and allow multiple ontological frameworks to coexist.<ref name=":8">{{Cite web|url=https://arxiv.org/abs/2512.12260v1|title=A Multi-Axial Mindset for Ontology Design Lessons from Wikidata's Polyhierarchical Structure|last=Doğan|first=Ege Atacan|last2=Patel-Schneider|first2=Peter F.|date=2025-12-13|website=arXiv.org|language=en|access-date=2026-05-26}}</ref> It also supports epistemic pluralism : different worldviews can be represented in wikidata, even though scientific knowledge is preferred.<ref name=":8" /> See more on membership properties : https://www.wikidata.org/wiki/Help:Basic_membership_properties See the discussion on cause modelling : https://www.wikidata.org/wiki/Help:Modeling_causes/en ==== Conceptual modelling ==== We first reflected on what kind of wikidata properties could be used to represent concepts and theories in wikidata. Capturing the content of a concept is not straightforward and there are various approaches coming from psychology and philosophy on the matter<ref>{{Cite book|title=The Origin of Concepts|last=Carey|first=Susan|date=2011|publisher=Oxford University Press USA - OSO|isbn=978-0-19-536763-8|series=Oxford Series in Cognitive Development Ser|location=Cary}}</ref> we summarize these approaches below and examine which wikidata properties exist to represent them. * Definition: the content of a concept can be formed by its decomposition into other concepts. Many Wikidata properties can be relevant to model definitions, for example: {{Wikidata entity link|P1269}}, {{Wikidata entity link|P361}}/{{Wikidata entity link|P527}}, {{Wikidata entity link|P2670}}, {{Wikidata entity link|P1552}}/{{Wikidata entity link|P6477}}, {{Wikidata entity link|P3712}}... * Categorization: the content of a concept is formed by its illustration by an exemplar (a [[wikipedia:Prototype_theory|prototype]]) that best represent the concept. Apart from the inclusion of images to illustrate an item, Wikidata structure do not highlight exemplars. However, properties signifying relations of categorizations are among the most used with {{Wikidata entity link|P31}} and {{Wikidata entity link|P279}}. * Theory: the content of a concept is formed by its role in providing explanation of the world. Wikidata includes several properties to describe causal relationships: {{Wikidata entity link|P828}}/{{Wikidata entity link|P1542}}, {{Wikidata entity link|P1537}}/{{Wikidata entity link|P1479}}. * Essence: the content of a concept is "something" deep explaning the entity's existence and its properties. We can use concepts before knowing what they mean, and this is what allows us to revise our knowledge about it. The idea of essence is well represented by the QID of Wikidata entities: it is independent of language and definitions and we can create it before really knowing what all its properties will be. * Origin: the content of the concept is determined causally by social and historial factors (e.g. someone inventing the concept and introducing its use in a language community). This can be represented by the property {{Wikidata entity link|P3938}}. ==== Thematic networks ==== [[File:Thematic network example.jpg|thumb|547x547px|Structure of a thematic network (Source: Attride-Stirling 2001)]] A thematic network is “simply a way of organizing a thematic analysis of qualitative data”<ref name=":7">{{Cite journal|last=Attride-Stirling|first=Jennifer|date=2001-12|title=Thematic networks: an analytic tool for qualitative research|url=https://journals.sagepub.com/doi/10.1177/146879410100100307|journal=Qualitative Research|language=en|volume=1|issue=3|pages=385–405|doi=10.1177/146879410100100307|issn=1468-7941}}</ref>. It is compatible with classical coding strategies such as [[grounded theory]]<ref>{{Cite journal|last=Corbin|first=Juliet|last2=Strauss|first2=Anselm|date=1990-12-01|title=Grounded Theory Research: Procedures, Canons and Evaluative Criteria|url=https://www.degruyter.com/document/doi/10.1515/zfsoz-1990-0602/html|journal=Zeitschrift für Soziologie|language=en|volume=19|issue=6|pages=418–427|doi=10.1515/zfsoz-1990-0602|issn=2366-0325}}</ref>. Thematic networks can be used to visualise the data structure after identifying themes and help structure and interpret the data<ref name=":7" />. The principle is to assemble basic themes into more general themes. Qualitative researchers usually use {{Wikidata entity link|Q4550939}} and qualitative coding (e.g. grounded theory) to identify themes and sub-themes. However, the nature of the relationship between these various themes and sub-themes is often not specified. * ==== Causal networks ==== The use of diagrams to represent causal relationship exist in various research practices. In statistics, researchers sometime present models with boxes and arrows representing correlations and/or causations<ref>{{Cite book|url=https://mirror.vcu.edu/pub/mx/doc/mxmang10.pdf|title=Statistical Modeling|last=Neale|first=Michael C.|last2=Boker|first2=Steven M.|last3=Xie|first3=Gary|last4=Maes|first4=Hermine H.|publisher=Richmond, VA: Department of Psychiatry|year=1999|location=Virginia Commonwealth University}}</ref>. In qualitative research, building grounded theory models is about "[accounting] for not only all the major emergent concepts, themes, and dimensions, but also for their dynamic interrelationships. Speaking in classic boxes-and-arrows terms, this process amounts to assembling the constellation of boxes with a special focus on the arrows."<ref>{{Cite journal|last=Gioia|first=Dennis A.|last2=Corley|first2=Kevin G.|last3=Hamilton|first3=Aimee L.|date=2013-01|title=Seeking Qualitative Rigor in Inductive Research: Notes on the Gioia Methodology|url=https://journals.sagepub.com/doi/10.1177/1094428112452151|journal=Organizational Research Methods|language=en|volume=16|issue=1|pages=15–31|doi=10.1177/1094428112452151|issn=1094-4281}}</ref> Researchers relying on system theory also use causal loop diagram where boxes represent variables and arrows represent causal influence (positive or negative), causal relationship can "feedback" (two variables can influence each other)<ref>{{Cite book|url=https://link.springer.com/10.1007/978-3-031-01919-7_4|title=Causal Loop Diagrams|last=Barbrook-Johnson|first=Pete|last2=Penn|first2=Alexandra S.|date=2022|publisher=Springer International Publishing|isbn=978-3-031-01833-6|location=Cham|pages=47–59|language=en|doi=10.1007/978-3-031-01919-7_4}}</ref>. Wikidata includes several properties to describe causal relationships: * {{Wikidata entity link|P828}} * {{Wikidata entity link|P1542}} * {{Wikidata entity link|P1537}} * {{Wikidata entity link|P1479}} : it is difficult to identify single causes for social phenomenons, many factors having an effect on the subject item will likely be contributing factors ==== Modelling concepts ==== To model concepts related to just transition. We read the selected papers and used them as source to build a knowledge graph in wikidata. For example, the paper {{Wikidata entity link|Q137901182}} mention "Energy democracy is both an ideal and a process"<ref>{{Cite journal|last=Droubi|first=Sufyan|last2=Heffron|first2=Raphael|last3=McCauley|first3=Darren|date=2022-04-01|title=A critical review of energy democracy: A failure to deliver justice?|url=https://www.wikidata.org/wiki/Q137901182|journal=Energy Research & Social Science|volume=86|pages=4|doi=10.1016/J.ERSS.2021.102444}}</ref>, we thus entered the wikidata statement {{Wikidata entity link|Q14944319}} is an {{Wikidata entity link|P31}} {{Wikidata entity link|Q840396}}, using the paper as source. Ontology challenges: *{{Wikidata entity link|P31}}: concepts may have a dual nature because they designate at the same time an idea and the entity that this idea represent. Energy democracy is a concept, an ideal, a process and an outcome. *'''Process versus outcome :''' For material processes, the distinction between process and outcome is rather simple. For example, in Wikidata, {{Wikidata entity link|Q11629}} (practice of applying paint) is different from {{Wikidata entity link|Q3305213}} (visual artwork), and this distinction is based on the criterion "{{Wikidata entity link|Q127270577}}". However, this distinction is less straightforward for social processes that do not have an end. Such processes are ongoing and outcomes cannot be separated as clearly. * '''Ideal versus reality :''' Concepts do not have goals in themselves, but the reality they represent can have goals. To distinguish goals from the process to reach it, we used {{Wikidata entity link|P3712}} to describe ideals and {{Wikidata entity link|P2670}} to describe processes. * '''Phenomenon versus theory :''' Wikidata current items are not really suited to model "meta-research" statements. For example, modelling the idea tha the literature on energy democracy is fragmented would require creating an item representing the energy democracy literature, not just energy democracy in general. Similarly, it can be difficult to model the chronological evolution of the definition of an idea (although it could be technically possible). It is hard to represent in Wikidata affirmations related to missing knowlege, propositions of untested hypothesis, critique of existing research or research agenda recommandations * '''Origin of discourses versus origin of practices :''' To distinguish the causes of the concepts/discourses and the causes of the phenomenon itself, we used {{Wikidata entity link|P3938}} to indicate the origins of the concept or the movements promoting it. Other challenges * Wikidata does not seem to be the best tool to model quantitative statements, for example, the paper {{Wikidata entity link|Q137901196}} states that "9.8% of the final energy consumed in developing countries comes from modern renewable energy sources". Including energy data in Wikidata require using or creating specific properties (e.g. {{Wikidata entity link|P6826}}) * When concepts are not precisely defined, statements cannot be modelled correctly. For example, in the sentence "management of social affairs by voluntary and self-governing associations is deemed to ensure that both citizen choice and public welfare are best served"<ref>{{Cite journal|last=Veelen|first=Bregje van|last2=Horst|first2=Dan van der|date=2018-12-01|title=What is energy democracy? Connecting social science energy research and political theory|url=https://www.wikidata.org/wiki/Q129652515|journal=Energy Research & Social Science|language=English|volume=46|pages=19–28|doi=10.1016/J.ERSS.2018.06.010}}</ref>, "choice" could refer to {{Wikidata entity link|Q111986453}}, {{Wikidata entity link|Q1331926}}, or {{Wikidata entity link|Q12888920}} as "choice" can refer to the availability of different options, or the decision process to chose among them. Advantages : * Link toward unique identifiers for concepts, but also laws (e.g. {{Wikidata entity link|Q139764294}}) == Data visualisation == === Filter statements === * Visualize only statements using a specitic source. Example : https://w.wiki/PFqH * Visualize only items which are part to the present project (require that all items of the project include the statement {{Wikidata entity link|P6104}} {{Wikidata entity link|Q134545539}}). === Mapping a concept === Scholia request "topic in context" === Mapping sources consensus === Visualise graphs and use the number of references to determine edge thickness/weight. == Writing == To cite articles we used the [[Template:Cite Q|Cite Q template.]] Each reference is an item in Wikidata and the template retrieve the necessary data to generate the citation references below. == Future research == The analysis of knowledge graph could in theory allow to make logical deduction to generate new data<ref name=":9" />. == Data == {| class="wikitable sortable" ! QID !! Year !! DOI !! Title |- | [[d:Q137901191|Q137901191]] || 2025 || [https://doi.org/10.1002/GEO2.70040 10.1002/GEO2.70040] || Place-Based Sustainability Transformations for Just Futures: A Systematic Review |- | [[d:Q137901187|Q137901187]] || 2025 || [https://doi.org/10.1002/WCC.932 10.1002/WCC.932] || Public Communication of Climate and Justice: A Scoping Review |- | [[d:Q135979013|Q135979013]] || 2025 || [https://doi.org/10.1007/S13280-025-02202-Z 10.1007/S13280-025-02202-Z] || Participatory approaches to climate adaptation, resilience, and mitigation: A systematic review |- | [[d:Q137901223|Q137901223]] || 2022 || [https://doi.org/10.1007/S13412-021-00726-W 10.1007/S13412-021-00726-W] || A review of stakeholder participation studies in renewable electricity and water: does the resource context matter? |- | [[d:Q137901184|Q137901184]] || 2021 || [https://doi.org/10.1007/S40518-021-00184-6 10.1007/S40518-021-00184-6] || Energy Storage as an Equity Asset. |- | [[d:Q114204627|Q114204627]] || 2021 || [https://doi.org/10.1007/S43621-021-00024-Z 10.1007/S43621-021-00024-Z] || Can public awareness, knowledge and engagement improve climate change adaptation policies? |- | [[d:Q137901209|Q137901209]] || 2026 || [https://doi.org/10.1016/J.AGSY.2025.104512 10.1016/J.AGSY.2025.104512] || Designing with non-humans for agricultural systems transformation: An interdisciplinary review and framework for reflection |- | [[d:Q137901201|Q137901201]] || 2025 || [https://doi.org/10.1016/J.COPSYC.2024.101987 10.1016/J.COPSYC.2024.101987] || Individual and community catalysts for Renewable Energy Communities (RECs) development |- | [[d:Q114197507|Q114197507]] || 2022 || [https://doi.org/10.1016/J.CRM.2022.100438 10.1016/J.CRM.2022.100438] || Advancements of sustainable development goals in co-production for climate change adaptation research |- | [[d:Q129203992|Q129203992]] || 2024 || [https://doi.org/10.1016/J.EGYR.2024.01.040 10.1016/J.EGYR.2024.01.040] || Empowering energy citizenship: Exploring dimensions and drivers in citizen engagement during the energy transition |- | [[d:Q137901216|Q137901216]] || 2026 || [https://doi.org/10.1016/J.EIAR.2025.108187 10.1016/J.EIAR.2025.108187] || From participation to partnership: A systematic review of public engagement in sustainable urban planning |- | [[d:Q137210566|Q137210566]] || 2016 || [https://doi.org/10.1016/J.ERSS.2015.10.004 10.1016/J.ERSS.2015.10.004] || Energy justice: A conceptual review |- | [[d:Q115448818|Q115448818]] || 2016 || [https://doi.org/10.1016/J.ERSS.2016.04.001 10.1016/J.ERSS.2016.04.001] || Stakeholder involvement in sustainability science—A critical view |- | [[d:Q129652515|Q129652515]] || 2018 || [https://doi.org/10.1016/J.ERSS.2018.06.010 10.1016/J.ERSS.2018.06.010] || What is energy democracy? Connecting social science energy research and political theory |- | [[d:Q137901196|Q137901196]] || 2020 || [https://doi.org/10.1016/J.ERSS.2020.101716 10.1016/J.ERSS.2020.101716] || Of renewable energy, energy democracy, and sustainable development: A roadmap to accelerate the energy transition in developing countries |- | [[d:Q136447761|Q136447761]] || 2020 || [https://doi.org/10.1016/J.ERSS.2020.101768 10.1016/J.ERSS.2020.101768] || Energy democracy as a process, an outcome and a goal: A conceptual review |- | [[d:Q137901204|Q137901204]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101834 10.1016/J.ERSS.2020.101834] || Identities, innovation, and governance: A systematic review of co-creation in wind energy transitions |- | [[d:Q137901183|Q137901183]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101837 10.1016/J.ERSS.2020.101837] || Renewable energy for whom? A global systematic review of the environmental justice implications of renewable energy technologies |- | [[d:Q137901207|Q137901207]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101871 10.1016/J.ERSS.2020.101871] || Rethinking community empowerment in the energy transformation: A critical review of the definitions, drivers and outcomes |- | [[d:Q137901215|Q137901215]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101876 10.1016/J.ERSS.2020.101876] || Co-production in the wind energy sector: A systematic literature review of public engagement beyond invited stakeholder participation |- | [[d:Q114306511|Q114306511]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101907 10.1016/J.ERSS.2020.101907] || From consultation toward co-production in science and policy: A critical systematic review of participatory climate and energy initiatives |- | [[d:Q137901221|Q137901221]] || 2021 || [https://doi.org/10.1016/J.ERSS.2021.102257 10.1016/J.ERSS.2021.102257] || The challenges of engaging island communities: Lessons on renewable energy from a review of 17 case studies |- | [[d:Q137901218|Q137901218]] || 2022 || [https://doi.org/10.1016/J.ERSS.2021.102333 10.1016/J.ERSS.2021.102333] || The (in)justices of smart local energy systems: A systematic review, integrated framework, and future research agenda |- | [[d:Q137901182|Q137901182]] || 2022 || [https://doi.org/10.1016/J.ERSS.2021.102444 10.1016/J.ERSS.2021.102444] || A critical review of energy democracy: A failure to deliver justice? |- | [[d:Q114306483|Q114306483]] || 2022 || [https://doi.org/10.1016/J.ERSS.2021.102482 10.1016/J.ERSS.2021.102482] || The role of energy democracy and energy citizenship for participatory energy transitions: A comprehensive review |- | [[d:Q114306476|Q114306476]] || 2022 || [https://doi.org/10.1016/J.ERSS.2022.102714 10.1016/J.ERSS.2022.102714] || What about citizens? A literature review of citizen engagement in sustainability transitions research |- | [[d:Q137901193|Q137901193]] || 2022 || [https://doi.org/10.1016/J.ERSS.2022.102862 10.1016/J.ERSS.2022.102862] || When energy justice is contested: A systematic review of a decade of research on Sweden?s conflicted energy landscape |- | [[d:Q137901219|Q137901219]] || 2023 || [https://doi.org/10.1016/J.ERSS.2022.102913 10.1016/J.ERSS.2022.102913] || Can we optimise for justice? Reviewing the inclusion of energy justice in energy system optimisation models |- | [[d:Q137901186|Q137901186]] || 2023 || [https://doi.org/10.1016/J.ERSS.2023.103010 10.1016/J.ERSS.2023.103010] || Analysing intersections of justice with energy transitions in India- A systematic literature review |- | [[d:Q137901181|Q137901181]] || 2023 || [https://doi.org/10.1016/J.ERSS.2023.103053 10.1016/J.ERSS.2023.103053] || Fostering justice through engagement: A literature review of public engagement in energy transitions |- | [[d:Q137211155|Q137211155]] || 2023 || [https://doi.org/10.1016/J.ERSS.2023.103213 10.1016/J.ERSS.2023.103213] || A fairway to fairness: Toward a richer conceptualization of fairness perceptions for just energy transitions |- | [[d:Q137901217|Q137901217]] || 2023 || [https://doi.org/10.1016/J.ERSS.2023.103221 10.1016/J.ERSS.2023.103221] || Powering just energy transitions: A review of the justice implications of community choice aggregation |- | [[d:Q137901199|Q137901199]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104016 10.1016/J.ERSS.2025.104016] || Making energy renovations equitable: A literature review of decision-making criteria for a just energy transition in residential buildings |- | [[d:Q137901188|Q137901188]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104036 10.1016/J.ERSS.2025.104036] || Community energy justice: A review of origins, convergence, and a research agenda |- | [[d:Q137901211|Q137901211]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104067 10.1016/J.ERSS.2025.104067] || Psychological and social factors driving citizen involvement in renewable energy communities: A systematic review |- | [[d:Q137901192|Q137901192]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104149 10.1016/J.ERSS.2025.104149] || Assessing social impacts and Energy Justice along green hydrogen supply chains: a capability-based framework |- | [[d:Q137901195|Q137901195]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104422 10.1016/J.ERSS.2025.104422] || Out of place, scale and time? Navigating injustices across mission arenas of the German Energiewende |- | [[d:Q137901185|Q137901185]] || 2024 || [https://doi.org/10.1016/J.ESD.2024.101546 10.1016/J.ESD.2024.101546] || Characterizing 'injustices' in clean energy transitions in Africa |- | [[d:Q137901226|Q137901226]] || 2024 || [https://doi.org/10.1016/J.JCLEPRO.2024.143470 10.1016/J.JCLEPRO.2024.143470] || Energy justice and sustainable urban renewal: A systematic review of low-income old town communities |- | [[d:Q137901222|Q137901222]] || 2024 || [https://doi.org/10.1016/J.JENVMAN.2024.120804 10.1016/J.JENVMAN.2024.120804] || Forest, climate, and policy literature lacks acknowledgement of environmental justice, diversity, equity, and inclusion |- | [[d:Q115441381|Q115441381]] || 2021 || [https://doi.org/10.1016/J.RSER.2021.111504 10.1016/J.RSER.2021.111504] || Participatory methods in energy system modelling and planning – A review |- | [[d:Q137901205|Q137901205]] || 2025 || [https://doi.org/10.1016/J.RSER.2025.115892 10.1016/J.RSER.2025.115892] || A systematic review of the intersection between energy justice and human rights |- | [[d:Q137901225|Q137901225]] || 2024 || [https://doi.org/10.1017/SUS.2024.24 10.1017/SUS.2024.24] || Blue carbon as just transition? A structured literature review |- | [[d:Q137901220|Q137901220]] || 2025 || [https://doi.org/10.1017/SUS.2025.2 10.1017/SUS.2025.2] || Toward an intersectional equity approach in social-ecological transformations |- | [[d:Q137901203|Q137901203]] || 2024 || [https://doi.org/10.1080/14693062.2023.2256697 10.1080/14693062.2023.2256697] || Exploring the democracy-climate nexus: a review of correlations between democracy and climate policy performance |- | [[d:Q137901164|Q137901164]] || 2022 || [https://doi.org/10.1111/GEC3.12662 10.1111/GEC3.12662] || Creating fairer futures for sustainability transitions |- | [[d:Q137901227|Q137901227]] || 2025 || [https://doi.org/10.1139/ER-2024-0018 10.1139/ER-2024-0018] || Community engagement in nature-positive food systems programming and research in East and Southern Africa: a review |- | [[d:Q119955266|Q119955266]] || 2019 || [https://doi.org/10.1146/ANNUREV-ENVIRON-101718-033103 10.1146/ANNUREV-ENVIRON-101718-033103] || Co-Producing Sustainability: Reordering the Governance of Science, Policy, and Practice |- | [[d:Q137901206|Q137901206]] || 2023 || [https://doi.org/10.1146/ANNUREV-ENVIRON-112621-063400 10.1146/ANNUREV-ENVIRON-112621-063400] || Metrics for Decision-Making in Energy Justice |- | [[d:Q137901213|Q137901213]] || 2022 || [https://doi.org/10.1186/S13705-021-00330-4 10.1186/S13705-021-00330-4] || Mapping emergent public engagement in societal transitions: a scoping review |- | [[d:Q137901163|Q137901163]] || 2025 || [https://doi.org/10.17573/CEPAR.2025.2.09 10.17573/CEPAR.2025.2.09] || From Co-Creation to Circular Cities: Exploring Living Labs in EU Governance Frameworks - A Literature Review |- | [[d:Q137901197|Q137901197]] || 2024 || [https://doi.org/10.3390/EN17143512 10.3390/EN17143512] || A Systematic Review on the Path to Inclusive and Sustainable Energy Transitions |- | [[d:Q104887325|Q104887325]] || 2019 || [https://doi.org/10.3390/SU11041023 10.3390/SU11041023] || Deliberation and the Promise of a Deeply Democratic Sustainability Transition |- | [[d:Q137901202|Q137901202]] || 2021 || [https://doi.org/10.3390/SU13042128 10.3390/SU13042128] || A Review of Energy Communities in Sub-Saharan Africa as a Transition Pathway to Energy Democracy |- | [[d:Q137901210|Q137901210]] || 2023 || [https://doi.org/10.3390/SU15032441 10.3390/SU15032441] || Sustainable Project Governance: Scientometric Analysis and Emerging Trends |- | [[d:Q137901224|Q137901224]] || 2024 || [https://doi.org/10.3390/SU16198700 10.3390/SU16198700] || Empowering Communities to Act for a Change: A Review of the Community Empowerment Programs towards Sustainability and Resilience |} == References == {{References}} gm3rxcovq32q3m8guqnwd4pquucegdw 2812873 2812862 2026-06-05T15:49:17Z Jeanne Noiraud 1366702 /* Future research */ 2812873 wikitext text/x-wiki == Contributors == {| class="wikitable" |+ !Name !Affiliation !ORCID !Contribution |- |Adélie Ranville |IAE de Grenoble, CERAG lab (https://ror.org/0509qp208) |https://orcid.org/0000-0002-3993-6135 |Research design, database search, article screening, knowledge modelling |- |Amélie Pereira | | |Meta-data enrichement |- | | | | |} == Introduction == === Definition of living review === The concept of living systematic reviews is recent (2014), so the definition has been regularly reworked<ref name="Why1">{{Cite Q |Q40040379 }}</ref>. Living systematic reviews complement the older concept of [[literature review]]. Its objective is the same : obtain an accurate overview of the state of scientific knowledge on a subject<ref name="Why1" /><ref name="Why4">{{Cite journal |last=Akl |first=Elie A. |last2=Meerpohl |first2=Joerg J. |last3=Elliott |first3=Julian |last4=Kahale |first4=Lara A. |last5=Schünemann |first5=Holger J. |last6=Agoritsas |first6=Thomas |last7=Hilton |first7=John |last8=Perron |first8=Caroline |last9=Akl |first9=Elie |last10=Hodder |first10=Rebecca |last11=Pestridge |first11=Charlotte |last12=Albrecht |first12=Lauren |last13=Horsley |first13=Tanya |last14=Platt |first14=Joanne |last15=Armstrong |first15=Rebecca |date=2017-11 |title=Living systematic reviews: 4. Living guideline recommendations |url=https://www.wikidata.org/wiki/Q50084143 |journal=Journal of Clinical Epidemiology |language=en |volume=91 |pages=47–53 |doi=10.1016/j.jclinepi.2017.08.009}}</ref><ref name=":6">{{Citation|title=Living Systematic Reviews|url=https://doi.org/10.1007/978-1-0716-1566-9_7|publisher=Springer US|work=Meta-Research: Methods and Protocols|date=2022|access-date=2026-01-16|place=New York, NY|isbn=978-1-0716-1566-9|pages=121–134|doi=10.1007/978-1-0716-1566-9_7|language=en|first=Mark|last=Simmonds|first2=Julian H.|last2=Elliott|first3=Anneliese|last3=Synnot|first4=Tari|last4=Turner|editor-first=Evangelos|editor-last=Evangelou|editor2-first=Areti Angeliki|editor2-last=Veroniki}}</ref>. A traditional review may be obsolete by the time it is published, as new studies have emerged between the submission of the manuscript and its publication<ref name="Why1"/><ref name="Why4" /><ref name=":6" />. Living systematic reviews exists to address this common problem<ref name="Why1" /><ref name="Why4" /><ref name=":6" /><ref name=":2">https://blogs.lse.ac.uk/impactofsocialsciences/2019/05/14/the-death-of-the-literature-review-and-the-rise-of-the-dynamic-knowledge-map/</ref>. It is therefore particularly useful in rapidly evolving fields of research<ref name="Why1" /><ref name=":6" />, such as just transition. [[wikidata:Q33002955|Knowledge graphs]], a structured representation of knowledge in the form of a graph, linked together by relationships that encode explicit meanings between these entities, are very suitable for conducting living systematic reviews<ref name=":2" /><ref name="Fotopoulou">{{Cite journal|first1=Eleni |last1=Fotopoulou|first2=Ioanna|last2=Mandilara|first3=Anastasios|last3=Zafeiropoulos|first4=Chrysi|last4=Laspidou|first5=Giannis |last5=Adamos|first6=Phoebe|last6=Koundouri|first7=Symeon|last7=Papavassiliou|title=SustainGraph: A knowledge graph for tracking the progress and the interlinking among the sustainable development goals’ targets|journal=Frontiers in environmental science, Frontiers|volume=10|date=2022-10-26|issn=2296-665X|doi=10.3389/FENVS.2022.1003599|url=https://www.wikidata.org/wiki/Q117837999}}.</ref>. Advances in AI could render certain older methodological types of living systematic reviews obsoletes<ref>{{Cite journal|last=Krlev|first=Gorgi|last2=Hannigan|first2=Tim|last3=Spicer|first3=André|date=2025-01|title=What Makes a Good Review Article? Empirical Evidence From Management and Organization Research|url=https://journals.aom.org/doi/abs/10.5465/annals.2021.0051|journal=Academy of Management Annals|volume=19|issue=1|pages=376–403|doi=10.5465/annals.2021.0051|issn=1941-6520}}</ref>, as IA are useful to extract, filter and classify datas<ref>{{Cite web|url=https://arxiv.org/abs/2504.20276v1|title=Enhancing Systematic Reviews with Large Language Models: Using GPT-4 and Kimi|last=Kaptur|first=Dandan Chen|last2=Huang|first2=Yue|date=2025-04-28|website=arXiv.org|language=en|doi=10.48550/arXiv.2504.20276|access-date=2026-01-21|last3=Ji|first3=Xuejun Ryan|last4=Guo|first4=Yanhui|last5=Kaptur|first5=Bradley}}</ref><ref>{{Cite web|url=https://arxiv.org/abs/2504.20276v1|title=Enhancing Systematic Reviews with Large Language Models: Using GPT-4 and Kimi|last=Kaptur|first=Dandan Chen|last2=Huang|first2=Yue|date=2025-04-28|website=arXiv.org|language=en|doi=10.48550/arXiv.2504.20276|access-date=2026-01-21|last3=Ji|first3=Xuejun Ryan|last4=Guo|first4=Yanhui|last5=Kaptur|first5=Bradley}}</ref>. [[Large language models]] (LLM) are "on the rise" (2025), but "not yet ready for use"<ref>{{Cite journal |last=Lieberum |first=Judith-Lisa |last2=Toews |first2=Markus |last3=Metzendorf |first3=Maria-Inti |last4=Heilmeyer |first4=Felix |last5=Siemens |first5=Waldemar |last6=Haverkamp |first6=Christian |last7=Böhringer |first7=Daniel |last8=Meerpohl |first8=Joerg J. |last9=Eisele-Metzger |first9=Angelika |date=2025-05 |title=Large language models for conducting systematic reviews: on the rise, but not yet ready for use—a scoping review |url=https://www.wikidata.org/wiki/Q134545593|journal=Journal of Clinical Epidemiology |language=en |volume=181 |pages=111746 |doi=10.1016/j.jclinepi.2025.111746}}</ref>. The living review method relevant for just transition because it includes topic such as energy democracy which necessitate transdisciplinarity and consolidation of fragmented literature<ref>{{Cite journal|last=Droubi|first=Sufyan|last2=Heffron|first2=Raphael|last3=McCauley|first3=Darren|date=2022-04-01|title=A critical review of energy democracy: A failure to deliver justice?|url=https://www.wikidata.org/wiki/Q137901182|journal=Energy Research & Social Science|volume=86|doi=10.1016/J.ERSS.2021.102444}}</ref>. === Definitions of just transition : === * «a fair and equitable process of moving towards a post-carbon society’. »<ref name=":0">{{Cite journal|last=McCauley|first=Darren|last2=Heffron|first2=Raphael|date=2018-08-01|title=Just transition: Integrating climate, energy and environmental justice|url=https://www.wikidata.org/wiki/Q129947262|journal=Energy Policy|language=English|volume=119|pages=1–7|doi=10.1016/J.ENPOL.2018.04.014}}</ref>. The concept of just transition originated from global trade unions in the 1980s to promote green jobs creation as a key element of sustainability transitions<ref name=":0" />. However, scholars have broadened the use of this term to develop frameworks for analysing issues of fairness in these transitions<ref name=":0" />. The concept of just transition can be used to bridge various bodies of scholarship : climate justice, environmental justiceand energy justice<ref name=":3">{{Cite journal|last=Wang|first=Xinxin|last2=Lo|first2=Kevin|date=2021-12-01|title=Just transition: A conceptual review|url=https://www.wikidata.org/wiki/Q137209041|journal=Energy Research & Social Science|volume=82|pages=102291|doi=10.1016/J.ERSS.2021.102291}}</ref><ref name=":1">{{Cite book|url=https://www.wikidata.org/wiki/Q134545572|title=What is the “Just Transition”?|last=Heffron|first=Raphael J.|date=2021-01-01|pages=9–19|language=English}}</ref> and take into account various aspects of justice including distributional justice, procedural justice, restorative justice, recognition justice<ref name=":0" /><ref name=":3" /><ref name=":1" /><ref name=":4">{{Cite journal|last=Jenkins|first=Kirsten|last2=McCauley|first2=Darren|last3=Heffron|first3=Raphael|last4=Stephan|first4=Hannes|last5=Rehner|first5=Robert|date=2016-01-01|title=Energy justice: A conceptual review|url=https://www.wikidata.org/wiki/Q137210566|journal=Energy Research & Social Science|volume=11|pages=174–182|doi=10.1016/J.ERSS.2015.10.004}}</ref>. === Definition of Procedural justice === Procedural justice is about the fairness of decision-making processes related to transitions<ref name=":4" /> such as the inclusion of those impacted by these decisions<ref name=":5">{{Cite journal|last=Stark|first=Anthony|last2=Gale|first2=Fred|last3=Murphy-Gregory|first3=Hannah|date=2023-05-05|title=Just Transitions’ Meanings: A Systematic Review|url=https://www.wikidata.org/wiki/Q137210229|journal=Society and Natural Resources|volume=36|issue=10|pages=1277–1297|doi=10.1080/08941920.2023.2207166}}</ref>. Procedural justice can include issues of community and citizen participation in decision making, their political representation their consultation or the integration of their knowledge, with a focus on neglected population (indigenous people, women, gender and ethnic minorities<ref>{{Cite journal|last=Jenkins|first=Kirsten|last2=McCauley|first2=Darren|last3=Heffron|first3=Raphael|last4=Stephan|first4=Hannes|last5=Rehner|first5=Robert|date=2016-01-01|title=Energy justice: A conceptual review|url=https://www.wikidata.org/wiki/Q137210566|journal=Energy Research & Social Science|volume=11|pages=174–182|doi=10.1016/J.ERSS.2015.10.004}}</ref>. For example, the participation of affected communities in decisions related to the construction of new infrastructures<ref name=":0" />. == Methodology == === Wikidata and the semantic web ===<!-- Add introduction to what wikidata is and how the triplet works in a pedagogical manner Example of good description here : https://pubs.rsc.org/en/content/articlelanding/2025/np/d4np00008k#fig1 --> "A knowledge graph is a structured representation of knowledge that captures information in a machine-readable format.<ref name=":9">{{Cite journal|last=Hogan|first=Aidan|last2=Blomqvist|first2=Eva|last3=Cochez|first3=Michael|last4=D’amato|first4=Claudia|last5=Melo|first5=Gerard De|last6=Gutierrez|first6=Claudio|last7=Kirrane|first7=Sabrina|last8=Gayo|first8=José Emilio Labra|last9=Navigli|first9=Roberto|date=2022-05-31|title=Knowledge Graphs|url=https://dl.acm.org/doi/10.1145/3447772|journal=ACM Computing Surveys|language=en|volume=54|issue=4|pages=1–37|doi=10.1145/3447772|issn=0360-0300}}</ref> A knowledge graph consists of a graph or network of interconnected data points, where each data point represents a piece of information or a concept, and the relationships between them are explicitly defined. Knowledge graphs organize and store data in a format that facilitates information retrieval, data analysis, and reasoning."<ref>{{Cite journal|last=Meijer|first=David|last2=Beniddir|first2=Mehdi A.|last3=Coley|first3=Connor W.|last4=Mejri|first4=Yassine M.|last5=Öztürk|first5=Meltem|last6=Hooft|first6=Justin J. J. van der|last7=Medema|first7=Marnix H.|last8=Skiredj|first8=Adam|date=2025-04-16|title=Empowering natural product science with AI: leveraging multimodal data and knowledge graphs|url=https://pubs.rsc.org/en/content/articlelanding/2025/np/d4np00008k|journal=Natural Product Reports|language=en|volume=42|issue=4|pages=654–662|doi=10.1039/D4NP00008K|issn=1460-4752}}</ref> == Building a corpus and enriching bibliographic metadata == === Database search === We conducted preliminary searches in various databases including Web of science, Go Triple, Dimensions and OpenAlex. Web of Science was the database offering the most relevant restults and included the possibility to filter results to display only litterature reviews. Articles metadata were exported (in .ris format) and then imported into the reference manager software Zotero. {| class="wikitable" |+ !Keywords search !Database !Search date !Filters !Number of results |- |(((TS=(procedural justice OR procedural fairness OR democracy OR participation OR participatory)) AND TS=(sustainability OR energy OR climate)) AND TS=(transition OR transitions)) AND TS=(review OR reviews) |Web of Science (all databases, all dates) |December 2025 |Document type: Review Article |362 |} === Article screening === Articles abstract were then screened and we selected only articles which were litterature reviews focusing on concepts related to procedural justice as their main topics. We excluded article which were * Not related to sustainability transition (e.g. sustainable shift in..., hard science papers...) * Not literature reviews (e.g. review of policies, initiatives, cases, review notes, book review...) * Not related to procedural justice but to participation into markets, participation in eco-friendly behaviors or included justice consideration only in “future research” suggestions * Discussing participatory research methodologies (e.g. participatory modelling) without approaching it as an issue of justice, power or democracy * Discussing procedural justice concepts as key variables or key results without it being the main focus of the paper === Importing selected articles into Wikidata === To import the selected articles meta-data into Wikidata, we first ran [https://gist.github.com/zuphilip/aa9f59271fcb0807fb20c7d0110d26e4 a script] to check if any article was already present in the database. Next we used [https://gist.github.com/zuphilip/90acdc3eac4109830db1b3ab855fcb24 another script] that checks the ISSN of the publication in Wikidata and add P-Q-pairs in the extra field of Zotero. Then we exported the articles data using the "export to Wikidata QuickStatements" function of Zotero and use the QuickStatements tool to add them to Wikidata. Next we used the [[wikidata:Wikidata:Zotero/Cita|Cita]] (V1.0.0-beta.17) Zotero add-on to add articles QID in Zotero. At this point we identified that duplicates had been created in Wikidata (possibly because the initial [https://gist.github.com/zuphilip/aa9f59271fcb0807fb20c7d0110d26e4 script] did not work that well because of the recent [[wikidata:Wikidata:SPARQL_query_service/WDQS_graph_split|Graph Split]] on Wikidata). We merged duplicates on wikidata using the [[wikidata:Help:Merge|"Merge" gadget]] on Wikidata. We checked manually for duplicated statments in those items. === Article classification through meta-data enrichement ===<!-- Add : What is meta-data enrichement --> Existing review try to classify existing articles according to various criteria such as industry focus, academic discipline, geography of research sites (countries), stakeholder focus (community, consumer, worker...), type of study (case study, theory development) or methodology (quantitative, qualitative, mixt).<ref name=":5" /> We selected the most relevant properties in Wikidata to reflect these classifications : {{Wikidata entity link|P921}} to describe what the article is about, {{Wikidata entity link|P8363}} to describe its main methodology/research design and {{Wikidata entity link|P6153}} to describe its geographical focus. ==== Main subjects ==== We first read the articles abstracts and listed relevant topics and their Wikidata ID in a shared spreadsheet. These topics were : {| class="wikitable" |+ !Qid !Main topic !Description |- |[[d:Q42377797|Q42377797]] |acceptability |characteristic of a thing being subject to acceptance for some purpose |- |[[d:Q2798912|Q2798912]] |accountability |concept of responsibility in ethics, governance and decision-making |- |[[d:Q421953|Q421953]] |actor–network theory |theory within social science |- |[[d:Q84459973|Q84459973]] |affordability | |- |[[d:Q185836|Q185836]] |age of a person |time elapsed since a person was born |- |[[d:Q4764988|Q4764988]] |animal studies |field in which animals are studied in a variety of cross-disciplinary ways |- |[[d:Q4338318|Q4338318]] |awareness |state or ability to perceive, to feel, or to be conscious of events, objects, or sensory patterns |- |[[d:Q4930066|Q4930066]] |blue carbon |carbon captured by the world's coastal ocean ecosystems |- |[[d:Q430460|Q430460]] |capability approach |economic theory |- |[[d:Q7569|Q7569]] |child |human between birth and puberty |- |[[d:Q4116870|Q4116870]] |civic engagement |individual or group activity addressing issues of public concern |- |[[d:Q125928|Q125928]] |climate change |human-caused changes to climate on Earth |- |[[d:Q260607|Q260607]] |climate change adaptation |process of adjustment to actual or expected climate change and its effects, seeking to moderate or avoid harm or exploit beneficial opportunities |- |[[d:Q1291678|Q1291678]] |climate justice |term linking the climate crisis with environmental and social justice |- |[[d:Q2270945|Q2270945]] |co-creation |product or service design process in which input from consumers plays a central role |- |[[d:Q16972712|Q16972712]] |co-design |approach to design attempting to actively involve all stakeholders |- |[[d:Q16324410|Q16324410]] |coproduction |product or service design process in which input from consumers plays a central role |- |[[d:Q11024|Q11024]] |communication |act of conveying intended meaning |- |[[d:Q177634|Q177634]] |community |social unit of human organisms who share common values |- |[[d:Q5154673|Q5154673]] |community choice aggregation |alternative energy supply system |- |[[d:Q113514984|Q113514984]] |community energy |delivery of community-led renewable energy, energy demand reduction and energy supply projects |- |[[d:Q65807646|Q65807646]] |community participation |The taking part by members of a community in decisionmaking processes related to the development of their community |- |[[d:Q188843|Q188843]] |cosmopolitanism |ideology that all human beings belong to a single community, based on a shared morality |- |[[d:Q11693783|Q11693783]] |decarbonization |change of economy, especially of energy industries, towards lower carbon dioxide emissions |- |[[d:Q284289|Q284289]] |deliberative democracy |form of democracy focusing on consensus |- |[[d:Q7174|Q7174]] |democracy |form of government |- |[[d:Q552284|Q552284]] |distributive justice |concept of the socially just allocation of goods |- |[[d:Q1230584|Q1230584]] |diversity |concept in sociology and political studies |- |[[d:Q1049066|Q1049066]] |ecological economics |research field on the interdependence of human economies and natural ecosystems |- |[[d:Q8134|Q8134]] |economics |social science that studies the production, distribution, and consumption of goods and services |- |[[d:Q868575|Q868575]] |empowerment |providing increased autonomy |- |[[d:Q295865|Q295865]] |ecosystem service |benefits created by nature, forests and environmental systems |- |[[d:Q138359220|Q138359220]] |energy citizenship |involvement of citizens in energy-related decisions |- |[https://www.wikidata.org/w/index.php?title=Q131444737&redirect=no Q131444737] |community energy |[redirection] |- |[[d:Q16869822|Q16869822]] |energy consumption |amount of energy or power used |- |[[d:Q1358789|Q1358789]] |senior |elderly person |- |[[d:Q14944319|Q14944319]] |energy democracy |concept in environmental justice movement |- |[[d:Q192704|Q192704]] |energy efficiency |ratio between the useful energy output and the input of a machine |- |[[d:Q24965464|Q24965464]] |energy modeling |process of building computer models of energy systems in order to analyze them |- |[[d:Q1805337|Q1805337]] |energy policy |policy addressing energy issues |- |[[d:Q1341244|Q1341244]] |energy poverty |lack of access to modern energy services |- |[[d:Q3406659|Q3406659]] |energy production |conversion of energy from a primary source into a form useful to humans |- |[[d:Q117091181|Q117091181]] |energy justice |subconcept of economic equality |- |[[d:Q3456219|Q3456219]] |energy renovation |building works aimed at reducing energy consumption and decarbonising the energy sources used |- |[[d:Q2700433|Q2700433]] |energy security |national security considerations of energy availability |- |[[d:Q837718|Q837718]] |energy storage |capture of energy produced at one time for use at a later time |- |[[d:Q795757|Q795757]] |energy transition |long-term structural change towards sustainable energy systems |- |[[d:Q1479527|Q1479527]] |environmental justice |system of fairness |- |[[d:Q771773|Q771773]] |fairness |concept in sociology and generally the interaction of society |- |[[d:Q56395513|Q56395513]] |farming system |method of agricultural production defined by its physical practices and economic characteristics |- |[[d:Q5465532|Q5465532]] |food system |all processes and infrastructure involved in feeding a population |- |[[d:Q4421|Q4421]] |forest |dense collection of trees covering a relatively large area |- |[[d:Q48277|Q48277]] |gender |social concept which distinguish the different gender categories |- |[[d:Q1553864|Q1553864]] |governance |all of the processes of governing, whether undertaken by a government, market or network, whether over a family, tribe, formal or informal organization or territory and whether through the laws, norms, power or language of an organized society |- |[[d:Q8458|Q8458]] |human rights |inalienable fundamental rights to which a person is inherently entitled |- |[[d:Q11376059|Q11376059]] |human rights violation |act or omission which contravene the principles of human rights |- |[[d:Q103817|Q103817]] |indigenous people |first inhabitants of an area and their descendants |- |[[d:Q113561794|Q113561794]] |indigenous science |indigenous knowledge applied to the scientific method |- |[[d:Q770480|Q770480]] |injustice |quality relating to unfairness or undeserved outcomes |- |[[d:Q17142211|Q17142211]] |interactional justice |the perceived appropriateness of interpersonal treatment |- |[[d:Q1516555|Q1516555]] |intersectionnality |theoretical framework of multidimensional oppression |- |[[d:Q6316391|Q6316391]] |just transition |Framework developed by the trade union movement to encompass wide range of social interventions needed to secure decent work opportunities and a greener economy. |- |[[d:Q366139|Q366139]] |legitimation |the process of making something acceptable and normative to a group |- |[[d:Q3027857|Q3027857]] |living lab |user-centered, open innovation ecosystem integrating research and innovation in real life communities |- |[[d:Q59679511|Q59679511]] |low income |home with little money |- |[[d:Q43619|Q43619]] |natural environment |all living and non-living things occurring naturally on Earth or some region thereof |- |[[d:Q127514833|Q127514833]] |nature-positive |global goal to halt and reverse nature loss by 2030 |- |[[d:Q13023682|Q13023682]] |non-human |organism not in the genus Homo |- |[[d:Q728646|Q728646]] |partnership |arrangement in which parties agree to cooperate to advance their mutual interests |- |[[d:Q3907287|Q3907287]] |policy making |the act of developing policy |- |[[d:Q9357091|Q9357091]] |political theory |class of theory |- |[[d:Q265425|Q265425]] |postcolonialism |academic discipline |- |[[d:Q25107|Q25107]] |power |ability to influence the behavior of others |- |[[d:Q442100|Q442100]] |procedural justice |fairness in the processes that resolve disputes and allocate resources |- |[[d:Q7249406|Q7249406]] |project governance |management framework |- |[[d:Q7257735|Q7257735]] |public engagement |Policy-making practice |- |[[d:Q541936|Q541936]] |public participation |participation of citizens in various policy decisions and planning processes |- |[[d:Q6142016|Q6142016]] |recognition justice |social philosophy theory |- |[[d:Q10509953|Q10509953]] |renewable electricity |electricity from renweable sources |- |[[d:Q12705|Q12705]] |renewable energy |energy collected from renewable resources |- |[[d:Q56510941|Q56510941]] |renewable energy policy | |- |[[d:Q1165392|Q1165392]] |restorative justice |approach to justice where victims and perpetrators mediate a restitution agreement |- |[[d:Q4414036|Q4414036]] |rural population |inhabitants of rural areas or of small towns classified as rural |- |[[d:Q17152351|Q17152351]] |smart system |adaptive intelligent systems |- |[[d:Q187588|Q187588]] |social class |group of people categorized in a hierarchy based on socioeconomic factors |- |[[d:Q264892|Q264892]] |social justice |concept that discrimination recognized in society should be remedied |- |[[d:Q34749|Q34749]] |social science |academic disciplines concerned with society and the relationships between individuals in society |- |[[d:Q2930198|Q2930198]] |stakeholder participation |involvement of groups or individuals affected by the actions of an entity |- |[[d:Q125359881|Q125359881]] |sustainability transition | |- |[[d:Q219416|Q219416]] |sustainability |ability of human civilization to coexist with the biosphere in a steady state |- |[[d:Q131201|Q131201]] |sustainable development |mode of human development that meets current demands without compromising the needs of future generations |- |[[d:Q7649586|Q7649586]] |Sustainable Development Goals |set of United Nations-defined global development goals and climate change |- |[[d:Q69883|Q69883]] |urban planning |technical and political process concerned with the use of land and design of the urban environment |- |[[d:Q920600|Q920600]] |urban renewal |program of land redevelopment in cities, often where there is urban decay |- |[[d:Q3376054|Q3376054]] |vulnerable population |group of persons whose range of options is severely limited, are subjected to coercion, or who may be compromised in their ability to give informed consent |- |[[d:Q107389921|Q107389921]] |water-management | |- |[[d:Q7981051|Q7981051]] |well-being |measure of how well life is to someone or a group with factors such as health, happiness and satisfaction |- |[[d:Q467|Q467]] |woman |female adult human |- |[[d:Q188867|Q188867]] |future studies |study of possible, probable, and preferable social, technological and political futures |- |[[d:Q1038171|Q1038171]] |participatory design |active involvement of all stakeholders in the design process |} <!-- include all below items using the wikidata link template --> Then, for each article, we inferred what the {{Wikidata entity link|P921}} was from the abstracts and author provided keywords. ==== Study types ==== Our review included only litterature reviews. We first read abstracts to identify all the [https://angryloki.github.io/wikidata-graph-builder/?item=Q2412849&property=P279&mode=reverse different types of litterature reviews] present in the corpus and created wikidata items which did not exist, for example {{Wikidata entity link|Q137209848}} and {{Wikidata entity link|Q137174203}}. We improved these method items using the methodological references cited in the reviewed papers. The types of reviews were : {| class="wikitable" |+ !Qid !Study type !Description |- |[[d:Q603441|Q603441]] |bibliometrics |statistical analysis of written publications, such as books or articles |- |[[d:Q472342|Q472342]] |scientometrics |study of measuring and analysing science, technology and innovation |- |[[d:Q815382|Q815382]] |meta-analysis |statistical method that summarizes data from multiple sources |- |[[d:Q1504425|Q1504425]] |systematic review |publication type, study that gathers, analyzes, and communicates the results of research and information on a topic |- |[[d:Q2412849|Q2412849]] |literature review |process of information search and text of a review article (Q7318358), which includes the current knowledge including substantive findings, as well as theoretical and methodological contributions to a particular topic |- |[[d:Q6822263|Q6822263]] |meta-regression |statistical tool used in meta-analyses |- |[[d:Q7301211|Q7301211]] |realist evaluation |[...] |- |[[d:Q17007303|Q17007303]] |combinatorial meta-analysis |[...] |- |[[d:Q70470634|Q70470634]] |network meta-analysis |meta-analysis of randomized trials in which estimates of comparative treatment effects are visualized and interpreted from a network of interventions |- |[[d:Q101116078|Q101116078]] |scoping review |search for concepts by mapping the language and data which surrounds those concepts and adjusting the search method iteratively to synthesize evidence and assess the scope of an area of inquiry |- |[[d:Q110665014|Q110665014]] |narrative review |type of literature review, without structured method of retrieval and analysis |- |[[d:Q137174203|Q137174203]] |conceptual review |academic research aiming to review existing concepts and definitions in the litterature |- |[[d:Q137174450|Q137174450]] |critical review |type of literature review analysing strenghts, major contributions, mistakes and neglected issues in an academic field of research |- |[[d:Q137209848|Q137209848]] |integrative literature review |type of literature review |- |[[d:Q110665014|Q137211242]] |narrative review |type of literature review, without structured method of retrieval and analysis |}<!-- include all below items using the wikidata link template --> [Include list and description of types of litterature reviews] Then, we added the {{Wikidata entity link|P8363}} of each articles based on the abstract and method sections. In case of doubt, we compared our interpretation. ==== Research site ==== When an article had a specific geographical focus, we used the property {{Wikidata entity link|P6153}} to describe it. For example, the article "{{Wikidata entity link|Q137901202}}" focused on {{Wikidata entity link|Q132959}}. ==== Results ==== The table listing all the papers in the sample can be visualized [https://tabernacle.toolforge.org/?#/tab/manual/Q137211155%0A%0A%0A%0A%0A%0A%0AQ114306483%0A%0A%0A%0A%0AQ137901181%0A%0A%0A%0AQ137901182%0A%0A%0A%0A%0A%0A%0A%0AQ137901183%0A%0A%0AQ114306476%0A%0A%0A%0A%0AQ137901184%0A%0A%0A%0A%0AQ137901185%0A%0A%0A%0A%0A%0AQ137901186%0A%0A%0A%0A%0A%0A%0AQ137901187%0A%0A%0A%0A%0A%0A%0AQ137901188%0A%0A%0A%0A%0AQ137210566%0A%0A%0A%0A%0AQ114306511%0A%0A%0A%0A%0A%0AQ137901191%0A%0A%0A%0A%0AQ137901192%0A%0A%0A%0A%0AQ137901193%0A%0A%0A%0A%0AQ135979013%0A%0A%0A%0A%0A%0A%0A%0AQ137901195%0A%0A%0A%0A%0A%0AQ137901196%0A%0A%0A%0A%0A%0A%0AQ137901197%0A%0A%0A%0A%0AQ136447761%0A%0A%0A%0AQ137901199%0A%0A%0A%0A%0A%0A%0AQ129652515%0A%0A%0A%0A%0A%0A%0AQ137901201%0A%0A%0A%0A%0A%0AQ137901202%0A%0A%0A%0A%0AQ137901203%0A%0A%0A%0AQ137901204%0A%0A%0A%0A%0A%0A%0A%0AQ137901205%0A%0A%0A%0A%0AQ137901206%0A%0A%0A%0A%0A%0A%0A%0A%0AQ137901207%0A%0A%0A%0A%0AQ129203992%0A%0A%0A%0A%0A%0A%0AQ114197507%0A%0A%0A%0AQ137901161%0A%0A%0A%0A%0A%0A%0A%0AQ137901209%0A%0A%0A%0A%0A%0AQ137901210%0A%0A%0A%0A%0A%0AQ137901211%0A%0A%0A%0A%0AQ11420462%0A%0AQ137901213%0A%0A%0A%0A%0A%0A%0A%0A%0A%0A%0A%0A%0A%0A%0AQ104887325%0A%0A%0A%0A%0A%0AQ137901162%0A%0A%0AQ137901163%0A%0A%0A%0A%0AQ137901164%0A%0A%0A%0A%0A%0AQ137901215%0A%0A%0A%0A%0AQ137901216%0A%0A%0A%0A%0A%0A%0A%0A%0AQ137901217%0A%0A%0A%0A%0AQ115448818%0A%0A%0A%0A%0AQ137901218%0A%0A%0A%0AQ137901219%0A%0A%0A%0A%0AQ137901220%0A%0A%0A%0A%0A%0AQ137901221%0A%0A%0A%0A%0A%0AQ137901222%0A%0A%0A%0A%0AQ137901223%0A%0A%0AQ137901224%0A%0A%0A%0AQ137901225%0A%0A%0A%0A%0A%0A%0AQ137901226%0A%0A%0A%0AQ137901227%0A%0A%0AQ137901182/Len%3BP921%3BP6153%3BP8363%3BP50 here]. == Modelling knowledge == Concept maps can be a powerful literature review tool<ref>{{Cite journal|last=Lewis|first=John Kennedy|date=2016|title=Using ATLAS.ti to Facilitate Data Analysis for a Systematic Review of Leadership Competencies in the Completion of a Doctoral Dissertation|url=https://www.ssrn.com/abstract=2850726|journal=SSRN Electronic Journal|language=en|doi=10.2139/ssrn.2850726|issn=1556-5068}}</ref> allowing to synthetize theoretical statements about relationship between concepts<ref>{{Cite journal|last=Panniers|first=Teresa L|last2=Feuerbach|first2=Renee Daiuta|last3=Soeken|first3=Karen L|date=2003-08-01|title=Methods in informatics: using data derived from a systematic review of health care texts to develop a concept map for use in the neonatal intensive care setting|url=https://www.sciencedirect.com/science/article/pii/S1532046403000911|journal=Journal of Biomedical Informatics|series=Building Nursing Knowledge through Informatics: From Concept Representation to Data Mining|volume=36|issue=4|pages=232–239|doi=10.1016/j.jbi.2003.09.010|issn=1532-0464}}</ref>. In the present study, we explored how concept map can be used to model the knowledge present in the paper we selected. [define knowledge modelling] ==== Wikidata ontology ==== Wikidata "supports multiple coexisting classification" and allow multiple ontological frameworks to coexist.<ref name=":8">{{Cite web|url=https://arxiv.org/abs/2512.12260v1|title=A Multi-Axial Mindset for Ontology Design Lessons from Wikidata's Polyhierarchical Structure|last=Doğan|first=Ege Atacan|last2=Patel-Schneider|first2=Peter F.|date=2025-12-13|website=arXiv.org|language=en|access-date=2026-05-26}}</ref> It also supports epistemic pluralism : different worldviews can be represented in wikidata, even though scientific knowledge is preferred.<ref name=":8" /> See more on membership properties : https://www.wikidata.org/wiki/Help:Basic_membership_properties See the discussion on cause modelling : https://www.wikidata.org/wiki/Help:Modeling_causes/en ==== Conceptual modelling ==== We first reflected on what kind of wikidata properties could be used to represent concepts and theories in wikidata. Capturing the content of a concept is not straightforward and there are various approaches coming from psychology and philosophy on the matter<ref>{{Cite book|title=The Origin of Concepts|last=Carey|first=Susan|date=2011|publisher=Oxford University Press USA - OSO|isbn=978-0-19-536763-8|series=Oxford Series in Cognitive Development Ser|location=Cary}}</ref> we summarize these approaches below and examine which wikidata properties exist to represent them. * Definition: the content of a concept can be formed by its decomposition into other concepts. Many Wikidata properties can be relevant to model definitions, for example: {{Wikidata entity link|P1269}}, {{Wikidata entity link|P361}}/{{Wikidata entity link|P527}}, {{Wikidata entity link|P2670}}, {{Wikidata entity link|P1552}}/{{Wikidata entity link|P6477}}, {{Wikidata entity link|P3712}}... * Categorization: the content of a concept is formed by its illustration by an exemplar (a [[wikipedia:Prototype_theory|prototype]]) that best represent the concept. Apart from the inclusion of images to illustrate an item, Wikidata structure do not highlight exemplars. However, properties signifying relations of categorizations are among the most used with {{Wikidata entity link|P31}} and {{Wikidata entity link|P279}}. * Theory: the content of a concept is formed by its role in providing explanation of the world. Wikidata includes several properties to describe causal relationships: {{Wikidata entity link|P828}}/{{Wikidata entity link|P1542}}, {{Wikidata entity link|P1537}}/{{Wikidata entity link|P1479}}. * Essence: the content of a concept is "something" deep explaning the entity's existence and its properties. We can use concepts before knowing what they mean, and this is what allows us to revise our knowledge about it. The idea of essence is well represented by the QID of Wikidata entities: it is independent of language and definitions and we can create it before really knowing what all its properties will be. * Origin: the content of the concept is determined causally by social and historial factors (e.g. someone inventing the concept and introducing its use in a language community). This can be represented by the property {{Wikidata entity link|P3938}}. ==== Thematic networks ==== [[File:Thematic network example.jpg|thumb|547x547px|Structure of a thematic network (Source: Attride-Stirling 2001)]] A thematic network is “simply a way of organizing a thematic analysis of qualitative data”<ref name=":7">{{Cite journal|last=Attride-Stirling|first=Jennifer|date=2001-12|title=Thematic networks: an analytic tool for qualitative research|url=https://journals.sagepub.com/doi/10.1177/146879410100100307|journal=Qualitative Research|language=en|volume=1|issue=3|pages=385–405|doi=10.1177/146879410100100307|issn=1468-7941}}</ref>. It is compatible with classical coding strategies such as [[grounded theory]]<ref>{{Cite journal|last=Corbin|first=Juliet|last2=Strauss|first2=Anselm|date=1990-12-01|title=Grounded Theory Research: Procedures, Canons and Evaluative Criteria|url=https://www.degruyter.com/document/doi/10.1515/zfsoz-1990-0602/html|journal=Zeitschrift für Soziologie|language=en|volume=19|issue=6|pages=418–427|doi=10.1515/zfsoz-1990-0602|issn=2366-0325}}</ref>. Thematic networks can be used to visualise the data structure after identifying themes and help structure and interpret the data<ref name=":7" />. The principle is to assemble basic themes into more general themes. Qualitative researchers usually use {{Wikidata entity link|Q4550939}} and qualitative coding (e.g. grounded theory) to identify themes and sub-themes. However, the nature of the relationship between these various themes and sub-themes is often not specified. * ==== Causal networks ==== The use of diagrams to represent causal relationship exist in various research practices. In statistics, researchers sometime present models with boxes and arrows representing correlations and/or causations<ref>{{Cite book|url=https://mirror.vcu.edu/pub/mx/doc/mxmang10.pdf|title=Statistical Modeling|last=Neale|first=Michael C.|last2=Boker|first2=Steven M.|last3=Xie|first3=Gary|last4=Maes|first4=Hermine H.|publisher=Richmond, VA: Department of Psychiatry|year=1999|location=Virginia Commonwealth University}}</ref>. In qualitative research, building grounded theory models is about "[accounting] for not only all the major emergent concepts, themes, and dimensions, but also for their dynamic interrelationships. Speaking in classic boxes-and-arrows terms, this process amounts to assembling the constellation of boxes with a special focus on the arrows."<ref>{{Cite journal|last=Gioia|first=Dennis A.|last2=Corley|first2=Kevin G.|last3=Hamilton|first3=Aimee L.|date=2013-01|title=Seeking Qualitative Rigor in Inductive Research: Notes on the Gioia Methodology|url=https://journals.sagepub.com/doi/10.1177/1094428112452151|journal=Organizational Research Methods|language=en|volume=16|issue=1|pages=15–31|doi=10.1177/1094428112452151|issn=1094-4281}}</ref> Researchers relying on system theory also use causal loop diagram where boxes represent variables and arrows represent causal influence (positive or negative), causal relationship can "feedback" (two variables can influence each other)<ref>{{Cite book|url=https://link.springer.com/10.1007/978-3-031-01919-7_4|title=Causal Loop Diagrams|last=Barbrook-Johnson|first=Pete|last2=Penn|first2=Alexandra S.|date=2022|publisher=Springer International Publishing|isbn=978-3-031-01833-6|location=Cham|pages=47–59|language=en|doi=10.1007/978-3-031-01919-7_4}}</ref>. Wikidata includes several properties to describe causal relationships: * {{Wikidata entity link|P828}} * {{Wikidata entity link|P1542}} * {{Wikidata entity link|P1537}} * {{Wikidata entity link|P1479}} : it is difficult to identify single causes for social phenomenons, many factors having an effect on the subject item will likely be contributing factors ==== Modelling concepts ==== To model concepts related to just transition. We read the selected papers and used them as source to build a knowledge graph in wikidata. For example, the paper {{Wikidata entity link|Q137901182}} mention "Energy democracy is both an ideal and a process"<ref>{{Cite journal|last=Droubi|first=Sufyan|last2=Heffron|first2=Raphael|last3=McCauley|first3=Darren|date=2022-04-01|title=A critical review of energy democracy: A failure to deliver justice?|url=https://www.wikidata.org/wiki/Q137901182|journal=Energy Research & Social Science|volume=86|pages=4|doi=10.1016/J.ERSS.2021.102444}}</ref>, we thus entered the wikidata statement {{Wikidata entity link|Q14944319}} is an {{Wikidata entity link|P31}} {{Wikidata entity link|Q840396}}, using the paper as source. Ontology challenges: *{{Wikidata entity link|P31}}: concepts may have a dual nature because they designate at the same time an idea and the entity that this idea represent. Energy democracy is a concept, an ideal, a process and an outcome. *'''Process versus outcome :''' For material processes, the distinction between process and outcome is rather simple. For example, in Wikidata, {{Wikidata entity link|Q11629}} (practice of applying paint) is different from {{Wikidata entity link|Q3305213}} (visual artwork), and this distinction is based on the criterion "{{Wikidata entity link|Q127270577}}". However, this distinction is less straightforward for social processes that do not have an end. Such processes are ongoing and outcomes cannot be separated as clearly. * '''Ideal versus reality :''' Concepts do not have goals in themselves, but the reality they represent can have goals. To distinguish goals from the process to reach it, we used {{Wikidata entity link|P3712}} to describe ideals and {{Wikidata entity link|P2670}} to describe processes. * '''Phenomenon versus theory :''' Wikidata current items are not really suited to model "meta-research" statements. For example, modelling the idea tha the literature on energy democracy is fragmented would require creating an item representing the energy democracy literature, not just energy democracy in general. Similarly, it can be difficult to model the chronological evolution of the definition of an idea (although it could be technically possible). It is hard to represent in Wikidata affirmations related to missing knowlege, propositions of untested hypothesis, critique of existing research or research agenda recommandations * '''Origin of discourses versus origin of practices :''' To distinguish the causes of the concepts/discourses and the causes of the phenomenon itself, we used {{Wikidata entity link|P3938}} to indicate the origins of the concept or the movements promoting it. Other challenges * Wikidata does not seem to be the best tool to model quantitative statements, for example, the paper {{Wikidata entity link|Q137901196}} states that "9.8% of the final energy consumed in developing countries comes from modern renewable energy sources". Including energy data in Wikidata require using or creating specific properties (e.g. {{Wikidata entity link|P6826}}) * When concepts are not precisely defined, statements cannot be modelled correctly. For example, in the sentence "management of social affairs by voluntary and self-governing associations is deemed to ensure that both citizen choice and public welfare are best served"<ref>{{Cite journal|last=Veelen|first=Bregje van|last2=Horst|first2=Dan van der|date=2018-12-01|title=What is energy democracy? Connecting social science energy research and political theory|url=https://www.wikidata.org/wiki/Q129652515|journal=Energy Research & Social Science|language=English|volume=46|pages=19–28|doi=10.1016/J.ERSS.2018.06.010}}</ref>, "choice" could refer to {{Wikidata entity link|Q111986453}}, {{Wikidata entity link|Q1331926}}, or {{Wikidata entity link|Q12888920}} as "choice" can refer to the availability of different options, or the decision process to chose among them. Advantages : * Link toward unique identifiers for concepts, but also laws (e.g. {{Wikidata entity link|Q139764294}}) == Data visualisation == === Filter statements === * Visualize only statements using a specitic source. Example : https://w.wiki/PFqH * Visualize only items which are part to the present project (require that all items of the project include the statement {{Wikidata entity link|P6104}} {{Wikidata entity link|Q134545539}}). === Mapping a concept === Scholia request "topic in context" === Mapping sources consensus === Visualise graphs and use the number of references to determine edge thickness/weight. == Writing == To cite articles we used the [[Template:Cite Q|Cite Q template.]] Each reference is an item in Wikidata and the template retrieve the necessary data to generate the citation references below. == Future research == The analysis of knowledge graph could in theory allow to make logical deduction to generate new data<ref name=":9" />. Reflect on the future of scholarly communication : https://hal.science/hal-03277615/file/OPERAS_Future_of_Scholarly_Communication_06.2021.pdf == Data == {| class="wikitable sortable" ! QID !! Year !! DOI !! Title |- | [[d:Q137901191|Q137901191]] || 2025 || [https://doi.org/10.1002/GEO2.70040 10.1002/GEO2.70040] || Place-Based Sustainability Transformations for Just Futures: A Systematic Review |- | [[d:Q137901187|Q137901187]] || 2025 || [https://doi.org/10.1002/WCC.932 10.1002/WCC.932] || Public Communication of Climate and Justice: A Scoping Review |- | [[d:Q135979013|Q135979013]] || 2025 || [https://doi.org/10.1007/S13280-025-02202-Z 10.1007/S13280-025-02202-Z] || Participatory approaches to climate adaptation, resilience, and mitigation: A systematic review |- | [[d:Q137901223|Q137901223]] || 2022 || [https://doi.org/10.1007/S13412-021-00726-W 10.1007/S13412-021-00726-W] || A review of stakeholder participation studies in renewable electricity and water: does the resource context matter? |- | [[d:Q137901184|Q137901184]] || 2021 || [https://doi.org/10.1007/S40518-021-00184-6 10.1007/S40518-021-00184-6] || Energy Storage as an Equity Asset. |- | [[d:Q114204627|Q114204627]] || 2021 || [https://doi.org/10.1007/S43621-021-00024-Z 10.1007/S43621-021-00024-Z] || Can public awareness, knowledge and engagement improve climate change adaptation policies? |- | [[d:Q137901209|Q137901209]] || 2026 || [https://doi.org/10.1016/J.AGSY.2025.104512 10.1016/J.AGSY.2025.104512] || Designing with non-humans for agricultural systems transformation: An interdisciplinary review and framework for reflection |- | [[d:Q137901201|Q137901201]] || 2025 || [https://doi.org/10.1016/J.COPSYC.2024.101987 10.1016/J.COPSYC.2024.101987] || Individual and community catalysts for Renewable Energy Communities (RECs) development |- | [[d:Q114197507|Q114197507]] || 2022 || [https://doi.org/10.1016/J.CRM.2022.100438 10.1016/J.CRM.2022.100438] || Advancements of sustainable development goals in co-production for climate change adaptation research |- | [[d:Q129203992|Q129203992]] || 2024 || [https://doi.org/10.1016/J.EGYR.2024.01.040 10.1016/J.EGYR.2024.01.040] || Empowering energy citizenship: Exploring dimensions and drivers in citizen engagement during the energy transition |- | [[d:Q137901216|Q137901216]] || 2026 || [https://doi.org/10.1016/J.EIAR.2025.108187 10.1016/J.EIAR.2025.108187] || From participation to partnership: A systematic review of public engagement in sustainable urban planning |- | [[d:Q137210566|Q137210566]] || 2016 || [https://doi.org/10.1016/J.ERSS.2015.10.004 10.1016/J.ERSS.2015.10.004] || Energy justice: A conceptual review |- | [[d:Q115448818|Q115448818]] || 2016 || [https://doi.org/10.1016/J.ERSS.2016.04.001 10.1016/J.ERSS.2016.04.001] || Stakeholder involvement in sustainability science—A critical view |- | [[d:Q129652515|Q129652515]] || 2018 || [https://doi.org/10.1016/J.ERSS.2018.06.010 10.1016/J.ERSS.2018.06.010] || What is energy democracy? Connecting social science energy research and political theory |- | [[d:Q137901196|Q137901196]] || 2020 || [https://doi.org/10.1016/J.ERSS.2020.101716 10.1016/J.ERSS.2020.101716] || Of renewable energy, energy democracy, and sustainable development: A roadmap to accelerate the energy transition in developing countries |- | [[d:Q136447761|Q136447761]] || 2020 || [https://doi.org/10.1016/J.ERSS.2020.101768 10.1016/J.ERSS.2020.101768] || Energy democracy as a process, an outcome and a goal: A conceptual review |- | [[d:Q137901204|Q137901204]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101834 10.1016/J.ERSS.2020.101834] || Identities, innovation, and governance: A systematic review of co-creation in wind energy transitions |- | [[d:Q137901183|Q137901183]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101837 10.1016/J.ERSS.2020.101837] || Renewable energy for whom? A global systematic review of the environmental justice implications of renewable energy technologies |- | [[d:Q137901207|Q137901207]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101871 10.1016/J.ERSS.2020.101871] || Rethinking community empowerment in the energy transformation: A critical review of the definitions, drivers and outcomes |- | [[d:Q137901215|Q137901215]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101876 10.1016/J.ERSS.2020.101876] || Co-production in the wind energy sector: A systematic literature review of public engagement beyond invited stakeholder participation |- | [[d:Q114306511|Q114306511]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101907 10.1016/J.ERSS.2020.101907] || From consultation toward co-production in science and policy: A critical systematic review of participatory climate and energy initiatives |- | [[d:Q137901221|Q137901221]] || 2021 || [https://doi.org/10.1016/J.ERSS.2021.102257 10.1016/J.ERSS.2021.102257] || The challenges of engaging island communities: Lessons on renewable energy from a review of 17 case studies |- | [[d:Q137901218|Q137901218]] || 2022 || [https://doi.org/10.1016/J.ERSS.2021.102333 10.1016/J.ERSS.2021.102333] || The (in)justices of smart local energy systems: A systematic review, integrated framework, and future research agenda |- | [[d:Q137901182|Q137901182]] || 2022 || [https://doi.org/10.1016/J.ERSS.2021.102444 10.1016/J.ERSS.2021.102444] || A critical review of energy democracy: A failure to deliver justice? |- | [[d:Q114306483|Q114306483]] || 2022 || [https://doi.org/10.1016/J.ERSS.2021.102482 10.1016/J.ERSS.2021.102482] || The role of energy democracy and energy citizenship for participatory energy transitions: A comprehensive review |- | [[d:Q114306476|Q114306476]] || 2022 || [https://doi.org/10.1016/J.ERSS.2022.102714 10.1016/J.ERSS.2022.102714] || What about citizens? A literature review of citizen engagement in sustainability transitions research |- | [[d:Q137901193|Q137901193]] || 2022 || [https://doi.org/10.1016/J.ERSS.2022.102862 10.1016/J.ERSS.2022.102862] || When energy justice is contested: A systematic review of a decade of research on Sweden?s conflicted energy landscape |- | [[d:Q137901219|Q137901219]] || 2023 || [https://doi.org/10.1016/J.ERSS.2022.102913 10.1016/J.ERSS.2022.102913] || Can we optimise for justice? Reviewing the inclusion of energy justice in energy system optimisation models |- | [[d:Q137901186|Q137901186]] || 2023 || [https://doi.org/10.1016/J.ERSS.2023.103010 10.1016/J.ERSS.2023.103010] || Analysing intersections of justice with energy transitions in India- A systematic literature review |- | [[d:Q137901181|Q137901181]] || 2023 || [https://doi.org/10.1016/J.ERSS.2023.103053 10.1016/J.ERSS.2023.103053] || Fostering justice through engagement: A literature review of public engagement in energy transitions |- | [[d:Q137211155|Q137211155]] || 2023 || [https://doi.org/10.1016/J.ERSS.2023.103213 10.1016/J.ERSS.2023.103213] || A fairway to fairness: Toward a richer conceptualization of fairness perceptions for just energy transitions |- | [[d:Q137901217|Q137901217]] || 2023 || [https://doi.org/10.1016/J.ERSS.2023.103221 10.1016/J.ERSS.2023.103221] || Powering just energy transitions: A review of the justice implications of community choice aggregation |- | [[d:Q137901199|Q137901199]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104016 10.1016/J.ERSS.2025.104016] || Making energy renovations equitable: A literature review of decision-making criteria for a just energy transition in residential buildings |- | [[d:Q137901188|Q137901188]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104036 10.1016/J.ERSS.2025.104036] || Community energy justice: A review of origins, convergence, and a research agenda |- | [[d:Q137901211|Q137901211]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104067 10.1016/J.ERSS.2025.104067] || Psychological and social factors driving citizen involvement in renewable energy communities: A systematic review |- | [[d:Q137901192|Q137901192]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104149 10.1016/J.ERSS.2025.104149] || Assessing social impacts and Energy Justice along green hydrogen supply chains: a capability-based framework |- | [[d:Q137901195|Q137901195]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104422 10.1016/J.ERSS.2025.104422] || Out of place, scale and time? Navigating injustices across mission arenas of the German Energiewende |- | [[d:Q137901185|Q137901185]] || 2024 || [https://doi.org/10.1016/J.ESD.2024.101546 10.1016/J.ESD.2024.101546] || Characterizing 'injustices' in clean energy transitions in Africa |- | [[d:Q137901226|Q137901226]] || 2024 || [https://doi.org/10.1016/J.JCLEPRO.2024.143470 10.1016/J.JCLEPRO.2024.143470] || Energy justice and sustainable urban renewal: A systematic review of low-income old town communities |- | [[d:Q137901222|Q137901222]] || 2024 || [https://doi.org/10.1016/J.JENVMAN.2024.120804 10.1016/J.JENVMAN.2024.120804] || Forest, climate, and policy literature lacks acknowledgement of environmental justice, diversity, equity, and inclusion |- | [[d:Q115441381|Q115441381]] || 2021 || [https://doi.org/10.1016/J.RSER.2021.111504 10.1016/J.RSER.2021.111504] || Participatory methods in energy system modelling and planning – A review |- | [[d:Q137901205|Q137901205]] || 2025 || [https://doi.org/10.1016/J.RSER.2025.115892 10.1016/J.RSER.2025.115892] || A systematic review of the intersection between energy justice and human rights |- | [[d:Q137901225|Q137901225]] || 2024 || [https://doi.org/10.1017/SUS.2024.24 10.1017/SUS.2024.24] || Blue carbon as just transition? A structured literature review |- | [[d:Q137901220|Q137901220]] || 2025 || [https://doi.org/10.1017/SUS.2025.2 10.1017/SUS.2025.2] || Toward an intersectional equity approach in social-ecological transformations |- | [[d:Q137901203|Q137901203]] || 2024 || [https://doi.org/10.1080/14693062.2023.2256697 10.1080/14693062.2023.2256697] || Exploring the democracy-climate nexus: a review of correlations between democracy and climate policy performance |- | [[d:Q137901164|Q137901164]] || 2022 || [https://doi.org/10.1111/GEC3.12662 10.1111/GEC3.12662] || Creating fairer futures for sustainability transitions |- | [[d:Q137901227|Q137901227]] || 2025 || [https://doi.org/10.1139/ER-2024-0018 10.1139/ER-2024-0018] || Community engagement in nature-positive food systems programming and research in East and Southern Africa: a review |- | [[d:Q119955266|Q119955266]] || 2019 || [https://doi.org/10.1146/ANNUREV-ENVIRON-101718-033103 10.1146/ANNUREV-ENVIRON-101718-033103] || Co-Producing Sustainability: Reordering the Governance of Science, Policy, and Practice |- | [[d:Q137901206|Q137901206]] || 2023 || [https://doi.org/10.1146/ANNUREV-ENVIRON-112621-063400 10.1146/ANNUREV-ENVIRON-112621-063400] || Metrics for Decision-Making in Energy Justice |- | [[d:Q137901213|Q137901213]] || 2022 || [https://doi.org/10.1186/S13705-021-00330-4 10.1186/S13705-021-00330-4] || Mapping emergent public engagement in societal transitions: a scoping review |- | [[d:Q137901163|Q137901163]] || 2025 || [https://doi.org/10.17573/CEPAR.2025.2.09 10.17573/CEPAR.2025.2.09] || From Co-Creation to Circular Cities: Exploring Living Labs in EU Governance Frameworks - A Literature Review |- | [[d:Q137901197|Q137901197]] || 2024 || [https://doi.org/10.3390/EN17143512 10.3390/EN17143512] || A Systematic Review on the Path to Inclusive and Sustainable Energy Transitions |- | [[d:Q104887325|Q104887325]] || 2019 || [https://doi.org/10.3390/SU11041023 10.3390/SU11041023] || Deliberation and the Promise of a Deeply Democratic Sustainability Transition |- | [[d:Q137901202|Q137901202]] || 2021 || [https://doi.org/10.3390/SU13042128 10.3390/SU13042128] || A Review of Energy Communities in Sub-Saharan Africa as a Transition Pathway to Energy Democracy |- | [[d:Q137901210|Q137901210]] || 2023 || [https://doi.org/10.3390/SU15032441 10.3390/SU15032441] || Sustainable Project Governance: Scientometric Analysis and Emerging Trends |- | [[d:Q137901224|Q137901224]] || 2024 || [https://doi.org/10.3390/SU16198700 10.3390/SU16198700] || Empowering Communities to Act for a Change: A Review of the Community Empowerment Programs towards Sustainability and Resilience |} == References == {{References}} tumhqd6up64m3qfftx8kocnzmpn4sn7 User:Dc.samizdat/Golden chords of the 120-cell 2 326765 2812903 2812707 2026-06-05T18:07:27Z Dc.samizdat 2856930 /* The 24-cell */ 2812903 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 5<sup>2</sup> 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 5<sup>2</sup> (75) disjoint instances of itself in the 600-point (120-cell), which contains 3<sup>2</sup> times 5<sup>2</sup> (225) distinct instances of the 24-point (24-cell), and 3<sup>3</sup> times 5<sup>2</sup> (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> |} == Complementary chord pairs == The list of 30 chords 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}} 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 concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius <math>c_1</math> is a dodecahedron cell, and the largest, central section of radius <math>c_{15}</math> is a non-uniform 60-point [[w:Rhombicosidodecahedron|rhombicosidodecahedron.]] At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal dodecahedron cell at distance <math>c_{29}</math>. ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[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|B<sub>4</sub> 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 geodesic 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 invariant planes containing its edges, 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 isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular double helix which visits each vertex 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> [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F<sub>4</sub> 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.]] 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>. In this system of unit-radius coordinates <math>r_1=1/r_5</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. 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> [[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.]] Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 30° turns. We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 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-cells. Three Clifford parallel octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular triple helix {24/9}=3{8/3} that visits each vertex once. We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. 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 Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of green <math>r_5</math> chords, visible in the orthogonal projection. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular double helix {24/10}=2{12/5} that visits each vertex once. Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math> is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an 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 a different 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 between the 24-cell's existing 24 vertices, in effect adding twenty-four more 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. [[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]] 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. The skew {30}-gons have these chords: :<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}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</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}</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> [[File:Regular_star_polygon_30-7.svg|thumb|Star polygon {30/7} of <math>r_7</math> edges. Fontaine and Hurley's rotation over the <math>r_7</math> chord is the rotation of the 600-cell by the characteristic rotation of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges.]] Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 12° turns. We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, 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, but it does not visit other 16-cell vertex positions. Fifteen Clifford parallel octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular helix of fifteen twisted strands that visits each vertex once. [Perhaps this is five {24/8)=8{3} as suggested by the text but I believe it can only be four {30/7} as suggested by the illustration.] [[File:Regular_star_figure_3(10,3).svg|thumb|Star polygon {30/9}=3{10/3} of <math>r_9</math> edges. Fontaine and Hurley's rotation over the <math>r_9</math> chord is the rotation of the 600-cell by the characteristic rotation of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, 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, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular helix of ten twisted strands that visits each vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.] [[File:Regular_star_figure_2(15,4).svg|thumb|Star polygon {30/8}=2{15/4} of <math>r_4</math> edges. Fontaine and Hurley's rotation over the <math>r_4</math> chord is the characteristic rotation of the 600-cell'','' its isoclinic rotation in invariant planes containing its edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of <math>r_5</math> chords. 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 pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>\sqrt{1}</math> chords form a circular helix of eight twisted strands that visits each vertex once. Fontaine and Hurley's rotation over the <math>r_4</math> star polygon {30/8}=2{15/4} which constructs <math>1/r_4</math> is the ''characteristic rotation of the 600-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 600-cell an 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, and every great square to a Clifford parallel great square. 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 over 15 <math>\sqrt{1}</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 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. == 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. In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-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 distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [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 higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to Clifford polygon sequences 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}} lv0ydqp2nqeqdg7u0wm7karq0sbp0nw 2812904 2812903 2026-06-05T18:08:05Z Dc.samizdat 2856930 /* The 24-cell */ 2812904 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 5<sup>2</sup> 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 5<sup>2</sup> (75) disjoint instances of itself in the 600-point (120-cell), which contains 3<sup>2</sup> times 5<sup>2</sup> (225) distinct instances of the 24-point (24-cell), and 3<sup>3</sup> times 5<sup>2</sup> (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> |} == Complementary chord pairs == The list of 30 chords 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}} 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 concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius <math>c_1</math> is a dodecahedron cell, and the largest, central section of radius <math>c_{15}</math> is a non-uniform 60-point [[w:Rhombicosidodecahedron|rhombicosidodecahedron.]] At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal dodecahedron cell at distance <math>c_{29}</math>. ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[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|B<sub>4</sub> 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 geodesic 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 invariant planes containing its edges, 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 isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular double helix which visits each vertex 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> [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F<sub>4</sub> 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.]] 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>. In this system of unit-radius coordinates <math>r_1=1/r_5</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. 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> [[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.]] Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 30° turns. [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 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-cells. Three Clifford parallel octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular triple helix {24/9}=3{8/3} that visits each vertex once. We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. 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 Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of green <math>r_5</math> chords, visible in the orthogonal projection. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular double helix {24/10}=2{12/5} that visits each vertex once. Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math> is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an 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 a different 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 between the 24-cell's existing 24 vertices, in effect adding twenty-four more 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. [[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]] 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. The skew {30}-gons have these chords: :<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}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</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}</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> [[File:Regular_star_polygon_30-7.svg|thumb|Star polygon {30/7} of <math>r_7</math> edges. Fontaine and Hurley's rotation over the <math>r_7</math> chord is the rotation of the 600-cell by the characteristic rotation of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges.]] Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 12° turns. We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, 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, but it does not visit other 16-cell vertex positions. Fifteen Clifford parallel octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular helix of fifteen twisted strands that visits each vertex once. [Perhaps this is five {24/8)=8{3} as suggested by the text but I believe it can only be four {30/7} as suggested by the illustration.] [[File:Regular_star_figure_3(10,3).svg|thumb|Star polygon {30/9}=3{10/3} of <math>r_9</math> edges. Fontaine and Hurley's rotation over the <math>r_9</math> chord is the rotation of the 600-cell by the characteristic rotation of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, 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, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular helix of ten twisted strands that visits each vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.] [[File:Regular_star_figure_2(15,4).svg|thumb|Star polygon {30/8}=2{15/4} of <math>r_4</math> edges. Fontaine and Hurley's rotation over the <math>r_4</math> chord is the characteristic rotation of the 600-cell'','' its isoclinic rotation in invariant planes containing its edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of <math>r_5</math> chords. 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 pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>\sqrt{1}</math> chords form a circular helix of eight twisted strands that visits each vertex once. Fontaine and Hurley's rotation over the <math>r_4</math> star polygon {30/8}=2{15/4} which constructs <math>1/r_4</math> is the ''characteristic rotation of the 600-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 600-cell an 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, and every great square to a Clifford parallel great square. 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 over 15 <math>\sqrt{1}</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 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. == 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. In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-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 distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [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 higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to Clifford polygon sequences 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}} budic06g7wezbr9e2ma6jzso2tgqq1y 2812905 2812904 2026-06-05T18:09:01Z Dc.samizdat 2856930 /* The 24-cell */ 2812905 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 5<sup>2</sup> 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 5<sup>2</sup> (75) disjoint instances of itself in the 600-point (120-cell), which contains 3<sup>2</sup> times 5<sup>2</sup> (225) distinct instances of the 24-point (24-cell), and 3<sup>3</sup> times 5<sup>2</sup> (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> |} == Complementary chord pairs == The list of 30 chords 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}} 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 concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius <math>c_1</math> is a dodecahedron cell, and the largest, central section of radius <math>c_{15}</math> is a non-uniform 60-point [[w:Rhombicosidodecahedron|rhombicosidodecahedron.]] At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal dodecahedron cell at distance <math>c_{29}</math>. ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[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|B<sub>4</sub> 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 geodesic 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 invariant planes containing its edges, 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 isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular double helix which visits each vertex 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> [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F<sub>4</sub> 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.]] 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>. In this system of unit-radius coordinates <math>r_1=1/r_5</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. 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. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 30° turns. [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 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-cells. Three Clifford parallel octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular triple helix {24/9}=3{8/3} that visits each vertex once. [[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.]] We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. 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 Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of green <math>r_5</math> chords, visible in the orthogonal projection. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular double helix {24/10}=2{12/5} that visits each vertex once. Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math> is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an 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 a different 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 between the 24-cell's existing 24 vertices, in effect adding twenty-four more 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. [[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]] 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. The skew {30}-gons have these chords: :<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}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</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}</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> [[File:Regular_star_polygon_30-7.svg|thumb|Star polygon {30/7} of <math>r_7</math> edges. Fontaine and Hurley's rotation over the <math>r_7</math> chord is the rotation of the 600-cell by the characteristic rotation of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges.]] Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 12° turns. We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, 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, but it does not visit other 16-cell vertex positions. Fifteen Clifford parallel octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular helix of fifteen twisted strands that visits each vertex once. [Perhaps this is five {24/8)=8{3} as suggested by the text but I believe it can only be four {30/7} as suggested by the illustration.] [[File:Regular_star_figure_3(10,3).svg|thumb|Star polygon {30/9}=3{10/3} of <math>r_9</math> edges. Fontaine and Hurley's rotation over the <math>r_9</math> chord is the rotation of the 600-cell by the characteristic rotation of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, 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, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular helix of ten twisted strands that visits each vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.] [[File:Regular_star_figure_2(15,4).svg|thumb|Star polygon {30/8}=2{15/4} of <math>r_4</math> edges. Fontaine and Hurley's rotation over the <math>r_4</math> chord is the characteristic rotation of the 600-cell'','' its isoclinic rotation in invariant planes containing its edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of <math>r_5</math> chords. 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 pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>\sqrt{1}</math> chords form a circular helix of eight twisted strands that visits each vertex once. Fontaine and Hurley's rotation over the <math>r_4</math> star polygon {30/8}=2{15/4} which constructs <math>1/r_4</math> is the ''characteristic rotation of the 600-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 600-cell an 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, and every great square to a Clifford parallel great square. 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 over 15 <math>\sqrt{1}</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 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. == 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. In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-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 distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [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 higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to Clifford polygon sequences 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}} o9zrtkoa3nwq1z88f3p89wymumnwhxl 2812906 2812905 2026-06-05T18:10:33Z Dc.samizdat 2856930 /* The 24-cell */ 2812906 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 5<sup>2</sup> 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 5<sup>2</sup> (75) disjoint instances of itself in the 600-point (120-cell), which contains 3<sup>2</sup> times 5<sup>2</sup> (225) distinct instances of the 24-point (24-cell), and 3<sup>3</sup> times 5<sup>2</sup> (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> |} == Complementary chord pairs == The list of 30 chords 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}} 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 concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius <math>c_1</math> is a dodecahedron cell, and the largest, central section of radius <math>c_{15}</math> is a non-uniform 60-point [[w:Rhombicosidodecahedron|rhombicosidodecahedron.]] At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal dodecahedron cell at distance <math>c_{29}</math>. ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[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|B<sub>4</sub> 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 geodesic 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 invariant planes containing its edges, 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 isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular double helix which visits each vertex 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>. In this system of unit-radius coordinates <math>r_1=1/r_5</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. 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. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 30° turns. [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 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-cells. Three Clifford parallel octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular triple helix {24/9}=3{8/3} that visits each vertex once. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F<sub>4</sub> 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.]] We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. 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 Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of green <math>r_5</math> chords, visible in the orthogonal projection. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular double helix {24/10}=2{12/5} that visits each vertex once. Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math> is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an 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 a different 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 between the 24-cell's existing 24 vertices, in effect adding twenty-four more 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. [[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]] 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. The skew {30}-gons have these chords: :<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}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</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}</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> [[File:Regular_star_polygon_30-7.svg|thumb|Star polygon {30/7} of <math>r_7</math> edges. Fontaine and Hurley's rotation over the <math>r_7</math> chord is the rotation of the 600-cell by the characteristic rotation of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges.]] Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 12° turns. We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, 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, but it does not visit other 16-cell vertex positions. Fifteen Clifford parallel octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular helix of fifteen twisted strands that visits each vertex once. [Perhaps this is five {24/8)=8{3} as suggested by the text but I believe it can only be four {30/7} as suggested by the illustration.] [[File:Regular_star_figure_3(10,3).svg|thumb|Star polygon {30/9}=3{10/3} of <math>r_9</math> edges. Fontaine and Hurley's rotation over the <math>r_9</math> chord is the rotation of the 600-cell by the characteristic rotation of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, 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, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular helix of ten twisted strands that visits each vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.] [[File:Regular_star_figure_2(15,4).svg|thumb|Star polygon {30/8}=2{15/4} of <math>r_4</math> edges. Fontaine and Hurley's rotation over the <math>r_4</math> chord is the characteristic rotation of the 600-cell'','' its isoclinic rotation in invariant planes containing its edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of <math>r_5</math> chords. 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 pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>\sqrt{1}</math> chords form a circular helix of eight twisted strands that visits each vertex once. Fontaine and Hurley's rotation over the <math>r_4</math> star polygon {30/8}=2{15/4} which constructs <math>1/r_4</math> is the ''characteristic rotation of the 600-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 600-cell an 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, and every great square to a Clifford parallel great square. 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 over 15 <math>\sqrt{1}</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 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. == 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. In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-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 distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [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 higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to Clifford polygon sequences 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}} mnnypk318i6gkmawv6yss71bidzj0oo 2812907 2812906 2026-06-05T18:11:00Z Dc.samizdat 2856930 /* The 24-cell */ 2812907 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 5<sup>2</sup> 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 5<sup>2</sup> (75) disjoint instances of itself in the 600-point (120-cell), which contains 3<sup>2</sup> times 5<sup>2</sup> (225) distinct instances of the 24-point (24-cell), and 3<sup>3</sup> times 5<sup>2</sup> (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> |} == Complementary chord pairs == The list of 30 chords 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}} 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 concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius <math>c_1</math> is a dodecahedron cell, and the largest, central section of radius <math>c_{15}</math> is a non-uniform 60-point [[w:Rhombicosidodecahedron|rhombicosidodecahedron.]] At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal dodecahedron cell at distance <math>c_{29}</math>. ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[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|B<sub>4</sub> 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 geodesic 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 invariant planes containing its edges, 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 isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular double helix which visits each vertex 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>. In this system of unit-radius coordinates <math>r_1=1/r_5</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. 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. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 30° turns. [[File:Regular_star_figure_3(8,3).svg|thumb|150px]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 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-cells. Three Clifford parallel octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular triple helix {24/9}=3{8/3} that visits each vertex once. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F<sub>4</sub> 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.]] We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. 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 Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of green <math>r_5</math> chords, visible in the orthogonal projection. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular double helix {24/10}=2{12/5} that visits each vertex once. Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math> is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an 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 a different 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 between the 24-cell's existing 24 vertices, in effect adding twenty-four more 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. [[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]] 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. The skew {30}-gons have these chords: :<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}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</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}</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> [[File:Regular_star_polygon_30-7.svg|thumb|Star polygon {30/7} of <math>r_7</math> edges. Fontaine and Hurley's rotation over the <math>r_7</math> chord is the rotation of the 600-cell by the characteristic rotation of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges.]] Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 12° turns. We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, 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, but it does not visit other 16-cell vertex positions. Fifteen Clifford parallel octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular helix of fifteen twisted strands that visits each vertex once. [Perhaps this is five {24/8)=8{3} as suggested by the text but I believe it can only be four {30/7} as suggested by the illustration.] [[File:Regular_star_figure_3(10,3).svg|thumb|Star polygon {30/9}=3{10/3} of <math>r_9</math> edges. Fontaine and Hurley's rotation over the <math>r_9</math> chord is the rotation of the 600-cell by the characteristic rotation of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, 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, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular helix of ten twisted strands that visits each vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.] [[File:Regular_star_figure_2(15,4).svg|thumb|Star polygon {30/8}=2{15/4} of <math>r_4</math> edges. Fontaine and Hurley's rotation over the <math>r_4</math> chord is the characteristic rotation of the 600-cell'','' its isoclinic rotation in invariant planes containing its edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of <math>r_5</math> chords. 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 pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>\sqrt{1}</math> chords form a circular helix of eight twisted strands that visits each vertex once. Fontaine and Hurley's rotation over the <math>r_4</math> star polygon {30/8}=2{15/4} which constructs <math>1/r_4</math> is the ''characteristic rotation of the 600-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 600-cell an 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, and every great square to a Clifford parallel great square. 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 over 15 <math>\sqrt{1}</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 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. == 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. In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-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 distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [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 higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to Clifford polygon sequences 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}} l45cgbvoaiowuamzu16m3tfxdkfvh3a 2812910 2812907 2026-06-05T18:14:56Z Dc.samizdat 2856930 /* The 24-cell */ 2812910 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 5<sup>2</sup> 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 5<sup>2</sup> (75) disjoint instances of itself in the 600-point (120-cell), which contains 3<sup>2</sup> times 5<sup>2</sup> (225) distinct instances of the 24-point (24-cell), and 3<sup>3</sup> times 5<sup>2</sup> (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> |} == Complementary chord pairs == The list of 30 chords 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}} 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 concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius <math>c_1</math> is a dodecahedron cell, and the largest, central section of radius <math>c_{15}</math> is a non-uniform 60-point [[w:Rhombicosidodecahedron|rhombicosidodecahedron.]] At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal dodecahedron cell at distance <math>c_{29}</math>. ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[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|B<sub>4</sub> 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 geodesic 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 invariant planes containing its edges, 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 isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular double helix which visits each vertex 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>. In this system of unit-radius coordinates <math>r_1=1/r_5</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. 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. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 30° turns. [[File:Regular_star_figure_3(8,3).svg|thumb|The 24-cell contains three 16-cells which can rotate in parallel on their characteristic {8/3} octagram Clifford polygons of <math>\sqrt{2}</math> chords.]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 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-cells. Three Clifford parallel octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular triple helix {24/9}=3{8/3} that visits each vertex once. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F<sub>4</sub> 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.]] We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. 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 Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of green <math>r_5</math> chords, visible in the orthogonal projection. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular double helix {24/10}=2{12/5} that visits each vertex once. Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math> is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an 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 a different 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 between the 24-cell's existing 24 vertices, in effect adding twenty-four more 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. [[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]] 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. The skew {30}-gons have these chords: :<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}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</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}</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> [[File:Regular_star_polygon_30-7.svg|thumb|Star polygon {30/7} of <math>r_7</math> edges. Fontaine and Hurley's rotation over the <math>r_7</math> chord is the rotation of the 600-cell by the characteristic rotation of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges.]] Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 12° turns. We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, 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, but it does not visit other 16-cell vertex positions. Fifteen Clifford parallel octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular helix of fifteen twisted strands that visits each vertex once. [Perhaps this is five {24/8)=8{3} as suggested by the text but I believe it can only be four {30/7} as suggested by the illustration.] [[File:Regular_star_figure_3(10,3).svg|thumb|Star polygon {30/9}=3{10/3} of <math>r_9</math> edges. Fontaine and Hurley's rotation over the <math>r_9</math> chord is the rotation of the 600-cell by the characteristic rotation of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, 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, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular helix of ten twisted strands that visits each vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.] [[File:Regular_star_figure_2(15,4).svg|thumb|Star polygon {30/8}=2{15/4} of <math>r_4</math> edges. Fontaine and Hurley's rotation over the <math>r_4</math> chord is the characteristic rotation of the 600-cell'','' its isoclinic rotation in invariant planes containing its edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of <math>r_5</math> chords. 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 pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>\sqrt{1}</math> chords form a circular helix of eight twisted strands that visits each vertex once. Fontaine and Hurley's rotation over the <math>r_4</math> star polygon {30/8}=2{15/4} which constructs <math>1/r_4</math> is the ''characteristic rotation of the 600-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 600-cell an 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, and every great square to a Clifford parallel great square. 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 over 15 <math>\sqrt{1}</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 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. == 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. In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-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 distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [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 higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to Clifford polygon sequences 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}} lz4pd383areu2p511172enocksd8b5n 2812911 2812910 2026-06-05T18:18:02Z Dc.samizdat 2856930 /* The 24-cell */ 2812911 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 5<sup>2</sup> 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 5<sup>2</sup> (75) disjoint instances of itself in the 600-point (120-cell), which contains 3<sup>2</sup> times 5<sup>2</sup> (225) distinct instances of the 24-point (24-cell), and 3<sup>3</sup> times 5<sup>2</sup> (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> |} == Complementary chord pairs == The list of 30 chords 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}} 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 concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius <math>c_1</math> is a dodecahedron cell, and the largest, central section of radius <math>c_{15}</math> is a non-uniform 60-point [[w:Rhombicosidodecahedron|rhombicosidodecahedron.]] At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal dodecahedron cell at distance <math>c_{29}</math>. ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[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|B<sub>4</sub> 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 geodesic 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 invariant planes containing its edges, 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 isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular double helix which visits each vertex 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>. In this system of unit-radius coordinates <math>r_1=1/r_5</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. [[File:Regular_star_figure_3(8,3).svg|thumb|The 24-cell contains three 16-cells which can rotate in parallel on their characteristic {8/3} octagram Clifford polygons of <math>\sqrt{2}</math> chords.]] 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. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 30° turns. We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 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-cells. Three Clifford parallel octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular triple helix {24/9}=3{8/3} that visits each vertex once. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F<sub>4</sub> 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.]] We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. 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 Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of green <math>r_5</math> chords, visible in the orthogonal projection. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular double helix {24/10}=2{12/5} that visits each vertex once. Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math> is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an 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 a different 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 between the 24-cell's existing 24 vertices, in effect adding twenty-four more 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. [[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]] 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. The skew {30}-gons have these chords: :<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}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</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}</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> [[File:Regular_star_polygon_30-7.svg|thumb|Star polygon {30/7} of <math>r_7</math> edges. Fontaine and Hurley's rotation over the <math>r_7</math> chord is the rotation of the 600-cell by the characteristic rotation of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges.]] Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 12° turns. We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, 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, but it does not visit other 16-cell vertex positions. Fifteen Clifford parallel octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular helix of fifteen twisted strands that visits each vertex once. [Perhaps this is five {24/8)=8{3} as suggested by the text but I believe it can only be four {30/7} as suggested by the illustration.] [[File:Regular_star_figure_3(10,3).svg|thumb|Star polygon {30/9}=3{10/3} of <math>r_9</math> edges. Fontaine and Hurley's rotation over the <math>r_9</math> chord is the rotation of the 600-cell by the characteristic rotation of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, 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, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular helix of ten twisted strands that visits each vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.] [[File:Regular_star_figure_2(15,4).svg|thumb|Star polygon {30/8}=2{15/4} of <math>r_4</math> edges. Fontaine and Hurley's rotation over the <math>r_4</math> chord is the characteristic rotation of the 600-cell'','' its isoclinic rotation in invariant planes containing its edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of <math>r_5</math> chords. 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 pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>\sqrt{1}</math> chords form a circular helix of eight twisted strands that visits each vertex once. Fontaine and Hurley's rotation over the <math>r_4</math> star polygon {30/8}=2{15/4} which constructs <math>1/r_4</math> is the ''characteristic rotation of the 600-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 600-cell an 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, and every great square to a Clifford parallel great square. 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 over 15 <math>\sqrt{1}</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 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. == 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. In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-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 distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [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 higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to Clifford polygon sequences 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}} cvjmmteme514ueptkwwvw23969wox2t 2812912 2812911 2026-06-05T18:18:58Z Dc.samizdat 2856930 /* The 24-cell */ 2812912 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 5<sup>2</sup> 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 5<sup>2</sup> (75) disjoint instances of itself in the 600-point (120-cell), which contains 3<sup>2</sup> times 5<sup>2</sup> (225) distinct instances of the 24-point (24-cell), and 3<sup>3</sup> times 5<sup>2</sup> (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> |} == Complementary chord pairs == The list of 30 chords 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}} 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 concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius <math>c_1</math> is a dodecahedron cell, and the largest, central section of radius <math>c_{15}</math> is a non-uniform 60-point [[w:Rhombicosidodecahedron|rhombicosidodecahedron.]] At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal dodecahedron cell at distance <math>c_{29}</math>. ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[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|B<sub>4</sub> 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 geodesic 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 invariant planes containing its edges, 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 isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular double helix which visits each vertex 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>. In this system of unit-radius coordinates <math>r_1=1/r_5</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. 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. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 30° turns. [[File:Regular_star_figure_3(8,3).svg|thumb|150px|The 24-cell contains three 16-cells which can rotate in parallel on their characteristic {8/3} octagram Clifford polygons of <math>\sqrt{2}</math> chords.]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 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-cells. Three Clifford parallel octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular triple helix {24/9}=3{8/3} that visits each vertex once. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F<sub>4</sub> 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.]] We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. 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 Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of green <math>r_5</math> chords, visible in the orthogonal projection. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular double helix {24/10}=2{12/5} that visits each vertex once. Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math> is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an 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 a different 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 between the 24-cell's existing 24 vertices, in effect adding twenty-four more 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. [[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]] 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. The skew {30}-gons have these chords: :<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}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</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}</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> [[File:Regular_star_polygon_30-7.svg|thumb|Star polygon {30/7} of <math>r_7</math> edges. Fontaine and Hurley's rotation over the <math>r_7</math> chord is the rotation of the 600-cell by the characteristic rotation of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges.]] Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 12° turns. We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, 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, but it does not visit other 16-cell vertex positions. Fifteen Clifford parallel octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular helix of fifteen twisted strands that visits each vertex once. [Perhaps this is five {24/8)=8{3} as suggested by the text but I believe it can only be four {30/7} as suggested by the illustration.] [[File:Regular_star_figure_3(10,3).svg|thumb|Star polygon {30/9}=3{10/3} of <math>r_9</math> edges. Fontaine and Hurley's rotation over the <math>r_9</math> chord is the rotation of the 600-cell by the characteristic rotation of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, 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, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular helix of ten twisted strands that visits each vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.] [[File:Regular_star_figure_2(15,4).svg|thumb|Star polygon {30/8}=2{15/4} of <math>r_4</math> edges. Fontaine and Hurley's rotation over the <math>r_4</math> chord is the characteristic rotation of the 600-cell'','' its isoclinic rotation in invariant planes containing its edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of <math>r_5</math> chords. 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 pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>\sqrt{1}</math> chords form a circular helix of eight twisted strands that visits each vertex once. Fontaine and Hurley's rotation over the <math>r_4</math> star polygon {30/8}=2{15/4} which constructs <math>1/r_4</math> is the ''characteristic rotation of the 600-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 600-cell an 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, and every great square to a Clifford parallel great square. 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 over 15 <math>\sqrt{1}</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 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. == 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. In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-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 distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [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 higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to Clifford polygon sequences 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}} 58djv2a2qdkkofpto45tbm381z7oys7 2812913 2812912 2026-06-05T18:24:06Z Dc.samizdat 2856930 /* The 24-cell */ 2812913 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 5<sup>2</sup> 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 5<sup>2</sup> (75) disjoint instances of itself in the 600-point (120-cell), which contains 3<sup>2</sup> times 5<sup>2</sup> (225) distinct instances of the 24-point (24-cell), and 3<sup>3</sup> times 5<sup>2</sup> (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> |} == Complementary chord pairs == The list of 30 chords 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}} 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 concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius <math>c_1</math> is a dodecahedron cell, and the largest, central section of radius <math>c_{15}</math> is a non-uniform 60-point [[w:Rhombicosidodecahedron|rhombicosidodecahedron.]] At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal dodecahedron cell at distance <math>c_{29}</math>. ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[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|B<sub>4</sub> 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 geodesic 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 invariant planes containing its edges, 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 isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular double helix which visits each vertex 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>. In this system of unit-radius coordinates <math>r_1=1/r_5</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. 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. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 30° turns. [[File:Regular_star_figure_3(8,3).svg|thumb|150px|{24/9}=3{8/3} Clifford polygons of <math>\sqrt{2}</math> 16-cell edges in the 24-cell.]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 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-cells. Three Clifford parallel octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular triple helix {24/9}=3{8/3} that visits each vertex once. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F<sub>4</sub> 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.]] We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. 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 Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of green <math>r_5</math> chords, visible in the orthogonal projection. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular double helix {24/10}=2{12/5} that visits each vertex once. Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math> is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an 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 a different 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 between the 24-cell's existing 24 vertices, in effect adding twenty-four more 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. [[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]] 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. The skew {30}-gons have these chords: :<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}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</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}</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> [[File:Regular_star_polygon_30-7.svg|thumb|Star polygon {30/7} of <math>r_7</math> edges. Fontaine and Hurley's rotation over the <math>r_7</math> chord is the rotation of the 600-cell by the characteristic rotation of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges.]] Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 12° turns. We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, 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, but it does not visit other 16-cell vertex positions. Fifteen Clifford parallel octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular helix of fifteen twisted strands that visits each vertex once. [Perhaps this is five {24/8)=8{3} as suggested by the text but I believe it can only be four {30/7} as suggested by the illustration.] [[File:Regular_star_figure_3(10,3).svg|thumb|Star polygon {30/9}=3{10/3} of <math>r_9</math> edges. Fontaine and Hurley's rotation over the <math>r_9</math> chord is the rotation of the 600-cell by the characteristic rotation of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, 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, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular helix of ten twisted strands that visits each vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.] [[File:Regular_star_figure_2(15,4).svg|thumb|Star polygon {30/8}=2{15/4} of <math>r_4</math> edges. Fontaine and Hurley's rotation over the <math>r_4</math> chord is the characteristic rotation of the 600-cell'','' its isoclinic rotation in invariant planes containing its edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of <math>r_5</math> chords. 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 pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>\sqrt{1}</math> chords form a circular helix of eight twisted strands that visits each vertex once. Fontaine and Hurley's rotation over the <math>r_4</math> star polygon {30/8}=2{15/4} which constructs <math>1/r_4</math> is the ''characteristic rotation of the 600-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 600-cell an 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, and every great square to a Clifford parallel great square. 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 over 15 <math>\sqrt{1}</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 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. == 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. In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-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 distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [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 higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to Clifford polygon sequences 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}} fmwnof5b6ln0pzk2pnpauxo1enp1e8l 2812914 2812913 2026-06-05T18:24:55Z Dc.samizdat 2856930 /* The 24-cell */ 2812914 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 5<sup>2</sup> 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 5<sup>2</sup> (75) disjoint instances of itself in the 600-point (120-cell), which contains 3<sup>2</sup> times 5<sup>2</sup> (225) distinct instances of the 24-point (24-cell), and 3<sup>3</sup> times 5<sup>2</sup> (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> |} == Complementary chord pairs == The list of 30 chords 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}} 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 concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius <math>c_1</math> is a dodecahedron cell, and the largest, central section of radius <math>c_{15}</math> is a non-uniform 60-point [[w:Rhombicosidodecahedron|rhombicosidodecahedron.]] At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal dodecahedron cell at distance <math>c_{29}</math>. ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[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|B<sub>4</sub> 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 geodesic 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 invariant planes containing its edges, 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 isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular double helix which visits each vertex 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>. In this system of unit-radius coordinates <math>r_1=1/r_5</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. 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> [[File:Regular_star_figure_3(8,3).svg|thumb|150px|{24/9}=3{8/3} Clifford polygons of <math>\sqrt{2}</math> 16-cell edges in the 24-cell.]] Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 30° turns. We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 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-cells. Three Clifford parallel octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular triple helix {24/9}=3{8/3} that visits each vertex once. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F<sub>4</sub> 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.]] We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. 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 Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of green <math>r_5</math> chords, visible in the orthogonal projection. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular double helix {24/10}=2{12/5} that visits each vertex once. Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math> is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an 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 a different 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 between the 24-cell's existing 24 vertices, in effect adding twenty-four more 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. [[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]] 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. The skew {30}-gons have these chords: :<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}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</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}</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> [[File:Regular_star_polygon_30-7.svg|thumb|Star polygon {30/7} of <math>r_7</math> edges. Fontaine and Hurley's rotation over the <math>r_7</math> chord is the rotation of the 600-cell by the characteristic rotation of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges.]] Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 12° turns. We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, 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, but it does not visit other 16-cell vertex positions. Fifteen Clifford parallel octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular helix of fifteen twisted strands that visits each vertex once. [Perhaps this is five {24/8)=8{3} as suggested by the text but I believe it can only be four {30/7} as suggested by the illustration.] [[File:Regular_star_figure_3(10,3).svg|thumb|Star polygon {30/9}=3{10/3} of <math>r_9</math> edges. Fontaine and Hurley's rotation over the <math>r_9</math> chord is the rotation of the 600-cell by the characteristic rotation of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, 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, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular helix of ten twisted strands that visits each vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.] [[File:Regular_star_figure_2(15,4).svg|thumb|Star polygon {30/8}=2{15/4} of <math>r_4</math> edges. Fontaine and Hurley's rotation over the <math>r_4</math> chord is the characteristic rotation of the 600-cell'','' its isoclinic rotation in invariant planes containing its edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of <math>r_5</math> chords. 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 pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>\sqrt{1}</math> chords form a circular helix of eight twisted strands that visits each vertex once. Fontaine and Hurley's rotation over the <math>r_4</math> star polygon {30/8}=2{15/4} which constructs <math>1/r_4</math> is the ''characteristic rotation of the 600-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 600-cell an 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, and every great square to a Clifford parallel great square. 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 over 15 <math>\sqrt{1}</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 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. == 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. In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-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 distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [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 higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to Clifford polygon sequences 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}} mlphvn6loeibk3ux3jb4ozyxayuvovs 2812915 2812914 2026-06-05T18:45:59Z Dc.samizdat 2856930 /* The 24-cell */ 2812915 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 5<sup>2</sup> 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 5<sup>2</sup> (75) disjoint instances of itself in the 600-point (120-cell), which contains 3<sup>2</sup> times 5<sup>2</sup> (225) distinct instances of the 24-point (24-cell), and 3<sup>3</sup> times 5<sup>2</sup> (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> |} == Complementary chord pairs == The list of 30 chords 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}} 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 concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius <math>c_1</math> is a dodecahedron cell, and the largest, central section of radius <math>c_{15}</math> is a non-uniform 60-point [[w:Rhombicosidodecahedron|rhombicosidodecahedron.]] At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal dodecahedron cell at distance <math>c_{29}</math>. ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[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|B<sub>4</sub> 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 geodesic 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 invariant planes containing its edges, 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 isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular double helix which visits each vertex 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>. In this system of unit-radius coordinates <math>r_1=1/r_5</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. 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. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 30° turns. [[File:Regular_star_figure_3(8,3).svg|thumb|left|100px|{24/9}=3{8/3} Clifford polygons of <math>\sqrt{2}</math> 16-cell edges in the 24-cell.]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 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-cells. Three Clifford parallel octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular triple helix {24/9}=3{8/3} that visits each vertex once. [[File:Regular_star_figure_3(8,3).svg|thumb|left|100px|{24/9}=3{8/3} Clifford polygons of <math>\sqrt{2}</math> 16-cell edges in the 24-cell.]] We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. 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 Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of green <math>r_5</math> chords, visible in the orthogonal projection. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular double helix {24/10}=2{12/5} that visits each vertex once. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F<sub>4</sub> 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.]] Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math> is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an 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 a different 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 between the 24-cell's existing 24 vertices, in effect adding twenty-four more 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. [[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]] 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. The skew {30}-gons have these chords: :<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}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</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}</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> [[File:Regular_star_polygon_30-7.svg|thumb|Star polygon {30/7} of <math>r_7</math> edges. Fontaine and Hurley's rotation over the <math>r_7</math> chord is the rotation of the 600-cell by the characteristic rotation of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges.]] Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 12° turns. We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, 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, but it does not visit other 16-cell vertex positions. Fifteen Clifford parallel octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular helix of fifteen twisted strands that visits each vertex once. [Perhaps this is five {24/8)=8{3} as suggested by the text but I believe it can only be four {30/7} as suggested by the illustration.] [[File:Regular_star_figure_3(10,3).svg|thumb|Star polygon {30/9}=3{10/3} of <math>r_9</math> edges. Fontaine and Hurley's rotation over the <math>r_9</math> chord is the rotation of the 600-cell by the characteristic rotation of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, 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, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular helix of ten twisted strands that visits each vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.] [[File:Regular_star_figure_2(15,4).svg|thumb|Star polygon {30/8}=2{15/4} of <math>r_4</math> edges. Fontaine and Hurley's rotation over the <math>r_4</math> chord is the characteristic rotation of the 600-cell'','' its isoclinic rotation in invariant planes containing its edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of <math>r_5</math> chords. 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 pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>\sqrt{1}</math> chords form a circular helix of eight twisted strands that visits each vertex once. Fontaine and Hurley's rotation over the <math>r_4</math> star polygon {30/8}=2{15/4} which constructs <math>1/r_4</math> is the ''characteristic rotation of the 600-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 600-cell an 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, and every great square to a Clifford parallel great square. 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 over 15 <math>\sqrt{1}</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 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. == 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. In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-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 distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [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 higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to Clifford polygon sequences 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}} 002we4mhqgd6ec3npe8q2dcubr473i6 2812916 2812915 2026-06-05T19:06:09Z Dc.samizdat 2856930 /* The 24-cell */ 2812916 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 5<sup>2</sup> 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 5<sup>2</sup> (75) disjoint instances of itself in the 600-point (120-cell), which contains 3<sup>2</sup> times 5<sup>2</sup> (225) distinct instances of the 24-point (24-cell), and 3<sup>3</sup> times 5<sup>2</sup> (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> |} == Complementary chord pairs == The list of 30 chords 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}} 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 concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius <math>c_1</math> is a dodecahedron cell, and the largest, central section of radius <math>c_{15}</math> is a non-uniform 60-point [[w:Rhombicosidodecahedron|rhombicosidodecahedron.]] At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal dodecahedron cell at distance <math>c_{29}</math>. ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[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|B<sub>4</sub> 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 geodesic 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 invariant planes containing its edges, 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 isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular double helix which visits each vertex 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>. In this system of unit-radius coordinates <math>r_1=1/r_5</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. 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. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 30° turns. [[File:Regular_star_figure_3(8,3).svg|thumb|left|100px|{24/9}=3{8/3} Clifford polygons of <math>\sqrt{2}</math> great square edges in the 24-cell.]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 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-cells. Three Clifford parallel octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular triple helix {24/9}=3{8/3} that visits each vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|100px|{24/9}=3{8/3} Clifford polygons of <math>\sqrt{3}</math> great hexagon edges in the 24-cell.]] We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. 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 Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of green <math>r_5</math> chords, visible in the orthogonal projection. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular double helix {24/10}=2{12/5} that visits each vertex once. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F<sub>4</sub> 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.]] Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math> is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an 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 a different 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 between the 24-cell's existing 24 vertices, in effect adding twenty-four more 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. [[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]] 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. The skew {30}-gons have these chords: :<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}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</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}</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> [[File:Regular_star_polygon_30-7.svg|thumb|Star polygon {30/7} of <math>r_7</math> edges. Fontaine and Hurley's rotation over the <math>r_7</math> chord is the rotation of the 600-cell by the characteristic rotation of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges.]] Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 12° turns. We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, 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, but it does not visit other 16-cell vertex positions. Fifteen Clifford parallel octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular helix of fifteen twisted strands that visits each vertex once. [Perhaps this is five {24/8)=8{3} as suggested by the text but I believe it can only be four {30/7} as suggested by the illustration.] [[File:Regular_star_figure_3(10,3).svg|thumb|Star polygon {30/9}=3{10/3} of <math>r_9</math> edges. Fontaine and Hurley's rotation over the <math>r_9</math> chord is the rotation of the 600-cell by the characteristic rotation of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, 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, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular helix of ten twisted strands that visits each vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.] [[File:Regular_star_figure_2(15,4).svg|thumb|Star polygon {30/8}=2{15/4} of <math>r_4</math> edges. Fontaine and Hurley's rotation over the <math>r_4</math> chord is the characteristic rotation of the 600-cell'','' its isoclinic rotation in invariant planes containing its edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of <math>r_5</math> chords. 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 pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>\sqrt{1}</math> chords form a circular helix of eight twisted strands that visits each vertex once. Fontaine and Hurley's rotation over the <math>r_4</math> star polygon {30/8}=2{15/4} which constructs <math>1/r_4</math> is the ''characteristic rotation of the 600-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 600-cell an 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, and every great square to a Clifford parallel great square. 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 over 15 <math>\sqrt{1}</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 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. == 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. In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-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 distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [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 higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to Clifford polygon sequences 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}} scvrl48tkk598i402e9f01ajsecgo3j 2812917 2812916 2026-06-05T19:10:31Z Dc.samizdat 2856930 /* The 24-cell */ 2812917 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 5<sup>2</sup> 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 5<sup>2</sup> (75) disjoint instances of itself in the 600-point (120-cell), which contains 3<sup>2</sup> times 5<sup>2</sup> (225) distinct instances of the 24-point (24-cell), and 3<sup>3</sup> times 5<sup>2</sup> (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> |} == Complementary chord pairs == The list of 30 chords 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}} 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 concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius <math>c_1</math> is a dodecahedron cell, and the largest, central section of radius <math>c_{15}</math> is a non-uniform 60-point [[w:Rhombicosidodecahedron|rhombicosidodecahedron.]] At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal dodecahedron cell at distance <math>c_{29}</math>. ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[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|B<sub>4</sub> 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 geodesic 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 invariant planes containing its edges, 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 isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular double helix which visits each vertex 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>. In this system of unit-radius coordinates <math>r_1=1/r_5</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. 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. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 30° turns. [[File:Regular_star_figure_3(8,3).svg|thumb|left|100px|{24/9}=3{8/3} Clifford polygons of <math>\sqrt{2}</math> great square edges in the 24-cell.]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 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-cells. Three Clifford parallel octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular triple helix {24/9}=3{8/3} that visits each vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|100px|{24/10}=2{12/5} Clifford polygons of <math>\sqrt{3}</math> chords in the 24-cell.]] We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. 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 Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of green <math>r_5</math> chords, visible in the orthogonal projection. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular double helix {24/10}=2{12/5} that visits each vertex once. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F<sub>4</sub> 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.]] Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math> is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an 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 a different 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 between the 24-cell's existing 24 vertices, in effect adding twenty-four more 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. [[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]] 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. The skew {30}-gons have these chords: :<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}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</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}</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> [[File:Regular_star_polygon_30-7.svg|thumb|Star polygon {30/7} of <math>r_7</math> edges. Fontaine and Hurley's rotation over the <math>r_7</math> chord is the rotation of the 600-cell by the characteristic rotation of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges.]] Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 12° turns. We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, 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, but it does not visit other 16-cell vertex positions. Fifteen Clifford parallel octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular helix of fifteen twisted strands that visits each vertex once. [Perhaps this is five {24/8)=8{3} as suggested by the text but I believe it can only be four {30/7} as suggested by the illustration.] [[File:Regular_star_figure_3(10,3).svg|thumb|Star polygon {30/9}=3{10/3} of <math>r_9</math> edges. Fontaine and Hurley's rotation over the <math>r_9</math> chord is the rotation of the 600-cell by the characteristic rotation of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, 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, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular helix of ten twisted strands that visits each vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.] [[File:Regular_star_figure_2(15,4).svg|thumb|Star polygon {30/8}=2{15/4} of <math>r_4</math> edges. Fontaine and Hurley's rotation over the <math>r_4</math> chord is the characteristic rotation of the 600-cell'','' its isoclinic rotation in invariant planes containing its edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of <math>r_5</math> chords. 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 pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>\sqrt{1}</math> chords form a circular helix of eight twisted strands that visits each vertex once. Fontaine and Hurley's rotation over the <math>r_4</math> star polygon {30/8}=2{15/4} which constructs <math>1/r_4</math> is the ''characteristic rotation of the 600-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 600-cell an 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, and every great square to a Clifford parallel great square. 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 over 15 <math>\sqrt{1}</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 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. == 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. In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-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 distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [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 higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to Clifford polygon sequences 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}} cyyq9cl8j1wapu1fsagsz3foa24t2tg 2812918 2812917 2026-06-05T19:12:20Z Dc.samizdat 2856930 2812918 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 5<sup>2</sup> 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 5<sup>2</sup> (75) disjoint instances of itself in the 600-point (120-cell), which contains 3<sup>2</sup> times 5<sup>2</sup> (225) distinct instances of the 24-point (24-cell), and 3<sup>3</sup> times 5<sup>2</sup> (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> |} == Complementary chord pairs == The list of 30 chords 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}} 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 concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius <math>c_1</math> is a dodecahedron cell, and the largest, central section of radius <math>c_{15}</math> is a non-uniform 60-point [[w:Rhombicosidodecahedron|rhombicosidodecahedron.]] At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal dodecahedron cell at distance <math>c_{29}</math>. ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[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|B<sub>4</sub> 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 geodesic 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 invariant planes containing its edges, 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 isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular double helix which visits each vertex 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>. In this system of unit-radius coordinates <math>r_1=1/r_5</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. 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. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 30° turns. [[File:Regular_star_figure_3(8,3).svg|thumb|left|120px|{24/9}=3{8/3} Clifford polygons of <math>\sqrt{2}</math> great square edges in the 24-cell.]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 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-cells. Three Clifford parallel octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular triple helix {24/9}=3{8/3} that visits each vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|120px|{24/10}=2{12/5} Clifford polygons of <math>\sqrt{3}</math> chords in the 24-cell.]] We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. 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 Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of green <math>r_5</math> chords, visible in the orthogonal projection. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular double helix {24/10}=2{12/5} that visits each vertex once. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F<sub>4</sub> 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.]] Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math> is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an 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 a different 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 between the 24-cell's existing 24 vertices, in effect adding twenty-four more 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. [[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]] 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. The skew {30}-gons have these chords: :<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}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</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}</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> [[File:Regular_star_polygon_30-7.svg|thumb|Star polygon {30/7} of <math>r_7</math> edges. Fontaine and Hurley's rotation over the <math>r_7</math> chord is the rotation of the 600-cell by the characteristic rotation of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges.]] Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 12° turns. We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, 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, but it does not visit other 16-cell vertex positions. Fifteen Clifford parallel octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular helix of fifteen twisted strands that visits each vertex once. [Perhaps this is five {24/8)=8{3} as suggested by the text but I believe it can only be four {30/7} as suggested by the illustration.] [[File:Regular_star_figure_3(10,3).svg|thumb|Star polygon {30/9}=3{10/3} of <math>r_9</math> edges. Fontaine and Hurley's rotation over the <math>r_9</math> chord is the rotation of the 600-cell by the characteristic rotation of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, 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, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular helix of ten twisted strands that visits each vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.] [[File:Regular_star_figure_2(15,4).svg|thumb|Star polygon {30/8}=2{15/4} of <math>r_4</math> edges. Fontaine and Hurley's rotation over the <math>r_4</math> chord is the characteristic rotation of the 600-cell'','' its isoclinic rotation in invariant planes containing its edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of <math>r_5</math> chords. 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 pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>\sqrt{1}</math> chords form a circular helix of eight twisted strands that visits each vertex once. Fontaine and Hurley's rotation over the <math>r_4</math> star polygon {30/8}=2{15/4} which constructs <math>1/r_4</math> is the ''characteristic rotation of the 600-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 600-cell an 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, and every great square to a Clifford parallel great square. 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 over 15 <math>\sqrt{1}</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 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. == 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. In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-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 distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [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 higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to Clifford polygon sequences 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}} kj0wq9szuj584mctmwb1490yltrp2az 2812919 2812918 2026-06-05T19:13:23Z Dc.samizdat 2856930 /* The 24-cell */ 2812919 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 5<sup>2</sup> 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 5<sup>2</sup> (75) disjoint instances of itself in the 600-point (120-cell), which contains 3<sup>2</sup> times 5<sup>2</sup> (225) distinct instances of the 24-point (24-cell), and 3<sup>3</sup> times 5<sup>2</sup> (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> |} == Complementary chord pairs == The list of 30 chords 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}} 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 concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius <math>c_1</math> is a dodecahedron cell, and the largest, central section of radius <math>c_{15}</math> is a non-uniform 60-point [[w:Rhombicosidodecahedron|rhombicosidodecahedron.]] At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal dodecahedron cell at distance <math>c_{29}</math>. ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[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|B<sub>4</sub> 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 geodesic 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 invariant planes containing its edges, 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 isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular double helix which visits each vertex 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>. In this system of unit-radius coordinates <math>r_1=1/r_5</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. 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. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 30° turns. [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} Clifford polygons of <math>\sqrt{2}</math> great square edges in the 24-cell.]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 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-cells. Three Clifford parallel octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular triple helix {24/9}=3{8/3} that visits each vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} Clifford polygons of <math>\sqrt{3}</math> chords in the 24-cell.]] We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. 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 Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of green <math>r_5</math> chords, visible in the orthogonal projection. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular double helix {24/10}=2{12/5} that visits each vertex once. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F<sub>4</sub> 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.]] Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math> is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an 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 a different 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 between the 24-cell's existing 24 vertices, in effect adding twenty-four more 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. [[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]] 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. The skew {30}-gons have these chords: :<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}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</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}</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> [[File:Regular_star_polygon_30-7.svg|thumb|Star polygon {30/7} of <math>r_7</math> edges. Fontaine and Hurley's rotation over the <math>r_7</math> chord is the rotation of the 600-cell by the characteristic rotation of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges.]] Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 12° turns. We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, 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, but it does not visit other 16-cell vertex positions. Fifteen Clifford parallel octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular helix of fifteen twisted strands that visits each vertex once. [Perhaps this is five {24/8)=8{3} as suggested by the text but I believe it can only be four {30/7} as suggested by the illustration.] [[File:Regular_star_figure_3(10,3).svg|thumb|Star polygon {30/9}=3{10/3} of <math>r_9</math> edges. Fontaine and Hurley's rotation over the <math>r_9</math> chord is the rotation of the 600-cell by the characteristic rotation of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, 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, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular helix of ten twisted strands that visits each vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.] [[File:Regular_star_figure_2(15,4).svg|thumb|Star polygon {30/8}=2{15/4} of <math>r_4</math> edges. Fontaine and Hurley's rotation over the <math>r_4</math> chord is the characteristic rotation of the 600-cell'','' its isoclinic rotation in invariant planes containing its edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of <math>r_5</math> chords. 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 pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>\sqrt{1}</math> chords form a circular helix of eight twisted strands that visits each vertex once. Fontaine and Hurley's rotation over the <math>r_4</math> star polygon {30/8}=2{15/4} which constructs <math>1/r_4</math> is the ''characteristic rotation of the 600-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 600-cell an 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, and every great square to a Clifford parallel great square. 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 over 15 <math>\sqrt{1}</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 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. == 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. In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-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 distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [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 higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to Clifford polygon sequences 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}} nipzszojyvcfl7wr1ok1yhi2jr6inrc 2812920 2812919 2026-06-05T19:14:56Z Dc.samizdat 2856930 /* The 24-cell */ 2812920 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 5<sup>2</sup> 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 5<sup>2</sup> (75) disjoint instances of itself in the 600-point (120-cell), which contains 3<sup>2</sup> times 5<sup>2</sup> (225) distinct instances of the 24-point (24-cell), and 3<sup>3</sup> times 5<sup>2</sup> (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> |} == Complementary chord pairs == The list of 30 chords 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}} 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 concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius <math>c_1</math> is a dodecahedron cell, and the largest, central section of radius <math>c_{15}</math> is a non-uniform 60-point [[w:Rhombicosidodecahedron|rhombicosidodecahedron.]] At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal dodecahedron cell at distance <math>c_{29}</math>. ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[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|B<sub>4</sub> 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 geodesic 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 invariant planes containing its edges, 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 isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular double helix which visits each vertex 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>. In this system of unit-radius coordinates <math>r_1=1/r_5</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. 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. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 30° turns. [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} Clifford polygons of <math>\sqrt{2}</math> great square edges in the 24-cell.]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 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-cells. Three Clifford parallel octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular triple helix {24/9}=3{8/3} that visits each vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} Clifford polygons of <math>\sqrt{3}</math> great hexagon diagonals in the 24-cell.]] We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. 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 Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of green <math>r_5</math> chords, visible in the orthogonal projection. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular double helix {24/10}=2{12/5} that visits each vertex once. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F<sub>4</sub> 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.]] Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math> is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an 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 a different 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 between the 24-cell's existing 24 vertices, in effect adding twenty-four more 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. [[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]] 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. The skew {30}-gons have these chords: :<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}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</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}</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> [[File:Regular_star_polygon_30-7.svg|thumb|Star polygon {30/7} of <math>r_7</math> edges. Fontaine and Hurley's rotation over the <math>r_7</math> chord is the rotation of the 600-cell by the characteristic rotation of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges.]] Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 12° turns. We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, 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, but it does not visit other 16-cell vertex positions. Fifteen Clifford parallel octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular helix of fifteen twisted strands that visits each vertex once. [Perhaps this is five {24/8)=8{3} as suggested by the text but I believe it can only be four {30/7} as suggested by the illustration.] [[File:Regular_star_figure_3(10,3).svg|thumb|Star polygon {30/9}=3{10/3} of <math>r_9</math> edges. Fontaine and Hurley's rotation over the <math>r_9</math> chord is the rotation of the 600-cell by the characteristic rotation of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, 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, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular helix of ten twisted strands that visits each vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.] [[File:Regular_star_figure_2(15,4).svg|thumb|Star polygon {30/8}=2{15/4} of <math>r_4</math> edges. Fontaine and Hurley's rotation over the <math>r_4</math> chord is the characteristic rotation of the 600-cell'','' its isoclinic rotation in invariant planes containing its edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of <math>r_5</math> chords. 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 pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>\sqrt{1}</math> chords form a circular helix of eight twisted strands that visits each vertex once. Fontaine and Hurley's rotation over the <math>r_4</math> star polygon {30/8}=2{15/4} which constructs <math>1/r_4</math> is the ''characteristic rotation of the 600-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 600-cell an 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, and every great square to a Clifford parallel great square. 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 over 15 <math>\sqrt{1}</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 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. == 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. In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-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 distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [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 higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to Clifford polygon sequences 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}} otodm4b4nyjcgzttgbhkjx8jz4y8wky 2812921 2812920 2026-06-05T19:31:36Z Dc.samizdat 2856930 /* The 24-cell */ 2812921 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 5<sup>2</sup> 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 5<sup>2</sup> (75) disjoint instances of itself in the 600-point (120-cell), which contains 3<sup>2</sup> times 5<sup>2</sup> (225) distinct instances of the 24-point (24-cell), and 3<sup>3</sup> times 5<sup>2</sup> (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> |} == Complementary chord pairs == The list of 30 chords 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}} 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 concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius <math>c_1</math> is a dodecahedron cell, and the largest, central section of radius <math>c_{15}</math> is a non-uniform 60-point [[w:Rhombicosidodecahedron|rhombicosidodecahedron.]] At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal dodecahedron cell at distance <math>c_{29}</math>. ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[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|B<sub>4</sub> 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 geodesic 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 invariant planes containing its edges, 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 isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular double helix which visits each vertex 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>. In this system of unit-radius coordinates <math>r_1=1/r_5</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. 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. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 30° turns. [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} Clifford polygons of <math>\sqrt{2}</math> great square edges in the 24-cell.]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 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-cells. Three Clifford parallel octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular triple helix {24/9}=3{8/3} that visits each vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} Clifford polygons of <math>\sqrt{3}</math> great hexagon diagonals in the 24-cell.]] [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F<sub>4</sub> 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.]] We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. 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 Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of green <math>r_5</math> chords, visible in the orthogonal projection. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular double helix {24/10}=2{12/5} that visits each vertex once. Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math> is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an 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 a different 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 between the 24-cell's existing 24 vertices, in effect adding twenty-four more 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. [[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]] 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. The skew {30}-gons have these chords: :<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}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</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}</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> [[File:Regular_star_polygon_30-7.svg|thumb|Star polygon {30/7} of <math>r_7</math> edges. Fontaine and Hurley's rotation over the <math>r_7</math> chord is the rotation of the 600-cell by the characteristic rotation of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges.]] Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 12° turns. We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, 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, but it does not visit other 16-cell vertex positions. Fifteen Clifford parallel octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular helix of fifteen twisted strands that visits each vertex once. [Perhaps this is five {24/8)=8{3} as suggested by the text but I believe it can only be four {30/7} as suggested by the illustration.] [[File:Regular_star_figure_3(10,3).svg|thumb|Star polygon {30/9}=3{10/3} of <math>r_9</math> edges. Fontaine and Hurley's rotation over the <math>r_9</math> chord is the rotation of the 600-cell by the characteristic rotation of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, 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, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular helix of ten twisted strands that visits each vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.] [[File:Regular_star_figure_2(15,4).svg|thumb|Star polygon {30/8}=2{15/4} of <math>r_4</math> edges. Fontaine and Hurley's rotation over the <math>r_4</math> chord is the characteristic rotation of the 600-cell'','' its isoclinic rotation in invariant planes containing its edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of <math>r_5</math> chords. 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 pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>\sqrt{1}</math> chords form a circular helix of eight twisted strands that visits each vertex once. Fontaine and Hurley's rotation over the <math>r_4</math> star polygon {30/8}=2{15/4} which constructs <math>1/r_4</math> is the ''characteristic rotation of the 600-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 600-cell an 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, and every great square to a Clifford parallel great square. 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 over 15 <math>\sqrt{1}</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 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. == 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. In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-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 distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [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 higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to Clifford polygon sequences 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}} buog0f0oh3c3fiyiammnvx96uni0bta 2812922 2812921 2026-06-05T19:32:31Z Dc.samizdat 2856930 /* The 24-cell */ 2812922 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 5<sup>2</sup> 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 5<sup>2</sup> (75) disjoint instances of itself in the 600-point (120-cell), which contains 3<sup>2</sup> times 5<sup>2</sup> (225) distinct instances of the 24-point (24-cell), and 3<sup>3</sup> times 5<sup>2</sup> (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> |} == Complementary chord pairs == The list of 30 chords 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}} 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 concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius <math>c_1</math> is a dodecahedron cell, and the largest, central section of radius <math>c_{15}</math> is a non-uniform 60-point [[w:Rhombicosidodecahedron|rhombicosidodecahedron.]] At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal dodecahedron cell at distance <math>c_{29}</math>. ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[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|B<sub>4</sub> 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 geodesic 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 invariant planes containing its edges, 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 isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular double helix which visits each vertex 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>. In this system of unit-radius coordinates <math>r_1=1/r_5</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. 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. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 30° turns. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F<sub>4</sub> 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.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} Clifford polygons of <math>\sqrt{2}</math> great square edges in the 24-cell.]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 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-cells. Three Clifford parallel octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular triple helix {24/9}=3{8/3} that visits each vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} Clifford polygons of <math>\sqrt{3}</math> great hexagon diagonals in the 24-cell.]] We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. 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 Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of green <math>r_5</math> chords, visible in the orthogonal projection. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular double helix {24/10}=2{12/5} that visits each vertex once. Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math> is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an 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 a different 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 between the 24-cell's existing 24 vertices, in effect adding twenty-four more 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. [[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]] 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. The skew {30}-gons have these chords: :<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}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</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}</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> [[File:Regular_star_polygon_30-7.svg|thumb|Star polygon {30/7} of <math>r_7</math> edges. Fontaine and Hurley's rotation over the <math>r_7</math> chord is the rotation of the 600-cell by the characteristic rotation of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges.]] Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 12° turns. We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, 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, but it does not visit other 16-cell vertex positions. Fifteen Clifford parallel octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular helix of fifteen twisted strands that visits each vertex once. [Perhaps this is five {24/8)=8{3} as suggested by the text but I believe it can only be four {30/7} as suggested by the illustration.] [[File:Regular_star_figure_3(10,3).svg|thumb|Star polygon {30/9}=3{10/3} of <math>r_9</math> edges. Fontaine and Hurley's rotation over the <math>r_9</math> chord is the rotation of the 600-cell by the characteristic rotation of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, 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, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular helix of ten twisted strands that visits each vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.] [[File:Regular_star_figure_2(15,4).svg|thumb|Star polygon {30/8}=2{15/4} of <math>r_4</math> edges. Fontaine and Hurley's rotation over the <math>r_4</math> chord is the characteristic rotation of the 600-cell'','' its isoclinic rotation in invariant planes containing its edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of <math>r_5</math> chords. 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 pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>\sqrt{1}</math> chords form a circular helix of eight twisted strands that visits each vertex once. Fontaine and Hurley's rotation over the <math>r_4</math> star polygon {30/8}=2{15/4} which constructs <math>1/r_4</math> is the ''characteristic rotation of the 600-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 600-cell an 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, and every great square to a Clifford parallel great square. 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 over 15 <math>\sqrt{1}</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 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. == 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. In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-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 distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [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 higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to Clifford polygon sequences 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}} 6mzelk0y0u9cxmiw8sy00pap3ddnbpq 2812923 2812922 2026-06-05T19:33:14Z Dc.samizdat 2856930 /* The 24-cell */ 2812923 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 5<sup>2</sup> 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 5<sup>2</sup> (75) disjoint instances of itself in the 600-point (120-cell), which contains 3<sup>2</sup> times 5<sup>2</sup> (225) distinct instances of the 24-point (24-cell), and 3<sup>3</sup> times 5<sup>2</sup> (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> |} == Complementary chord pairs == The list of 30 chords 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}} 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 concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius <math>c_1</math> is a dodecahedron cell, and the largest, central section of radius <math>c_{15}</math> is a non-uniform 60-point [[w:Rhombicosidodecahedron|rhombicosidodecahedron.]] At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal dodecahedron cell at distance <math>c_{29}</math>. ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[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|B<sub>4</sub> 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 geodesic 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 invariant planes containing its edges, 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 isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular double helix which visits each vertex 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>. In this system of unit-radius coordinates <math>r_1=1/r_5</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. 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. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 30° turns. [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} Clifford polygons of <math>\sqrt{2}</math> great square edges in the 24-cell.]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 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-cells. Three Clifford parallel octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular triple helix {24/9}=3{8/3} that visits each vertex once. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F<sub>4</sub> 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.]] [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} Clifford polygons of <math>\sqrt{3}</math> great hexagon diagonals in the 24-cell.]] We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. 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 Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of green <math>r_5</math> chords, visible in the orthogonal projection. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular double helix {24/10}=2{12/5} that visits each vertex once. Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math> is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an 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 a different 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 between the 24-cell's existing 24 vertices, in effect adding twenty-four more 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. [[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]] 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. The skew {30}-gons have these chords: :<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}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</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}</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> [[File:Regular_star_polygon_30-7.svg|thumb|Star polygon {30/7} of <math>r_7</math> edges. Fontaine and Hurley's rotation over the <math>r_7</math> chord is the rotation of the 600-cell by the characteristic rotation of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges.]] Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 12° turns. We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, 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, but it does not visit other 16-cell vertex positions. Fifteen Clifford parallel octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular helix of fifteen twisted strands that visits each vertex once. [Perhaps this is five {24/8)=8{3} as suggested by the text but I believe it can only be four {30/7} as suggested by the illustration.] [[File:Regular_star_figure_3(10,3).svg|thumb|Star polygon {30/9}=3{10/3} of <math>r_9</math> edges. Fontaine and Hurley's rotation over the <math>r_9</math> chord is the rotation of the 600-cell by the characteristic rotation of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, 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, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular helix of ten twisted strands that visits each vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.] [[File:Regular_star_figure_2(15,4).svg|thumb|Star polygon {30/8}=2{15/4} of <math>r_4</math> edges. Fontaine and Hurley's rotation over the <math>r_4</math> chord is the characteristic rotation of the 600-cell'','' its isoclinic rotation in invariant planes containing its edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of <math>r_5</math> chords. 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 pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>\sqrt{1}</math> chords form a circular helix of eight twisted strands that visits each vertex once. Fontaine and Hurley's rotation over the <math>r_4</math> star polygon {30/8}=2{15/4} which constructs <math>1/r_4</math> is the ''characteristic rotation of the 600-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 600-cell an 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, and every great square to a Clifford parallel great square. 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 over 15 <math>\sqrt{1}</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 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. == 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. In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-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 distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [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 higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to Clifford polygon sequences 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}} rtc7pfa8nuji71pjxp0h86f1d6u8vvz 2812924 2812923 2026-06-05T19:34:43Z Dc.samizdat 2856930 /* The 24-cell */ 2812924 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 5<sup>2</sup> 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 5<sup>2</sup> (75) disjoint instances of itself in the 600-point (120-cell), which contains 3<sup>2</sup> times 5<sup>2</sup> (225) distinct instances of the 24-point (24-cell), and 3<sup>3</sup> times 5<sup>2</sup> (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> |} == Complementary chord pairs == The list of 30 chords 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}} 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 concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius <math>c_1</math> is a dodecahedron cell, and the largest, central section of radius <math>c_{15}</math> is a non-uniform 60-point [[w:Rhombicosidodecahedron|rhombicosidodecahedron.]] At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal dodecahedron cell at distance <math>c_{29}</math>. ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[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|B<sub>4</sub> 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 geodesic 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 invariant planes containing its edges, 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 isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular double helix which visits each vertex 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>. In this system of unit-radius coordinates <math>r_1=1/r_5</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. 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. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 30° turns. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F<sub>4</sub> 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.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} Clifford polygons of <math>\sqrt{2}</math> great square edges in the 24-cell.]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 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-cells. Three Clifford parallel octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular triple helix {24/9}=3{8/3} that visits each vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} Clifford polygons of <math>\sqrt{3}</math> great hexagon diagonals in the 24-cell.]] We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. 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 Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of green <math>r_5</math> chords, visible in the orthogonal projection. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular double helix {24/10}=2{12/5} that visits each vertex once. Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math> is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an 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 a different 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 between the 24-cell's existing 24 vertices, in effect adding twenty-four more 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. [[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]] 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. The skew {30}-gons have these chords: :<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}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</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}</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> [[File:Regular_star_polygon_30-7.svg|thumb|Star polygon {30/7} of <math>r_7</math> edges. Fontaine and Hurley's rotation over the <math>r_7</math> chord is the rotation of the 600-cell by the characteristic rotation of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges.]] Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 12° turns. We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, 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, but it does not visit other 16-cell vertex positions. Fifteen Clifford parallel octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular helix of fifteen twisted strands that visits each vertex once. [Perhaps this is five {24/8)=8{3} as suggested by the text but I believe it can only be four {30/7} as suggested by the illustration.] [[File:Regular_star_figure_3(10,3).svg|thumb|Star polygon {30/9}=3{10/3} of <math>r_9</math> edges. Fontaine and Hurley's rotation over the <math>r_9</math> chord is the rotation of the 600-cell by the characteristic rotation of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, 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, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular helix of ten twisted strands that visits each vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.] [[File:Regular_star_figure_2(15,4).svg|thumb|Star polygon {30/8}=2{15/4} of <math>r_4</math> edges. Fontaine and Hurley's rotation over the <math>r_4</math> chord is the characteristic rotation of the 600-cell'','' its isoclinic rotation in invariant planes containing its edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of <math>r_5</math> chords. 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 pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>\sqrt{1}</math> chords form a circular helix of eight twisted strands that visits each vertex once. Fontaine and Hurley's rotation over the <math>r_4</math> star polygon {30/8}=2{15/4} which constructs <math>1/r_4</math> is the ''characteristic rotation of the 600-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 600-cell an 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, and every great square to a Clifford parallel great square. 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 over 15 <math>\sqrt{1}</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 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. == 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. In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-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 distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [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 higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to Clifford polygon sequences 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}} 6mzelk0y0u9cxmiw8sy00pap3ddnbpq 2812925 2812924 2026-06-05T19:46:24Z Dc.samizdat 2856930 /* The 24-cell */ 2812925 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 5<sup>2</sup> 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 5<sup>2</sup> (75) disjoint instances of itself in the 600-point (120-cell), which contains 3<sup>2</sup> times 5<sup>2</sup> (225) distinct instances of the 24-point (24-cell), and 3<sup>3</sup> times 5<sup>2</sup> (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> |} == Complementary chord pairs == The list of 30 chords 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}} 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 concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius <math>c_1</math> is a dodecahedron cell, and the largest, central section of radius <math>c_{15}</math> is a non-uniform 60-point [[w:Rhombicosidodecahedron|rhombicosidodecahedron.]] At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal dodecahedron cell at distance <math>c_{29}</math>. ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[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|B<sub>4</sub> 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 geodesic 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 invariant planes containing its edges, 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 isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular double helix which visits each vertex 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>. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. 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. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 30° turns. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F<sub>4</sub> 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.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of <math>\sqrt{2}</math> great square edges in the 24-cell.]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 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-cells. Three Clifford parallel octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular triple helix {24/9}=3{8/3} that visits each vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of <math>\sqrt{3}</math> great hexagon diagonals in the 24-cell.]] We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. 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 Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of <math>r_5</math> chords. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular double helix {24/10}=2{12/5} that visits each vertex once. Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math> is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an 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 a different 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 between the 24-cell's existing 24 vertices, in effect adding twenty-four more 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. [[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]] 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. The skew {30}-gons have these chords: :<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}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</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}</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> [[File:Regular_star_polygon_30-7.svg|thumb|Star polygon {30/7} of <math>r_7</math> edges. Fontaine and Hurley's rotation over the <math>r_7</math> chord is the rotation of the 600-cell by the characteristic rotation of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges.]] Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 12° turns. We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, 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, but it does not visit other 16-cell vertex positions. Fifteen Clifford parallel octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular helix of fifteen twisted strands that visits each vertex once. [Perhaps this is five {24/8)=8{3} as suggested by the text but I believe it can only be four {30/7} as suggested by the illustration.] [[File:Regular_star_figure_3(10,3).svg|thumb|Star polygon {30/9}=3{10/3} of <math>r_9</math> edges. Fontaine and Hurley's rotation over the <math>r_9</math> chord is the rotation of the 600-cell by the characteristic rotation of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, 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, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular helix of ten twisted strands that visits each vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.] [[File:Regular_star_figure_2(15,4).svg|thumb|Star polygon {30/8}=2{15/4} of <math>r_4</math> edges. Fontaine and Hurley's rotation over the <math>r_4</math> chord is the characteristic rotation of the 600-cell'','' its isoclinic rotation in invariant planes containing its edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of <math>r_5</math> chords. 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 pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>\sqrt{1}</math> chords form a circular helix of eight twisted strands that visits each vertex once. Fontaine and Hurley's rotation over the <math>r_4</math> star polygon {30/8}=2{15/4} which constructs <math>1/r_4</math> is the ''characteristic rotation of the 600-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 600-cell an 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, and every great square to a Clifford parallel great square. 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 over 15 <math>\sqrt{1}</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 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. == 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. In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-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 distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [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 higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to Clifford polygon sequences 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}} f4swnerpgdmejnuryjzhrgb61mdbxl5 2812926 2812925 2026-06-05T19:51:11Z Dc.samizdat 2856930 /* The 600-cell */ 2812926 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 5<sup>2</sup> 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 5<sup>2</sup> (75) disjoint instances of itself in the 600-point (120-cell), which contains 3<sup>2</sup> times 5<sup>2</sup> (225) distinct instances of the 24-point (24-cell), and 3<sup>3</sup> times 5<sup>2</sup> (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> |} == Complementary chord pairs == The list of 30 chords 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}} 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 concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius <math>c_1</math> is a dodecahedron cell, and the largest, central section of radius <math>c_{15}</math> is a non-uniform 60-point [[w:Rhombicosidodecahedron|rhombicosidodecahedron.]] At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal dodecahedron cell at distance <math>c_{29}</math>. ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[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|B<sub>4</sub> 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 geodesic 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 invariant planes containing its edges, 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 isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular double helix which visits each vertex 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>. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. 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. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 30° turns. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F<sub>4</sub> 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.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of <math>\sqrt{2}</math> great square edges in the 24-cell.]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 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-cells. Three Clifford parallel octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular triple helix {24/9}=3{8/3} that visits each vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of <math>\sqrt{3}</math> great hexagon diagonals in the 24-cell.]] We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. 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 Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of <math>r_5</math> chords. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular double helix {24/10}=2{12/5} that visits each vertex once. Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math> is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an 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 a different 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 == 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 between the 24-cell's existing 24 vertices, in effect adding twenty-four more 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. [[File:Regular_star_polygon_30-7.svg|thumb|Star polygon {30/7} of <math>r_7</math> edges. Fontaine and Hurley's rotation over the <math>r_7</math> chord is the rotation of the 600-cell by the characteristic rotation of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges.]] 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. [[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]] 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. The skew {30}-gons have these chords: :<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}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</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}</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> [[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.]] Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 12° turns. We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, 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, but it does not visit other 16-cell vertex positions. Fifteen Clifford parallel octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular helix of fifteen twisted strands that visits each vertex once. [Perhaps this is five {24/8)=8{3} as suggested by the text but I believe it can only be four {30/7} as suggested by the illustration.] [[File:Regular_star_figure_3(10,3).svg|thumb|Star polygon {30/9}=3{10/3} of <math>r_9</math> edges. Fontaine and Hurley's rotation over the <math>r_9</math> chord is the rotation of the 600-cell by the characteristic rotation of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, 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, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular helix of ten twisted strands that visits each vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.] [[File:Regular_star_figure_2(15,4).svg|thumb|Star polygon {30/8}=2{15/4} of <math>r_4</math> edges. Fontaine and Hurley's rotation over the <math>r_4</math> chord is the characteristic rotation of the 600-cell'','' its isoclinic rotation in invariant planes containing its edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of <math>r_5</math> chords. 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 pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>\sqrt{1}</math> chords form a circular helix of eight twisted strands that visits each vertex once. Fontaine and Hurley's rotation over the <math>r_4</math> star polygon {30/8}=2{15/4} which constructs <math>1/r_4</math> is the ''characteristic rotation of the 600-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 600-cell an 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, and every great square to a Clifford parallel great square. 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 over 15 <math>\sqrt{1}</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 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. == 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. In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-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 distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [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 higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to Clifford polygon sequences 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}} q2u9txw9fcr8gt7u2n1hangfv55b67z 2812927 2812926 2026-06-05T19:53:31Z Dc.samizdat 2856930 /* The 600-cell */ 2812927 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 5<sup>2</sup> 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 5<sup>2</sup> (75) disjoint instances of itself in the 600-point (120-cell), which contains 3<sup>2</sup> times 5<sup>2</sup> (225) distinct instances of the 24-point (24-cell), and 3<sup>3</sup> times 5<sup>2</sup> (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> |} == Complementary chord pairs == The list of 30 chords 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}} 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 concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius <math>c_1</math> is a dodecahedron cell, and the largest, central section of radius <math>c_{15}</math> is a non-uniform 60-point [[w:Rhombicosidodecahedron|rhombicosidodecahedron.]] At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal dodecahedron cell at distance <math>c_{29}</math>. ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[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|B<sub>4</sub> 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 geodesic 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 invariant planes containing its edges, 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 isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular double helix which visits each vertex 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>. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. 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. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 30° turns. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F<sub>4</sub> 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.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of <math>\sqrt{2}</math> great square edges in the 24-cell.]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 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-cells. Three Clifford parallel octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular triple helix {24/9}=3{8/3} that visits each vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of <math>\sqrt{3}</math> great hexagon diagonals in the 24-cell.]] We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. 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 Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of <math>r_5</math> chords. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular double helix {24/10}=2{12/5} that visits each vertex once. Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math> is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an 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 a different 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 == 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 between the 24-cell's existing 24 vertices, in effect adding twenty-four more 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. [[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.]] 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. [[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]] 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. The skew {30}-gons have these chords: :<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}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</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}</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. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 12° turns. [[File:Regular_star_polygon_30-7.svg|thumb|left|150px|Star polygon {30/7} of <math>r_7</math> edges. Fontaine and Hurley's rotation over the <math>r_7</math> chord is the rotation of the 600-cell by the characteristic rotation of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges.]] We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, 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, but it does not visit other 16-cell vertex positions. Fifteen Clifford parallel octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular helix of fifteen twisted strands that visits each vertex once. [Perhaps this is five {24/8)=8{3} as suggested by the text but I believe it can only be four {30/7} as suggested by the illustration.] [[File:Regular_star_figure_3(10,3).svg|thumb|left|150px|Star polygon {30/9}=3{10/3} of <math>r_9</math> edges. Fontaine and Hurley's rotation over the <math>r_9</math> chord is the rotation of the 600-cell by the characteristic rotation of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, 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, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular helix of ten twisted strands that visits each vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.] [[File:Regular_star_figure_2(15,4).svg|left|150px|thumb|Star polygon {30/8}=2{15/4} of <math>r_4</math> edges. Fontaine and Hurley's rotation over the <math>r_4</math> chord is the characteristic rotation of the 600-cell'','' its isoclinic rotation in invariant planes containing its edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of <math>r_5</math> chords. 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 pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>\sqrt{1}</math> chords form a circular helix of eight twisted strands that visits each vertex once. Fontaine and Hurley's rotation over the <math>r_4</math> star polygon {30/8}=2{15/4} which constructs <math>1/r_4</math> is the ''characteristic rotation of the 600-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 600-cell an 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, and every great square to a Clifford parallel great square. 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 over 15 <math>\sqrt{1}</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 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. == 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. In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-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 distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [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 higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to Clifford polygon sequences 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}} qfpp4wjxkzqqovk11fhdtlapkxvni8o 2812928 2812927 2026-06-05T19:54:06Z Dc.samizdat 2856930 /* The 600-cell */ 2812928 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 5<sup>2</sup> 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 5<sup>2</sup> (75) disjoint instances of itself in the 600-point (120-cell), which contains 3<sup>2</sup> times 5<sup>2</sup> (225) distinct instances of the 24-point (24-cell), and 3<sup>3</sup> times 5<sup>2</sup> (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> |} == Complementary chord pairs == The list of 30 chords 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}} 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 concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius <math>c_1</math> is a dodecahedron cell, and the largest, central section of radius <math>c_{15}</math> is a non-uniform 60-point [[w:Rhombicosidodecahedron|rhombicosidodecahedron.]] At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal dodecahedron cell at distance <math>c_{29}</math>. ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[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|B<sub>4</sub> 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 geodesic 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 invariant planes containing its edges, 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 isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular double helix which visits each vertex 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>. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. 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. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 30° turns. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F<sub>4</sub> 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.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of <math>\sqrt{2}</math> great square edges in the 24-cell.]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 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-cells. Three Clifford parallel octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular triple helix {24/9}=3{8/3} that visits each vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of <math>\sqrt{3}</math> great hexagon diagonals in the 24-cell.]] We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. 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 Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of <math>r_5</math> chords. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular double helix {24/10}=2{12/5} that visits each vertex once. Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math> is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an 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 a different 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 == 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. [[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]] The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 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. [[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.]] 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. The skew {30}-gons have these chords: :<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}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</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}</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. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 12° turns. [[File:Regular_star_polygon_30-7.svg|thumb|left|150px|Star polygon {30/7} of <math>r_7</math> edges. Fontaine and Hurley's rotation over the <math>r_7</math> chord is the rotation of the 600-cell by the characteristic rotation of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges.]] We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, 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, but it does not visit other 16-cell vertex positions. Fifteen Clifford parallel octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular helix of fifteen twisted strands that visits each vertex once. [Perhaps this is five {24/8)=8{3} as suggested by the text but I believe it can only be four {30/7} as suggested by the illustration.] [[File:Regular_star_figure_3(10,3).svg|thumb|left|150px|Star polygon {30/9}=3{10/3} of <math>r_9</math> edges. Fontaine and Hurley's rotation over the <math>r_9</math> chord is the rotation of the 600-cell by the characteristic rotation of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, 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, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular helix of ten twisted strands that visits each vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.] [[File:Regular_star_figure_2(15,4).svg|left|150px|thumb|Star polygon {30/8}=2{15/4} of <math>r_4</math> edges. Fontaine and Hurley's rotation over the <math>r_4</math> chord is the characteristic rotation of the 600-cell'','' its isoclinic rotation in invariant planes containing its edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of <math>r_5</math> chords. 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 pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>\sqrt{1}</math> chords form a circular helix of eight twisted strands that visits each vertex once. Fontaine and Hurley's rotation over the <math>r_4</math> star polygon {30/8}=2{15/4} which constructs <math>1/r_4</math> is the ''characteristic rotation of the 600-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 600-cell an 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, and every great square to a Clifford parallel great square. 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 over 15 <math>\sqrt{1}</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 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. == 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. In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-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 distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [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 higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to Clifford polygon sequences 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}} s8asp1ss88udtmrsbniz3xe5x1spvgw 2812929 2812928 2026-06-05T19:55:38Z Dc.samizdat 2856930 /* The 600-cell */ 2812929 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 5<sup>2</sup> 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 5<sup>2</sup> (75) disjoint instances of itself in the 600-point (120-cell), which contains 3<sup>2</sup> times 5<sup>2</sup> (225) distinct instances of the 24-point (24-cell), and 3<sup>3</sup> times 5<sup>2</sup> (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> |} == Complementary chord pairs == The list of 30 chords 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}} 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 concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius <math>c_1</math> is a dodecahedron cell, and the largest, central section of radius <math>c_{15}</math> is a non-uniform 60-point [[w:Rhombicosidodecahedron|rhombicosidodecahedron.]] At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal dodecahedron cell at distance <math>c_{29}</math>. ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[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|B<sub>4</sub> 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 geodesic 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 invariant planes containing its edges, 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 isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular double helix which visits each vertex 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>. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. 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. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 30° turns. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F<sub>4</sub> 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.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of <math>\sqrt{2}</math> great square edges in the 24-cell.]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 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-cells. Three Clifford parallel octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular triple helix {24/9}=3{8/3} that visits each vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of <math>\sqrt{3}</math> great hexagon diagonals in the 24-cell.]] We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. 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 Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of <math>r_5</math> chords. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular double helix {24/10}=2{12/5} that visits each vertex once. Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math> is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an 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 a different 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 == 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. [[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]] The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 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. The skew {30}-gons have these chords: :<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}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</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}</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. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 12° turns. [[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.]] [[File:Regular_star_polygon_30-7.svg|thumb|left|150px|Star polygon {30/7} of <math>r_7</math> edges. Fontaine and Hurley's rotation over the <math>r_7</math> chord is the rotation of the 600-cell by the characteristic rotation of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges.]] We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, 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, but it does not visit other 16-cell vertex positions. Fifteen Clifford parallel octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular helix of fifteen twisted strands that visits each vertex once. [Perhaps this is five {24/8)=8{3} as suggested by the text but I believe it can only be four {30/7} as suggested by the illustration.] [[File:Regular_star_figure_3(10,3).svg|thumb|left|150px|Star polygon {30/9}=3{10/3} of <math>r_9</math> edges. Fontaine and Hurley's rotation over the <math>r_9</math> chord is the rotation of the 600-cell by the characteristic rotation of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, 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, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular helix of ten twisted strands that visits each vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.] [[File:Regular_star_figure_2(15,4).svg|left|150px|thumb|Star polygon {30/8}=2{15/4} of <math>r_4</math> edges. Fontaine and Hurley's rotation over the <math>r_4</math> chord is the characteristic rotation of the 600-cell'','' its isoclinic rotation in invariant planes containing its edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of <math>r_5</math> chords. 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 pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>\sqrt{1}</math> chords form a circular helix of eight twisted strands that visits each vertex once. Fontaine and Hurley's rotation over the <math>r_4</math> star polygon {30/8}=2{15/4} which constructs <math>1/r_4</math> is the ''characteristic rotation of the 600-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 600-cell an 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, and every great square to a Clifford parallel great square. 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 over 15 <math>\sqrt{1}</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 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. == 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. In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-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 distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [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 higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to Clifford polygon sequences 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}} cbmicimdexhme9uzrhodp2856my1nx8 2812930 2812929 2026-06-05T19:57:31Z Dc.samizdat 2856930 /* The 600-cell */ 2812930 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 5<sup>2</sup> 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 5<sup>2</sup> (75) disjoint instances of itself in the 600-point (120-cell), which contains 3<sup>2</sup> times 5<sup>2</sup> (225) distinct instances of the 24-point (24-cell), and 3<sup>3</sup> times 5<sup>2</sup> (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> |} == Complementary chord pairs == The list of 30 chords 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}} 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 concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius <math>c_1</math> is a dodecahedron cell, and the largest, central section of radius <math>c_{15}</math> is a non-uniform 60-point [[w:Rhombicosidodecahedron|rhombicosidodecahedron.]] At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal dodecahedron cell at distance <math>c_{29}</math>. ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[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|B<sub>4</sub> 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 geodesic 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 invariant planes containing its edges, 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 isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular double helix which visits each vertex 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>. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. 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. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 30° turns. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F<sub>4</sub> 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.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of <math>\sqrt{2}</math> great square edges in the 24-cell.]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 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-cells. Three Clifford parallel octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular triple helix {24/9}=3{8/3} that visits each vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of <math>\sqrt{3}</math> great hexagon diagonals in the 24-cell.]] We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. 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 Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of <math>r_5</math> chords. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular double helix {24/10}=2{12/5} that visits each vertex once. Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math> is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an 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 a different 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 == 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. [[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]] The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 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. [[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'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. The skew {30}-gons have these chords: :<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}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</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}</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. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 12° turns. [[File:Regular_star_polygon_30-7.svg|thumb|left|200px|Star polygon {30/7} of <math>r_7</math> edges. Fontaine and Hurley's rotation over the <math>r_7</math> chord is the rotation of the 600-cell by the characteristic rotation of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges.]] We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, 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, but it does not visit other 16-cell vertex positions. Fifteen Clifford parallel octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular helix of fifteen twisted strands that visits each vertex once. [Perhaps this is five {24/8)=8{3} as suggested by the text but I believe it can only be four {30/7} as suggested by the illustration.] [[File:Regular_star_figure_3(10,3).svg|thumb|left|200px|Star polygon {30/9}=3{10/3} of <math>r_9</math> edges. Fontaine and Hurley's rotation over the <math>r_9</math> chord is the rotation of the 600-cell by the characteristic rotation of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, 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, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular helix of ten twisted strands that visits each vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.] [[File:Regular_star_figure_2(15,4).svg|left|200px|thumb|Star polygon {30/8}=2{15/4} of <math>r_4</math> edges. Fontaine and Hurley's rotation over the <math>r_4</math> chord is the characteristic rotation of the 600-cell'','' its isoclinic rotation in invariant planes containing its edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of <math>r_5</math> chords. 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 pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>\sqrt{1}</math> chords form a circular helix of eight twisted strands that visits each vertex once. Fontaine and Hurley's rotation over the <math>r_4</math> star polygon {30/8}=2{15/4} which constructs <math>1/r_4</math> is the ''characteristic rotation of the 600-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 600-cell an 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, and every great square to a Clifford parallel great square. 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 over 15 <math>\sqrt{1}</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 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. == 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. In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-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 distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [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 higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to Clifford polygon sequences 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}} 2cuxawo0y10gqoawl2hgcb3iet3r7n3 2812931 2812930 2026-06-05T19:58:19Z Dc.samizdat 2856930 /* The 600-cell */ 2812931 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 5<sup>2</sup> 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 5<sup>2</sup> (75) disjoint instances of itself in the 600-point (120-cell), which contains 3<sup>2</sup> times 5<sup>2</sup> (225) distinct instances of the 24-point (24-cell), and 3<sup>3</sup> times 5<sup>2</sup> (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> |} == Complementary chord pairs == The list of 30 chords 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}} 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 concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius <math>c_1</math> is a dodecahedron cell, and the largest, central section of radius <math>c_{15}</math> is a non-uniform 60-point [[w:Rhombicosidodecahedron|rhombicosidodecahedron.]] At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal dodecahedron cell at distance <math>c_{29}</math>. ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[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|B<sub>4</sub> 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 geodesic 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 invariant planes containing its edges, 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 isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular double helix which visits each vertex 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>. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. 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. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 30° turns. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F<sub>4</sub> 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.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of <math>\sqrt{2}</math> great square edges in the 24-cell.]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 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-cells. Three Clifford parallel octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular triple helix {24/9}=3{8/3} that visits each vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of <math>\sqrt{3}</math> great hexagon diagonals in the 24-cell.]] We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. 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 Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of <math>r_5</math> chords. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular double helix {24/10}=2{12/5} that visits each vertex once. Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math> is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an 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 a different 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 == 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. [[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]] The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 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. [[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'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. The skew {30}-gons have these chords: :<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}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</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}</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. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 12° turns. [[File:Regular_star_polygon_30-7.svg|thumb|Star polygon {30/7} of <math>r_7</math> edges. Fontaine and Hurley's rotation over the <math>r_7</math> chord is the rotation of the 600-cell by the characteristic rotation of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges.]] We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, 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, but it does not visit other 16-cell vertex positions. Fifteen Clifford parallel octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular helix of fifteen twisted strands that visits each vertex once. [Perhaps this is five {24/8)=8{3} as suggested by the text but I believe it can only be four {30/7} as suggested by the illustration.] [[File:Regular_star_figure_3(10,3).svg|thumb|Star polygon {30/9}=3{10/3} of <math>r_9</math> edges. Fontaine and Hurley's rotation over the <math>r_9</math> chord is the rotation of the 600-cell by the characteristic rotation of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, 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, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular helix of ten twisted strands that visits each vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.] [[File:Regular_star_figure_2(15,4).svg|left|Star polygon {30/8}=2{15/4} of <math>r_4</math> edges. Fontaine and Hurley's rotation over the <math>r_4</math> chord is the characteristic rotation of the 600-cell'','' its isoclinic rotation in invariant planes containing its edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of <math>r_5</math> chords. 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 pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>\sqrt{1}</math> chords form a circular helix of eight twisted strands that visits each vertex once. Fontaine and Hurley's rotation over the <math>r_4</math> star polygon {30/8}=2{15/4} which constructs <math>1/r_4</math> is the ''characteristic rotation of the 600-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 600-cell an 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, and every great square to a Clifford parallel great square. 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 over 15 <math>\sqrt{1}</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 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. == 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. In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-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 distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [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 higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to Clifford polygon sequences 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}} 46g7dbgkpfweorq0x3ruxz5qbzepbdi 2812932 2812931 2026-06-05T19:59:01Z Dc.samizdat 2856930 /* The 600-cell */ 2812932 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 5<sup>2</sup> 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 5<sup>2</sup> (75) disjoint instances of itself in the 600-point (120-cell), which contains 3<sup>2</sup> times 5<sup>2</sup> (225) distinct instances of the 24-point (24-cell), and 3<sup>3</sup> times 5<sup>2</sup> (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> |} == Complementary chord pairs == The list of 30 chords 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}} 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 concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius <math>c_1</math> is a dodecahedron cell, and the largest, central section of radius <math>c_{15}</math> is a non-uniform 60-point [[w:Rhombicosidodecahedron|rhombicosidodecahedron.]] At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal dodecahedron cell at distance <math>c_{29}</math>. ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[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|B<sub>4</sub> 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 geodesic 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 invariant planes containing its edges, 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 isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular double helix which visits each vertex 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>. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. 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. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 30° turns. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F<sub>4</sub> 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.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of <math>\sqrt{2}</math> great square edges in the 24-cell.]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 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-cells. Three Clifford parallel octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular triple helix {24/9}=3{8/3} that visits each vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of <math>\sqrt{3}</math> great hexagon diagonals in the 24-cell.]] We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. 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 Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of <math>r_5</math> chords. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular double helix {24/10}=2{12/5} that visits each vertex once. Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math> is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an 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 a different 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 == 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. [[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]] The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 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. [[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'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. The skew {30}-gons have these chords: :<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}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</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}</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. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 12° turns. [[File:Regular_star_polygon_30-7.svg|thumb|Star polygon {30/7} of <math>r_7</math> edges. Fontaine and Hurley's rotation over the <math>r_7</math> chord is the rotation of the 600-cell by the characteristic rotation of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges.]] We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, 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, but it does not visit other 16-cell vertex positions. Fifteen Clifford parallel octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular helix of fifteen twisted strands that visits each vertex once. [Perhaps this is five {24/8)=8{3} as suggested by the text but I believe it can only be four {30/7} as suggested by the illustration.] [[File:Regular_star_figure_3(10,3).svg|thumb|Star polygon {30/9}=3{10/3} of <math>r_9</math> edges. Fontaine and Hurley's rotation over the <math>r_9</math> chord is the rotation of the 600-cell by the characteristic rotation of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, 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, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular helix of ten twisted strands that visits each vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.] [[File:Regular_star_figure_2(15,4).svg|thumb|Star polygon {30/8}=2{15/4} of <math>r_4</math> edges. Fontaine and Hurley's rotation over the <math>r_4</math> chord is the characteristic rotation of the 600-cell'','' its isoclinic rotation in invariant planes containing its edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of <math>r_5</math> chords. 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 pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>\sqrt{1}</math> chords form a circular helix of eight twisted strands that visits each vertex once. Fontaine and Hurley's rotation over the <math>r_4</math> star polygon {30/8}=2{15/4} which constructs <math>1/r_4</math> is the ''characteristic rotation of the 600-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 600-cell an 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, and every great square to a Clifford parallel great square. 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 over 15 <math>\sqrt{1}</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 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. == 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. In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-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 distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [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 higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to Clifford polygon sequences 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}} miinwsfmv40fyeis5tz9zje2bvjm6vt 2812933 2812932 2026-06-05T20:02:28Z Dc.samizdat 2856930 /* The 600-cell */ 2812933 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 5<sup>2</sup> 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 5<sup>2</sup> (75) disjoint instances of itself in the 600-point (120-cell), which contains 3<sup>2</sup> times 5<sup>2</sup> (225) distinct instances of the 24-point (24-cell), and 3<sup>3</sup> times 5<sup>2</sup> (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> |} == Complementary chord pairs == The list of 30 chords 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}} 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 concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius <math>c_1</math> is a dodecahedron cell, and the largest, central section of radius <math>c_{15}</math> is a non-uniform 60-point [[w:Rhombicosidodecahedron|rhombicosidodecahedron.]] At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal dodecahedron cell at distance <math>c_{29}</math>. ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[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|B<sub>4</sub> 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 geodesic 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 invariant planes containing its edges, 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 isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular double helix which visits each vertex 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>. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. 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. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 30° turns. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F<sub>4</sub> 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.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of <math>\sqrt{2}</math> great square edges in the 24-cell.]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 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-cells. Three Clifford parallel octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular triple helix {24/9}=3{8/3} that visits each vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of <math>\sqrt{3}</math> great hexagon diagonals in the 24-cell.]] We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. 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 Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of <math>r_5</math> chords. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular double helix {24/10}=2{12/5} that visits each vertex once. Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math> is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an 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 a different 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 == 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. [[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]] The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 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. [[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'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. The skew {30}-gons have these chords: :<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}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</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}</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. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 12° turns. [[File:Regular_star_polygon_30-7.svg|thumb|Star polygon {30/7} of <math>r_7</math> edges.]] We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, 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, but it does not visit other 16-cell vertex positions. Fifteen Clifford parallel octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular helix of fifteen twisted strands that visits each vertex once. [Perhaps this is five {24/8)=8{3} as suggested by the text but I believe it can only be four {30/7} as suggested by the illustration.] Fontaine and Hurley's rotation over the <math>r_7</math> chord is the rotation of the 600-cell by the characteristic rotation of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges. [[File:Regular_star_figure_3(10,3).svg|thumb|Star polygon {30/9}=3{10/3} of <math>r_9</math> edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, 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, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular helix of ten twisted strands that visits each vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.] Fontaine and Hurley's rotation over the <math>r_9</math> chord is the rotation of the 600-cell by the characteristic rotation of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges. [[File:Regular_star_figure_2(15,4).svg|thumb|Star polygon {30/8}=2{15/4} of <math>r_4</math> edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of <math>r_5</math> chords. 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 pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>\sqrt{1}</math> chords form a circular helix of eight twisted strands that visits each vertex once. Fontaine and Hurley's rotation over the <math>r_4</math> star polygon {30/8}=2{15/4} which constructs <math>1/r_4</math> is the ''characteristic rotation of the 600-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 600-cell an 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, and every great square to a Clifford parallel great square. 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 over 15 <math>\sqrt{1}</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 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. == 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. In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-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 distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [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 higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to Clifford polygon sequences 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}} 3x0gybi3qybkny1rm750drf15lmytf1 2812934 2812933 2026-06-05T20:03:25Z Dc.samizdat 2856930 /* The 600-cell */ 2812934 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 5<sup>2</sup> 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 5<sup>2</sup> (75) disjoint instances of itself in the 600-point (120-cell), which contains 3<sup>2</sup> times 5<sup>2</sup> (225) distinct instances of the 24-point (24-cell), and 3<sup>3</sup> times 5<sup>2</sup> (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> |} == Complementary chord pairs == The list of 30 chords 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}} 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 concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius <math>c_1</math> is a dodecahedron cell, and the largest, central section of radius <math>c_{15}</math> is a non-uniform 60-point [[w:Rhombicosidodecahedron|rhombicosidodecahedron.]] At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal dodecahedron cell at distance <math>c_{29}</math>. ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[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|B<sub>4</sub> 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 geodesic 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 invariant planes containing its edges, 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 isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular double helix which visits each vertex 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>. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. 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. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 30° turns. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F<sub>4</sub> 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.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of <math>\sqrt{2}</math> great square edges in the 24-cell.]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 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-cells. Three Clifford parallel octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular triple helix {24/9}=3{8/3} that visits each vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of <math>\sqrt{3}</math> great hexagon diagonals in the 24-cell.]] We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. 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 Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of <math>r_5</math> chords. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular double helix {24/10}=2{12/5} that visits each vertex once. Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math> is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an 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 a different 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 == 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. [[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]] The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 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. [[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'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. The skew {30}-gons have these chords: :<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}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</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}</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. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 12° turns. [[File:Regular_star_polygon_30-7.svg|thumb|left|100px|Star polygon {30/7} of <math>r_7</math> edges.]] We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, 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, but it does not visit other 16-cell vertex positions. Fifteen Clifford parallel octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular helix of fifteen twisted strands that visits each vertex once. [Perhaps this is five {24/8)=8{3} as suggested by the text but I believe it can only be four {30/7} as suggested by the illustration.] Fontaine and Hurley's rotation over the <math>r_7</math> chord is the rotation of the 600-cell by the characteristic rotation of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges. [[File:Regular_star_figure_3(10,3).svg|thumb|left|100px|Star polygon {30/9}=3{10/3} of <math>r_9</math> edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, 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, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular helix of ten twisted strands that visits each vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.] Fontaine and Hurley's rotation over the <math>r_9</math> chord is the rotation of the 600-cell by the characteristic rotation of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges. [[File:Regular_star_figure_2(15,4).svg|thumb|left|100px|Star polygon {30/8}=2{15/4} of <math>r_4</math> edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of <math>r_5</math> chords. 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 pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>\sqrt{1}</math> chords form a circular helix of eight twisted strands that visits each vertex once. Fontaine and Hurley's rotation over the <math>r_4</math> star polygon {30/8}=2{15/4} which constructs <math>1/r_4</math> is the ''characteristic rotation of the 600-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 600-cell an 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, and every great square to a Clifford parallel great square. 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 over 15 <math>\sqrt{1}</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 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. == 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. In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-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 distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [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 higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to Clifford polygon sequences 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}} jljiweppe0iwdzup8b55nzvdby9m8gj 2812935 2812934 2026-06-05T20:04:37Z Dc.samizdat 2856930 /* The 600-cell */ 2812935 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 5<sup>2</sup> 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 5<sup>2</sup> (75) disjoint instances of itself in the 600-point (120-cell), which contains 3<sup>2</sup> times 5<sup>2</sup> (225) distinct instances of the 24-point (24-cell), and 3<sup>3</sup> times 5<sup>2</sup> (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> |} == Complementary chord pairs == The list of 30 chords 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}} 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 concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius <math>c_1</math> is a dodecahedron cell, and the largest, central section of radius <math>c_{15}</math> is a non-uniform 60-point [[w:Rhombicosidodecahedron|rhombicosidodecahedron.]] At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal dodecahedron cell at distance <math>c_{29}</math>. ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[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|B<sub>4</sub> 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 geodesic 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 invariant planes containing its edges, 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 isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular double helix which visits each vertex 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>. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. 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. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 30° turns. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F<sub>4</sub> 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.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of <math>\sqrt{2}</math> great square edges in the 24-cell.]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 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-cells. Three Clifford parallel octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular triple helix {24/9}=3{8/3} that visits each vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of <math>\sqrt{3}</math> great hexagon diagonals in the 24-cell.]] We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. 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 Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of <math>r_5</math> chords. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular double helix {24/10}=2{12/5} that visits each vertex once. Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math> is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an 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 a different 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 == 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. [[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]] The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 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. [[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'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. The skew {30}-gons have these chords: :<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}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</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}</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. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 12° turns. [[File:Regular_star_polygon_30-7.svg|thumb|left|150px|Star polygon {30/7} of <math>r_7</math> edges.]] We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, 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, but it does not visit other 16-cell vertex positions. Fifteen Clifford parallel octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular helix of fifteen twisted strands that visits each vertex once. [Perhaps this is five {24/8)=8{3} as suggested by the text but I believe it can only be four {30/7} as suggested by the illustration.] Fontaine and Hurley's rotation over the <math>r_7</math> chord is the rotation of the 600-cell by the characteristic rotation of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges. [[File:Regular_star_figure_3(10,3).svg|thumb|left|150px|Star polygon {30/9}=3{10/3} of <math>r_9</math> edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, 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, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular helix of ten twisted strands that visits each vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.] Fontaine and Hurley's rotation over the <math>r_9</math> chord is the rotation of the 600-cell by the characteristic rotation of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges. [[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|Star polygon {30/8}=2{15/4} of <math>r_4</math> edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of <math>r_5</math> chords. 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 pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>\sqrt{1}</math> chords form a circular helix of eight twisted strands that visits each vertex once. Fontaine and Hurley's rotation over the <math>r_4</math> star polygon {30/8}=2{15/4} which constructs <math>1/r_4</math> is the ''characteristic rotation of the 600-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 600-cell an 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, and every great square to a Clifford parallel great square. 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 over 15 <math>\sqrt{1}</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 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. == 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. In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-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 distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [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 higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to Clifford polygon sequences 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}} kf4pgl3n2unp3xuobdrzfidba2iatcc 2812936 2812935 2026-06-05T20:05:12Z Dc.samizdat 2856930 /* The 600-cell */ 2812936 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 5<sup>2</sup> 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 5<sup>2</sup> (75) disjoint instances of itself in the 600-point (120-cell), which contains 3<sup>2</sup> times 5<sup>2</sup> (225) distinct instances of the 24-point (24-cell), and 3<sup>3</sup> times 5<sup>2</sup> (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> |} == Complementary chord pairs == The list of 30 chords 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}} 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 concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius <math>c_1</math> is a dodecahedron cell, and the largest, central section of radius <math>c_{15}</math> is a non-uniform 60-point [[w:Rhombicosidodecahedron|rhombicosidodecahedron.]] At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal dodecahedron cell at distance <math>c_{29}</math>. ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[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|B<sub>4</sub> 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 geodesic 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 invariant planes containing its edges, 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 isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular double helix which visits each vertex 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>. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. 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. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 30° turns. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F<sub>4</sub> 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.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of <math>\sqrt{2}</math> great square edges in the 24-cell.]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 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-cells. Three Clifford parallel octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular triple helix {24/9}=3{8/3} that visits each vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of <math>\sqrt{3}</math> great hexagon diagonals in the 24-cell.]] We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. 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 Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of <math>r_5</math> chords. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular double helix {24/10}=2{12/5} that visits each vertex once. Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math> is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an 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 a different 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 == 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. [[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]] The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 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. [[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'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. The skew {30}-gons have these chords: :<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}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</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}</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. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 12° turns. [[File:Regular_star_polygon_30-7.svg|thumb|left|120px|Star polygon {30/7} of <math>r_7</math> edges.]] We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, 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, but it does not visit other 16-cell vertex positions. Fifteen Clifford parallel octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular helix of fifteen twisted strands that visits each vertex once. [Perhaps this is five {24/8)=8{3} as suggested by the text but I believe it can only be four {30/7} as suggested by the illustration.] Fontaine and Hurley's rotation over the <math>r_7</math> chord is the rotation of the 600-cell by the characteristic rotation of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges. [[File:Regular_star_figure_3(10,3).svg|thumb|left|120px|Star polygon {30/9}=3{10/3} of <math>r_9</math> edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, 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, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular helix of ten twisted strands that visits each vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.] Fontaine and Hurley's rotation over the <math>r_9</math> chord is the rotation of the 600-cell by the characteristic rotation of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges. [[File:Regular_star_figure_2(15,4).svg|thumb|left|120px|Star polygon {30/8}=2{15/4} of <math>r_4</math> edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of <math>r_5</math> chords. 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 pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>\sqrt{1}</math> chords form a circular helix of eight twisted strands that visits each vertex once. Fontaine and Hurley's rotation over the <math>r_4</math> star polygon {30/8}=2{15/4} which constructs <math>1/r_4</math> is the ''characteristic rotation of the 600-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 600-cell an 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, and every great square to a Clifford parallel great square. 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 over 15 <math>\sqrt{1}</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 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. == 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. In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-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 distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [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 higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to Clifford polygon sequences 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}} qqtlat8a6p681362wm2r07djcrken3l 2812937 2812936 2026-06-05T20:07:06Z Dc.samizdat 2856930 /* The 600-cell */ 2812937 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 5<sup>2</sup> 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 5<sup>2</sup> (75) disjoint instances of itself in the 600-point (120-cell), which contains 3<sup>2</sup> times 5<sup>2</sup> (225) distinct instances of the 24-point (24-cell), and 3<sup>3</sup> times 5<sup>2</sup> (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> |} == Complementary chord pairs == The list of 30 chords 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}} 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 concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius <math>c_1</math> is a dodecahedron cell, and the largest, central section of radius <math>c_{15}</math> is a non-uniform 60-point [[w:Rhombicosidodecahedron|rhombicosidodecahedron.]] At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal dodecahedron cell at distance <math>c_{29}</math>. ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[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|B<sub>4</sub> 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 geodesic 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 invariant planes containing its edges, 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 isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular double helix which visits each vertex 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>. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. 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. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 30° turns. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F<sub>4</sub> 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.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of <math>\sqrt{2}</math> great square edges in the 24-cell.]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 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-cells. Three Clifford parallel octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular triple helix {24/9}=3{8/3} that visits each vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of <math>\sqrt{3}</math> great hexagon diagonals in the 24-cell.]] We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. 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 Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of <math>r_5</math> chords. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular double helix {24/10}=2{12/5} that visits each vertex once. Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math> is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an 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 a different 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 == 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. [[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]] The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 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. [[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'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. The skew {30}-gons have these chords: :<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}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</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}</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. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 12° turns. [[File:Regular_star_polygon_30-7.svg|thumb|left|120px|Star polygon {30/7} of <math>r_7</math> edges.]] We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, 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, but it does not visit other 16-cell vertex positions. Fifteen Clifford parallel octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular helix of fifteen twisted strands that visits each vertex once. [Perhaps this is five {24/8)=8{3} as suggested by the text but I believe it can only be four {30/7} as suggested by the illustration.] Fontaine and Hurley's rotation over the <math>r_7</math> chord is the rotation of the 600-cell by the characteristic rotation of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges. [[File:Regular_star_figure_3(10,3).svg|thumb|left|120px|Star {30/9}=3{10/3} of <math>r_9</math> edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, 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, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular helix of ten twisted strands that visits each vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.] Fontaine and Hurley's rotation over the <math>r_9</math> chord is the rotation of the 600-cell by the characteristic rotation of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges. [[File:Regular_star_figure_2(15,4).svg|thumb|left|120px|Star {30/8}=2{15/4} of <math>r_4</math> edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of <math>r_5</math> chords. 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 pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>\sqrt{1}</math> chords form a circular helix of eight twisted strands that visits each vertex once. Fontaine and Hurley's rotation over the <math>r_4</math> star polygon {30/8}=2{15/4} which constructs <math>1/r_4</math> is the ''characteristic rotation of the 600-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 600-cell an 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, and every great square to a Clifford parallel great square. 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 over 15 <math>\sqrt{1}</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 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. == 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. In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-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 distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [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 higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to Clifford polygon sequences 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}} mfji0mb4tk7z4sjc0kckgb7fv8z5n7i 2812938 2812937 2026-06-05T20:07:37Z Dc.samizdat 2856930 /* The 600-cell */ 2812938 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 5<sup>2</sup> 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 5<sup>2</sup> (75) disjoint instances of itself in the 600-point (120-cell), which contains 3<sup>2</sup> times 5<sup>2</sup> (225) distinct instances of the 24-point (24-cell), and 3<sup>3</sup> times 5<sup>2</sup> (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> |} == Complementary chord pairs == The list of 30 chords 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}} 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 concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius <math>c_1</math> is a dodecahedron cell, and the largest, central section of radius <math>c_{15}</math> is a non-uniform 60-point [[w:Rhombicosidodecahedron|rhombicosidodecahedron.]] At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal dodecahedron cell at distance <math>c_{29}</math>. ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[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|B<sub>4</sub> 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 geodesic 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 invariant planes containing its edges, 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 isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular double helix which visits each vertex 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>. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. 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. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 30° turns. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F<sub>4</sub> 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.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of <math>\sqrt{2}</math> great square edges in the 24-cell.]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 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-cells. Three Clifford parallel octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular triple helix {24/9}=3{8/3} that visits each vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of <math>\sqrt{3}</math> great hexagon diagonals in the 24-cell.]] We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. 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 Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of <math>r_5</math> chords. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular double helix {24/10}=2{12/5} that visits each vertex once. Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math> is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an 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 a different 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 == 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. [[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]] The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 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. [[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'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. The skew {30}-gons have these chords: :<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}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</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}</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. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 12° turns. [[File:Regular_star_polygon_30-7.svg|thumb|left|125px|Star {30/7} of <math>r_7</math> edges.]] We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, 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, but it does not visit other 16-cell vertex positions. Fifteen Clifford parallel octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular helix of fifteen twisted strands that visits each vertex once. [Perhaps this is five {24/8)=8{3} as suggested by the text but I believe it can only be four {30/7} as suggested by the illustration.] Fontaine and Hurley's rotation over the <math>r_7</math> chord is the rotation of the 600-cell by the characteristic rotation of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges. [[File:Regular_star_figure_3(10,3).svg|thumb|left|125px|Star {30/9}=3{10/3} of <math>r_9</math> edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, 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, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular helix of ten twisted strands that visits each vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.] Fontaine and Hurley's rotation over the <math>r_9</math> chord is the rotation of the 600-cell by the characteristic rotation of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges. [[File:Regular_star_figure_2(15,4).svg|thumb|left|125px|Star {30/8}=2{15/4} of <math>r_4</math> edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of <math>r_5</math> chords. 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 pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>\sqrt{1}</math> chords form a circular helix of eight twisted strands that visits each vertex once. Fontaine and Hurley's rotation over the <math>r_4</math> star polygon {30/8}=2{15/4} which constructs <math>1/r_4</math> is the ''characteristic rotation of the 600-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 600-cell an 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, and every great square to a Clifford parallel great square. 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 over 15 <math>\sqrt{1}</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 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. == 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. In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-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 distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [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 higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to Clifford polygon sequences 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}} j77hi5457bha63by91kdot2gwalm65v 2812939 2812938 2026-06-05T20:09:10Z Dc.samizdat 2856930 /* The 600-cell */ 2812939 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 5<sup>2</sup> 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 5<sup>2</sup> (75) disjoint instances of itself in the 600-point (120-cell), which contains 3<sup>2</sup> times 5<sup>2</sup> (225) distinct instances of the 24-point (24-cell), and 3<sup>3</sup> times 5<sup>2</sup> (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> |} == Complementary chord pairs == The list of 30 chords 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}} 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 concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius <math>c_1</math> is a dodecahedron cell, and the largest, central section of radius <math>c_{15}</math> is a non-uniform 60-point [[w:Rhombicosidodecahedron|rhombicosidodecahedron.]] At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal dodecahedron cell at distance <math>c_{29}</math>. ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[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|B<sub>4</sub> 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 geodesic 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 invariant planes containing its edges, 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 isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular double helix which visits each vertex 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>. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. 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. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 30° turns. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F<sub>4</sub> 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.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of <math>\sqrt{2}</math> great square edges in the 24-cell.]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 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-cells. Three Clifford parallel octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular triple helix {24/9}=3{8/3} that visits each vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of <math>\sqrt{3}</math> great hexagon diagonals in the 24-cell.]] We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. 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 Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of <math>r_5</math> chords. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular double helix {24/10}=2{12/5} that visits each vertex once. Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math> is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an 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 a different 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 == 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. [[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]] The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 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. [[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'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. The skew {30}-gons have these chords: :<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}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</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}</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. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 12° turns. [[File:Regular_star_polygon_30-7.svg|thumb|left|125px|{30/7} of <math>r_7</math> edges.]] We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, 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, but it does not visit other 16-cell vertex positions. Fifteen Clifford parallel octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular helix of fifteen twisted strands that visits each vertex once. [Perhaps this is five {24/8)=8{3} as suggested by the text but I believe it can only be four {30/7} as suggested by the illustration.] Fontaine and Hurley's rotation over the <math>r_7</math> chord is the rotation of the 600-cell by the characteristic rotation of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges. [[File:Regular_star_figure_3(10,3).svg|thumb|left|125px|{30/9}=3{10/3} of <math>r_9</math> edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, 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, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular helix of ten twisted strands that visits each vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.] Fontaine and Hurley's rotation over the <math>r_9</math> chord is the rotation of the 600-cell by the characteristic rotation of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges. [[File:Regular_star_figure_2(15,4).svg|thumb|left|125px|{30/8}=2{15/4} of <math>r_4</math> edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of <math>r_5</math> chords. 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 pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>\sqrt{1}</math> chords form a circular helix of eight twisted strands that visits each vertex once. Fontaine and Hurley's rotation over the <math>r_4</math> star polygon {30/8}=2{15/4} which constructs <math>1/r_4</math> is the ''characteristic rotation of the 600-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 600-cell an 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, and every great square to a Clifford parallel great square. 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 over 15 <math>\sqrt{1}</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 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. == 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. In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-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 distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [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 higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to Clifford polygon sequences 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}} sts288d8yvxyzbyt0sp034m8tfu01p4 2812940 2812939 2026-06-05T20:11:56Z Dc.samizdat 2856930 /* The 600-cell */ 2812940 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 5<sup>2</sup> 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 5<sup>2</sup> (75) disjoint instances of itself in the 600-point (120-cell), which contains 3<sup>2</sup> times 5<sup>2</sup> (225) distinct instances of the 24-point (24-cell), and 3<sup>3</sup> times 5<sup>2</sup> (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> |} == Complementary chord pairs == The list of 30 chords 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}} 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 concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius <math>c_1</math> is a dodecahedron cell, and the largest, central section of radius <math>c_{15}</math> is a non-uniform 60-point [[w:Rhombicosidodecahedron|rhombicosidodecahedron.]] At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal dodecahedron cell at distance <math>c_{29}</math>. ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[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|B<sub>4</sub> 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 geodesic 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 invariant planes containing its edges, 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 isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular double helix which visits each vertex 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>. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. 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. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 30° turns. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F<sub>4</sub> 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.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of <math>\sqrt{2}</math> great square edges in the 24-cell.]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 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-cells. Three Clifford parallel octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular triple helix {24/9}=3{8/3} that visits each vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of <math>\sqrt{3}</math> great hexagon diagonals in the 24-cell.]] We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. 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 Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of <math>r_5</math> chords. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular double helix {24/10}=2{12/5} that visits each vertex once. Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math> is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an 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 a different 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 == 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. [[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]] The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 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. [[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'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. The skew {30}-gons have these chords: :<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}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</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}</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. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 12° turns. [[File:Regular_star_polygon_30-7.svg|thumb|left|130px|{30/7} of <math>r_7</math> edges.]] We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, 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, but it does not visit other 16-cell vertex positions. Fifteen Clifford parallel octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular helix of fifteen twisted strands that visits each vertex once. [Perhaps this is five {24/8)=8{3} as suggested by the text but I believe it can only be four {30/7} as suggested by the illustration.] Fontaine and Hurley's rotation over the <math>r_7</math> chord is the rotation of the 600-cell by the characteristic rotation of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges. [[File:Regular_star_figure_3(10,3).svg|thumb|left|130px|{30/9}=3{10/3} of <math>r_9</math> edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, 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, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular helix of ten twisted strands that visits each vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.] Fontaine and Hurley's rotation over the <math>r_9</math> chord is the rotation of the 600-cell by the characteristic rotation of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges. [[File:Regular_star_figure_2(15,4).svg|thumb|left|130px|{30/8}=2{15/4} of <math>r_4</math> edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of <math>r_5</math> chords. 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 pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>\sqrt{1}</math> chords form a circular helix of eight twisted strands that visits each vertex once. Fontaine and Hurley's rotation over the <math>r_4</math> star polygon {30/8}=2{15/4} which constructs <math>1/r_4</math> is the ''characteristic rotation of the 600-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 600-cell an 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, and every great square to a Clifford parallel great square. 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 over 15 <math>\sqrt{1}</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 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. == 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. In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-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 distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [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 higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to Clifford polygon sequences 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}} g2guf3l9u171q2vx0azr37d33gaml6u 2812941 2812940 2026-06-05T20:19:00Z Dc.samizdat 2856930 /* The 600-cell */ 2812941 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 5<sup>2</sup> 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 5<sup>2</sup> (75) disjoint instances of itself in the 600-point (120-cell), which contains 3<sup>2</sup> times 5<sup>2</sup> (225) distinct instances of the 24-point (24-cell), and 3<sup>3</sup> times 5<sup>2</sup> (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> |} == Complementary chord pairs == The list of 30 chords 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}} 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 concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius <math>c_1</math> is a dodecahedron cell, and the largest, central section of radius <math>c_{15}</math> is a non-uniform 60-point [[w:Rhombicosidodecahedron|rhombicosidodecahedron.]] At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal dodecahedron cell at distance <math>c_{29}</math>. ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[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|B<sub>4</sub> 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 geodesic 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 invariant planes containing its edges, 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 isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular double helix which visits each vertex 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>. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. 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. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 30° turns. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F<sub>4</sub> 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.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of <math>\sqrt{2}</math> great square edges in the 24-cell.]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 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-cells. Three Clifford parallel octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular triple helix {24/9}=3{8/3} that visits each vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of <math>\sqrt{3}</math> great hexagon diagonals in the 24-cell.]] We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. 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 Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of <math>r_5</math> chords. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular double helix {24/10}=2{12/5} that visits each vertex once. Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math> is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an 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 a different 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 == 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. [[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]] The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 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. [[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'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. The skew {30}-gons have these chords: :<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}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</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}</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. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 12° turns. [[File:Regular_star_polygon_30-7.svg|thumb|left|130px|{30/7} of <math>r_7</math> edges.]] We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, 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, but it does not visit other 16-cell vertex positions. Fifteen Clifford parallel octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular helix of fifteen twisted strands that visits each vertex once. [Perhaps this is five {24/8)=8{3} as suggested by the text but I believe it can only be four {30/7} as suggested by the illustration.] Fontaine and Hurley's rotation over the <math>r_7</math> chord is the rotation of the 600-cell by the rotation characteristic of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges. [[File:Regular_star_figure_3(10,3).svg|thumb|left|130px|{30/9}=3{10/3} of <math>r_9</math> edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, 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, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular helix of ten twisted strands that visits each vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.] Fontaine and Hurley's rotation over the <math>r_9</math> chord is the rotation of the 600-cell by the rotation characteristic of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges. [[File:Regular_star_figure_2(15,4).svg|thumb|left|130px|{30/8}=2{15/4} of <math>r_4</math> edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of <math>r_4</math> chords. Successive <math>r_4</math> chords are edges of different 24-cells. The rotational curve over each <math>r_4</math> chord makes five 12° turns. Eight Clifford parallel pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>\sqrt{1}</math> chords form a circular helix of eight twisted strands that visits each vertex once. Fontaine and Hurley's rotation over the <math>r_4</math> star polygon {30/8}=2{15/4} which constructs <math>1/r_4</math> is the characteristic rotation of the 600-cell, the isoclinic rotation in invariant planes containing its edges. In the 600-cell an 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, and every great square to a Clifford parallel great square. 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 over 15 <math>\sqrt{1}</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 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. == 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. In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-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 distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [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 higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to Clifford polygon sequences 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}} 2vh552vgfgz9ueug0er24vs51kye90b 2812942 2812941 2026-06-05T20:19:30Z Dc.samizdat 2856930 /* The 600-cell */ 2812942 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 5<sup>2</sup> 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 5<sup>2</sup> (75) disjoint instances of itself in the 600-point (120-cell), which contains 3<sup>2</sup> times 5<sup>2</sup> (225) distinct instances of the 24-point (24-cell), and 3<sup>3</sup> times 5<sup>2</sup> (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> |} == Complementary chord pairs == The list of 30 chords 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}} 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 concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius <math>c_1</math> is a dodecahedron cell, and the largest, central section of radius <math>c_{15}</math> is a non-uniform 60-point [[w:Rhombicosidodecahedron|rhombicosidodecahedron.]] At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal dodecahedron cell at distance <math>c_{29}</math>. ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[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|B<sub>4</sub> 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 geodesic 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 invariant planes containing its edges, 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 isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular double helix which visits each vertex 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>. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. 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. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 30° turns. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F<sub>4</sub> 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.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of <math>\sqrt{2}</math> great square edges in the 24-cell.]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 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-cells. Three Clifford parallel octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular triple helix {24/9}=3{8/3} that visits each vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of <math>\sqrt{3}</math> great hexagon diagonals in the 24-cell.]] We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. 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 Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of <math>r_5</math> chords. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular double helix {24/10}=2{12/5} that visits each vertex once. Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math> is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an 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 a different 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 == 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. [[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]] The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 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. The skew {30}-gons have these chords: :<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}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</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}</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. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 12° turns. [[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.]] [[File:Regular_star_polygon_30-7.svg|thumb|left|130px|{30/7} of <math>r_7</math> edges.]] We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, 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, but it does not visit other 16-cell vertex positions. Fifteen Clifford parallel octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular helix of fifteen twisted strands that visits each vertex once. [Perhaps this is five {24/8)=8{3} as suggested by the text but I believe it can only be four {30/7} as suggested by the illustration.] Fontaine and Hurley's rotation over the <math>r_7</math> chord is the rotation of the 600-cell by the rotation characteristic of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges. [[File:Regular_star_figure_3(10,3).svg|thumb|left|130px|{30/9}=3{10/3} of <math>r_9</math> edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, 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, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular helix of ten twisted strands that visits each vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.] Fontaine and Hurley's rotation over the <math>r_9</math> chord is the rotation of the 600-cell by the rotation characteristic of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges. [[File:Regular_star_figure_2(15,4).svg|thumb|left|130px|{30/8}=2{15/4} of <math>r_4</math> edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of <math>r_4</math> chords. Successive <math>r_4</math> chords are edges of different 24-cells. The rotational curve over each <math>r_4</math> chord makes five 12° turns. Eight Clifford parallel pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>\sqrt{1}</math> chords form a circular helix of eight twisted strands that visits each vertex once. Fontaine and Hurley's rotation over the <math>r_4</math> star polygon {30/8}=2{15/4} which constructs <math>1/r_4</math> is the characteristic rotation of the 600-cell, the isoclinic rotation in invariant planes containing its edges. In the 600-cell an 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, and every great square to a Clifford parallel great square. 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 over 15 <math>\sqrt{1}</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 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. == 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. In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-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 distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [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 higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to Clifford polygon sequences 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}} c3h71mbi2vjkdtr8ikbav6sgw4kc4fl 2812943 2812942 2026-06-05T20:21:02Z Dc.samizdat 2856930 /* The 600-cell */ 2812943 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 5<sup>2</sup> 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 5<sup>2</sup> (75) disjoint instances of itself in the 600-point (120-cell), which contains 3<sup>2</sup> times 5<sup>2</sup> (225) distinct instances of the 24-point (24-cell), and 3<sup>3</sup> times 5<sup>2</sup> (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> |} == Complementary chord pairs == The list of 30 chords 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}} 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 concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius <math>c_1</math> is a dodecahedron cell, and the largest, central section of radius <math>c_{15}</math> is a non-uniform 60-point [[w:Rhombicosidodecahedron|rhombicosidodecahedron.]] At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal dodecahedron cell at distance <math>c_{29}</math>. ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[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|B<sub>4</sub> 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 geodesic 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 invariant planes containing its edges, 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 isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular double helix which visits each vertex 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>. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. 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. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 30° turns. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F<sub>4</sub> 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.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of <math>\sqrt{2}</math> great square edges in the 24-cell.]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 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-cells. Three Clifford parallel octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular triple helix {24/9}=3{8/3} that visits each vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of <math>\sqrt{3}</math> great hexagon diagonals in the 24-cell.]] We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. 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 Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of <math>r_5</math> chords. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular double helix {24/10}=2{12/5} that visits each vertex once. Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math> is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an 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 a different 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 == 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 between the 24-cell's existing 24 vertices, in effect adding twenty-four more 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. [[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]] 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. The skew {30}-gons have these chords: :<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}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</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}</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. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 12° turns. [[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.]] [[File:Regular_star_polygon_30-7.svg|thumb|left|130px|{30/7} of <math>r_7</math> edges.]] We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, 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, but it does not visit other 16-cell vertex positions. Fifteen Clifford parallel octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular helix of fifteen twisted strands that visits each vertex once. [Perhaps this is five {24/8)=8{3} as suggested by the text but I believe it can only be four {30/7} as suggested by the illustration.] Fontaine and Hurley's rotation over the <math>r_7</math> chord is the rotation of the 600-cell by the rotation characteristic of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges. [[File:Regular_star_figure_3(10,3).svg|thumb|left|130px|{30/9}=3{10/3} of <math>r_9</math> edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, 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, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular helix of ten twisted strands that visits each vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.] Fontaine and Hurley's rotation over the <math>r_9</math> chord is the rotation of the 600-cell by the rotation characteristic of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges. [[File:Regular_star_figure_2(15,4).svg|thumb|left|130px|{30/8}=2{15/4} of <math>r_4</math> edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of <math>r_4</math> chords. Successive <math>r_4</math> chords are edges of different 24-cells. The rotational curve over each <math>r_4</math> chord makes five 12° turns. Eight Clifford parallel pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>\sqrt{1}</math> chords form a circular helix of eight twisted strands that visits each vertex once. Fontaine and Hurley's rotation over the <math>r_4</math> star polygon {30/8}=2{15/4} which constructs <math>1/r_4</math> is the characteristic rotation of the 600-cell, the isoclinic rotation in invariant planes containing its edges. In the 600-cell an 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, and every great square to a Clifford parallel great square. 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 over 15 <math>\sqrt{1}</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 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. == 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. In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-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 distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [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 higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to Clifford polygon sequences 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}} 95e1anbxy60r31lw1h7tlw1yz4ffcep 2812944 2812943 2026-06-05T20:22:14Z Dc.samizdat 2856930 /* The 600-cell */ 2812944 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 5<sup>2</sup> 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 5<sup>2</sup> (75) disjoint instances of itself in the 600-point (120-cell), which contains 3<sup>2</sup> times 5<sup>2</sup> (225) distinct instances of the 24-point (24-cell), and 3<sup>3</sup> times 5<sup>2</sup> (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> |} == Complementary chord pairs == The list of 30 chords 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}} 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 concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius <math>c_1</math> is a dodecahedron cell, and the largest, central section of radius <math>c_{15}</math> is a non-uniform 60-point [[w:Rhombicosidodecahedron|rhombicosidodecahedron.]] At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal dodecahedron cell at distance <math>c_{29}</math>. ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[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|B<sub>4</sub> 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 geodesic 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 invariant planes containing its edges, 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 isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular double helix which visits each vertex 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>. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. 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. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 30° turns. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F<sub>4</sub> 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.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of <math>\sqrt{2}</math> great square edges in the 24-cell.]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 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-cells. Three Clifford parallel octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular triple helix {24/9}=3{8/3} that visits each vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of <math>\sqrt{3}</math> great hexagon diagonals in the 24-cell.]] We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. 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 Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of <math>r_5</math> chords. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular double helix {24/10}=2{12/5} that visits each vertex once. Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math> is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an 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 a different 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 == 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 between the 24-cell's existing 24 vertices, in effect adding twenty-four more 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. [[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]] 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> [[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.]] 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. The skew {30}-gons have these chords: :<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}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</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}</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. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 12° turns. [[File:Regular_star_polygon_30-7.svg|thumb|left|130px|{30/7} of <math>r_7</math> edges.]] We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, 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, but it does not visit other 16-cell vertex positions. Fifteen Clifford parallel octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular helix of fifteen twisted strands that visits each vertex once. [Perhaps this is five {24/8)=8{3} as suggested by the text but I believe it can only be four {30/7} as suggested by the illustration.] Fontaine and Hurley's rotation over the <math>r_7</math> chord is the rotation of the 600-cell by the rotation characteristic of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges. [[File:Regular_star_figure_3(10,3).svg|thumb|left|130px|{30/9}=3{10/3} of <math>r_9</math> edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, 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, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular helix of ten twisted strands that visits each vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.] Fontaine and Hurley's rotation over the <math>r_9</math> chord is the rotation of the 600-cell by the rotation characteristic of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges. [[File:Regular_star_figure_2(15,4).svg|thumb|left|130px|{30/8}=2{15/4} of <math>r_4</math> edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of <math>r_4</math> chords. Successive <math>r_4</math> chords are edges of different 24-cells. The rotational curve over each <math>r_4</math> chord makes five 12° turns. Eight Clifford parallel pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>\sqrt{1}</math> chords form a circular helix of eight twisted strands that visits each vertex once. Fontaine and Hurley's rotation over the <math>r_4</math> star polygon {30/8}=2{15/4} which constructs <math>1/r_4</math> is the characteristic rotation of the 600-cell, the isoclinic rotation in invariant planes containing its edges. In the 600-cell an 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, and every great square to a Clifford parallel great square. 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 over 15 <math>\sqrt{1}</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 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. == 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. In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-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 distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [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 higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to Clifford polygon sequences 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}} otgyxgdzdh3yf1jdddfuiymdx2y8lrt 2812945 2812944 2026-06-05T20:26:15Z Dc.samizdat 2856930 /* The 600-cell */ 2812945 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 5<sup>2</sup> 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 5<sup>2</sup> (75) disjoint instances of itself in the 600-point (120-cell), which contains 3<sup>2</sup> times 5<sup>2</sup> (225) distinct instances of the 24-point (24-cell), and 3<sup>3</sup> times 5<sup>2</sup> (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> |} == Complementary chord pairs == The list of 30 chords 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}} 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 concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius <math>c_1</math> is a dodecahedron cell, and the largest, central section of radius <math>c_{15}</math> is a non-uniform 60-point [[w:Rhombicosidodecahedron|rhombicosidodecahedron.]] At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal dodecahedron cell at distance <math>c_{29}</math>. ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[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|B<sub>4</sub> 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 geodesic 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 invariant planes containing its edges, 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 isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular double helix which visits each vertex 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>. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. 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. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 30° turns. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F<sub>4</sub> 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.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of <math>\sqrt{2}</math> great square edges in the 24-cell.]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 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-cells. Three Clifford parallel octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular triple helix {24/9}=3{8/3} that visits each vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of <math>\sqrt{3}</math> great hexagon diagonals in the 24-cell.]] We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. 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 Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of <math>r_5</math> chords. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular double helix {24/10}=2{12/5} that visits each vertex once. Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math> is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an 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 a different 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 == 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 between the 24-cell's existing 24 vertices, in effect adding twenty-four more 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. [[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]] 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> [[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.]] 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. The skew {30}-gons have these chords: :<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}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</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}</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. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 12° turns. [[File:Regular_star_polygon_30-7.svg|thumb|left|130px|{30/7} of <math>r_7</math> edges.]] We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, 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, but it does not visit other 16-cell vertex positions. Fifteen Clifford parallel octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular helix of fifteen twisted strands that visits each vertex once. [Perhaps this is five {24/8)=8{3} as suggested by the text but I believe it can only be four {30/7} as suggested by the illustration.] Fontaine and Hurley's rotation over the <math>r_7</math> chord is the rotation of the 600-cell by the rotation characteristic of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges. [[File:Regular_star_figure_3(10,3).svg|thumb|left|130px|{30/9}=3{10/3} of <math>r_9</math> edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, 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, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular helix of ten twisted strands that visits each vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.] Fontaine and Hurley's rotation over the <math>r_9</math> chord is the rotation of the 600-cell by the rotation characteristic of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges. [[File:Regular_star_figure_2(15,4).svg|thumb|left|130px|{30/8}=2{15/4} of <math>r_4</math> edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of <math>r_4</math> chords. Successive <math>r_4</math> chords are edges of different 24-cells. The rotational curve over each <math>r_4</math> chord makes five 12° turns. Eight Clifford parallel pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>\sqrt{1}</math> chords form a circular helix of eight twisted strands that visits each vertex once. Fontaine and Hurley's rotation over the <math>r_4</math> star polygon {30/8}=2{15/4} which constructs <math>1/r_4</math> is the characteristic rotation of the 600-cell, the isoclinic rotation in invariant planes containing its edges. In the 600-cell an 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, and every great square to a Clifford parallel great square. 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 over 15 <math>\sqrt{1}</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 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. == 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. In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-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 distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [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 higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to Clifford polygon sequences 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}} 3rc8q1svtyb3ar8vmwsios8zylqtx7q 2812946 2812945 2026-06-05T20:28:26Z Dc.samizdat 2856930 /* Finally the 120-cell */ 2812946 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 5<sup>2</sup> 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 5<sup>2</sup> (75) disjoint instances of itself in the 600-point (120-cell), which contains 3<sup>2</sup> times 5<sup>2</sup> (225) distinct instances of the 24-point (24-cell), and 3<sup>3</sup> times 5<sup>2</sup> (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> |} == Complementary chord pairs == The list of 30 chords 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}} 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 concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius <math>c_1</math> is a dodecahedron cell, and the largest, central section of radius <math>c_{15}</math> is a non-uniform 60-point [[w:Rhombicosidodecahedron|rhombicosidodecahedron.]] At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal dodecahedron cell at distance <math>c_{29}</math>. ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[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|B<sub>4</sub> 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 geodesic 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 invariant planes containing its edges, 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 isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular double helix which visits each vertex 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>. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. 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. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 30° turns. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F<sub>4</sub> 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.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of <math>\sqrt{2}</math> great square edges in the 24-cell.]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 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-cells. Three Clifford parallel octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular triple helix {24/9}=3{8/3} that visits each vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of <math>\sqrt{3}</math> great hexagon diagonals in the 24-cell.]] We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. 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 Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of <math>r_5</math> chords. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular double helix {24/10}=2{12/5} that visits each vertex once. Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math> is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an 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 a different 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 == 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 between the 24-cell's existing 24 vertices, in effect adding twenty-four more 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. [[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]] 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> [[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.]] 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. The skew {30}-gons have these chords: :<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}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</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}</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. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 12° turns. [[File:Regular_star_polygon_30-7.svg|thumb|left|130px|{30/7} of <math>r_7</math> edges.]] We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, 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, but it does not visit other 16-cell vertex positions. Fifteen Clifford parallel octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular helix of fifteen twisted strands that visits each vertex once. [Perhaps this is five {24/8)=8{3} as suggested by the text but I believe it can only be four {30/7} as suggested by the illustration.] Fontaine and Hurley's rotation over the <math>r_7</math> chord is the rotation of the 600-cell by the rotation characteristic of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges. [[File:Regular_star_figure_3(10,3).svg|thumb|left|130px|{30/9}=3{10/3} of <math>r_9</math> edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, 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, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular helix of ten twisted strands that visits each vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.] Fontaine and Hurley's rotation over the <math>r_9</math> chord is the rotation of the 600-cell by the rotation characteristic of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges. [[File:Regular_star_figure_2(15,4).svg|thumb|left|130px|{30/8}=2{15/4} of <math>r_4</math> edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of <math>r_4</math> chords. Successive <math>r_4</math> chords are edges of different 24-cells. The rotational curve over each <math>r_4</math> chord makes five 12° turns. Eight Clifford parallel pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>\sqrt{1}</math> chords form a circular helix of eight twisted strands that visits each vertex once. Fontaine and Hurley's rotation over the <math>r_4</math> star polygon {30/8}=2{15/4} which constructs <math>1/r_4</math> is the characteristic rotation of the 600-cell, the isoclinic rotation in invariant planes containing its edges. In the 600-cell an 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, and every great square to a Clifford parallel great square. 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 over 15 <math>\sqrt{1}</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 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. == 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. In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-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 distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [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 higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to Clifford polygon sequences 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}} 1bbub19n9m41r704piy0naks6hy7336 2812947 2812946 2026-06-05T20:30:46Z Dc.samizdat 2856930 /* Conclusions */ 2812947 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 5<sup>2</sup> 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 5<sup>2</sup> (75) disjoint instances of itself in the 600-point (120-cell), which contains 3<sup>2</sup> times 5<sup>2</sup> (225) distinct instances of the 24-point (24-cell), and 3<sup>3</sup> times 5<sup>2</sup> (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> |} == Complementary chord pairs == The list of 30 chords 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}} 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 concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius <math>c_1</math> is a dodecahedron cell, and the largest, central section of radius <math>c_{15}</math> is a non-uniform 60-point [[w:Rhombicosidodecahedron|rhombicosidodecahedron.]] At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal dodecahedron cell at distance <math>c_{29}</math>. ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[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|B<sub>4</sub> 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 geodesic 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 invariant planes containing its edges, 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 isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular double helix which visits each vertex 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>. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. 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. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 30° turns. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F<sub>4</sub> 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.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of <math>\sqrt{2}</math> great square edges in the 24-cell.]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 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-cells. Three Clifford parallel octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular triple helix {24/9}=3{8/3} that visits each vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of <math>\sqrt{3}</math> great hexagon diagonals in the 24-cell.]] We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. 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 Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of <math>r_5</math> chords. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular double helix {24/10}=2{12/5} that visits each vertex once. Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math> is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an 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 a different 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 == 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 between the 24-cell's existing 24 vertices, in effect adding twenty-four more 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. [[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]] 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> [[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.]] 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. The skew {30}-gons have these chords: :<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}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</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}</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. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 12° turns. [[File:Regular_star_polygon_30-7.svg|thumb|left|130px|{30/7} of <math>r_7</math> edges.]] We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, 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, but it does not visit other 16-cell vertex positions. Fifteen Clifford parallel octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular helix of fifteen twisted strands that visits each vertex once. [Perhaps this is five {24/8)=8{3} as suggested by the text but I believe it can only be four {30/7} as suggested by the illustration.] Fontaine and Hurley's rotation over the <math>r_7</math> chord is the rotation of the 600-cell by the rotation characteristic of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges. [[File:Regular_star_figure_3(10,3).svg|thumb|left|130px|{30/9}=3{10/3} of <math>r_9</math> edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, 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, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular helix of ten twisted strands that visits each vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.] Fontaine and Hurley's rotation over the <math>r_9</math> chord is the rotation of the 600-cell by the rotation characteristic of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges. [[File:Regular_star_figure_2(15,4).svg|thumb|left|130px|{30/8}=2{15/4} of <math>r_4</math> edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of <math>r_4</math> chords. Successive <math>r_4</math> chords are edges of different 24-cells. The rotational curve over each <math>r_4</math> chord makes five 12° turns. Eight Clifford parallel pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>\sqrt{1}</math> chords form a circular helix of eight twisted strands that visits each vertex once. Fontaine and Hurley's rotation over the <math>r_4</math> star polygon {30/8}=2{15/4} which constructs <math>1/r_4</math> is the characteristic rotation of the 600-cell, the isoclinic rotation in invariant planes containing its edges. In the 600-cell an 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, and every great square to a Clifford parallel great square. 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 over 15 <math>\sqrt{1}</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 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. == 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. In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-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 distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [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 higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to star Clifford polygon sequences 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}} r4lxylxanrewkl6rzgoomdt1k5atc1e 2812948 2812947 2026-06-05T20:40:53Z Dc.samizdat 2856930 /* The 600-cell */ 2812948 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 5<sup>2</sup> 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 5<sup>2</sup> (75) disjoint instances of itself in the 600-point (120-cell), which contains 3<sup>2</sup> times 5<sup>2</sup> (225) distinct instances of the 24-point (24-cell), and 3<sup>3</sup> times 5<sup>2</sup> (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> |} == Complementary chord pairs == The list of 30 chords 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}} 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 concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius <math>c_1</math> is a dodecahedron cell, and the largest, central section of radius <math>c_{15}</math> is a non-uniform 60-point [[w:Rhombicosidodecahedron|rhombicosidodecahedron.]] At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal dodecahedron cell at distance <math>c_{29}</math>. ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[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|B<sub>4</sub> 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 geodesic 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 invariant planes containing its edges, 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 isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular double helix which visits each vertex 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>. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. 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. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 30° turns. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F<sub>4</sub> 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.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of <math>\sqrt{2}</math> great square edges in the 24-cell.]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 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-cells. Three Clifford parallel octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular triple helix {24/9}=3{8/3} that visits each vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of <math>\sqrt{3}</math> great hexagon diagonals in the 24-cell.]] We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. 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 Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of <math>r_5</math> chords. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular double helix {24/10}=2{12/5} that visits each vertex once. Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math> is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an 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 a different 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 == 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 between the 24-cell's existing 24 vertices, in effect adding twenty-four more 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. [[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]] 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> [[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.]] 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. The skew {30}-gons have these chords: :<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}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</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}</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. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 12° turns. [[File:Regular_star_polygon_30-7.svg|thumb|left|130px|{30/7} of <math>r_7</math> edges.]] We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, 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, but it also visits vertex positions in other 16-cells. Four Clifford parallel {30}-gon geodesic isoclines of circumference <math>..\pi</math> over <math>r_7</math> chords form a circular quadruple helix that visits each vertex once. Fontaine and Hurley's rotation over the <math>r_7</math> chord is the rotation of the 600-cell by the rotation characteristic of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges. [[File:Regular_star_figure_3(10,3).svg|thumb|left|130px|{30/9}=3{10/3} of <math>r_9</math> edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, 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, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular helix of ten twisted strands that visits each vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.] Fontaine and Hurley's rotation over the <math>r_9</math> chord is the rotation of the 600-cell by the rotation characteristic of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges. [[File:Regular_star_figure_2(15,4).svg|thumb|left|130px|{30/8}=2{15/4} of <math>r_4</math> edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of <math>r_4</math> chords. Successive <math>r_4</math> chords are edges of different 24-cells. The rotational curve over each <math>r_4</math> chord makes five 12° turns. Eight Clifford parallel pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>\sqrt{1}</math> chords form a circular helix of eight twisted strands that visits each vertex once. Fontaine and Hurley's rotation over the <math>r_4</math> star polygon {30/8}=2{15/4} which constructs <math>1/r_4</math> is the characteristic rotation of the 600-cell, the isoclinic rotation in invariant planes containing its edges. In the 600-cell an 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, and every great square to a Clifford parallel great square. 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 over 15 <math>\sqrt{1}</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 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. == 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. In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-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 distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [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 higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to star Clifford polygon sequences 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}} 58xfxo7scos49n1oghghfoafyjue9br 2812949 2812948 2026-06-05T20:41:27Z Dc.samizdat 2856930 /* The 600-cell */ 2812949 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 5<sup>2</sup> 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 5<sup>2</sup> (75) disjoint instances of itself in the 600-point (120-cell), which contains 3<sup>2</sup> times 5<sup>2</sup> (225) distinct instances of the 24-point (24-cell), and 3<sup>3</sup> times 5<sup>2</sup> (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> |} == Complementary chord pairs == The list of 30 chords 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}} 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 concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius <math>c_1</math> is a dodecahedron cell, and the largest, central section of radius <math>c_{15}</math> is a non-uniform 60-point [[w:Rhombicosidodecahedron|rhombicosidodecahedron.]] At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal dodecahedron cell at distance <math>c_{29}</math>. ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[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|B<sub>4</sub> 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 geodesic 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 invariant planes containing its edges, 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 isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular double helix which visits each vertex 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>. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. 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. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 30° turns. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F<sub>4</sub> 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.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of <math>\sqrt{2}</math> great square edges in the 24-cell.]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 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-cells. Three Clifford parallel octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular triple helix {24/9}=3{8/3} that visits each vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of <math>\sqrt{3}</math> great hexagon diagonals in the 24-cell.]] We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. 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 Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of <math>r_5</math> chords. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular double helix {24/10}=2{12/5} that visits each vertex once. Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math> is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an 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 a different 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 == 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 between the 24-cell's existing 24 vertices, in effect adding twenty-four more 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. [[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]] 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. The skew {30}-gons have these chords: :<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}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</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}</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. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 12° turns. [[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.]] [[File:Regular_star_polygon_30-7.svg|thumb|left|130px|{30/7} of <math>r_7</math> edges.]] We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, 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, but it also visits vertex positions in other 16-cells. Four Clifford parallel {30}-gon geodesic isoclines of circumference <math>..\pi</math> over <math>r_7</math> chords form a circular quadruple helix that visits each vertex once. Fontaine and Hurley's rotation over the <math>r_7</math> chord is the rotation of the 600-cell by the rotation characteristic of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges. [[File:Regular_star_figure_3(10,3).svg|thumb|left|130px|{30/9}=3{10/3} of <math>r_9</math> edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, 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, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular helix of ten twisted strands that visits each vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.] Fontaine and Hurley's rotation over the <math>r_9</math> chord is the rotation of the 600-cell by the rotation characteristic of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges. [[File:Regular_star_figure_2(15,4).svg|thumb|left|130px|{30/8}=2{15/4} of <math>r_4</math> edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of <math>r_4</math> chords. Successive <math>r_4</math> chords are edges of different 24-cells. The rotational curve over each <math>r_4</math> chord makes five 12° turns. Eight Clifford parallel pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>\sqrt{1}</math> chords form a circular helix of eight twisted strands that visits each vertex once. Fontaine and Hurley's rotation over the <math>r_4</math> star polygon {30/8}=2{15/4} which constructs <math>1/r_4</math> is the characteristic rotation of the 600-cell, the isoclinic rotation in invariant planes containing its edges. In the 600-cell an 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, and every great square to a Clifford parallel great square. 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 over 15 <math>\sqrt{1}</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 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. == 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. In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-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 distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [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 higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to star Clifford polygon sequences 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}} gbrti2r4g5ze1pzmnmderp08rjllwql 2812950 2812949 2026-06-05T20:42:47Z Dc.samizdat 2856930 /* The 600-cell */ 2812950 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 5<sup>2</sup> 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 5<sup>2</sup> (75) disjoint instances of itself in the 600-point (120-cell), which contains 3<sup>2</sup> times 5<sup>2</sup> (225) distinct instances of the 24-point (24-cell), and 3<sup>3</sup> times 5<sup>2</sup> (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> |} == Complementary chord pairs == The list of 30 chords 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}} 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 concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius <math>c_1</math> is a dodecahedron cell, and the largest, central section of radius <math>c_{15}</math> is a non-uniform 60-point [[w:Rhombicosidodecahedron|rhombicosidodecahedron.]] At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal dodecahedron cell at distance <math>c_{29}</math>. ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[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|B<sub>4</sub> 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 geodesic 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 invariant planes containing its edges, 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 isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular double helix which visits each vertex 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>. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. 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. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 30° turns. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F<sub>4</sub> 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.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of <math>\sqrt{2}</math> great square edges in the 24-cell.]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 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-cells. Three Clifford parallel octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular triple helix {24/9}=3{8/3} that visits each vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of <math>\sqrt{3}</math> great hexagon diagonals in the 24-cell.]] We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. 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 Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of <math>r_5</math> chords. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular double helix {24/10}=2{12/5} that visits each vertex once. Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math> is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an 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 a different 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 == 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 between the 24-cell's existing 24 vertices, in effect adding twenty-four more 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. [[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]] 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. The skew {30}-gons have these chords: :<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}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</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}</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. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 12° turns. [[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.]] [[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} of <math>r_7</math> edges.]] We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, 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, but it also visits vertex positions in other 16-cells. Four Clifford parallel {30}-gon geodesic isoclines of circumference <math>..\pi</math> over <math>r_7</math> chords form a circular quadruple helix that visits each vertex once. Fontaine and Hurley's rotation over the <math>r_7</math> chord is the rotation of the 600-cell by the rotation characteristic of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges. [[File:Regular_star_figure_3(10,3).svg|thumb|left|150px|{30/9}=3{10/3} of <math>r_9</math> edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, 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, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular helix of ten twisted strands that visits each vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.] Fontaine and Hurley's rotation over the <math>r_9</math> chord is the rotation of the 600-cell by the rotation characteristic of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges. [[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} of <math>r_4</math> edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of <math>r_4</math> chords. Successive <math>r_4</math> chords are edges of different 24-cells. The rotational curve over each <math>r_4</math> chord makes five 12° turns. Eight Clifford parallel pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>\sqrt{1}</math> chords form a circular helix of eight twisted strands that visits each vertex once. Fontaine and Hurley's rotation over the <math>r_4</math> star polygon {30/8}=2{15/4} which constructs <math>1/r_4</math> is the characteristic rotation of the 600-cell, the isoclinic rotation in invariant planes containing its edges. In the 600-cell an 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, and every great square to a Clifford parallel great square. 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 over 15 <math>\sqrt{1}</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 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. == 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. In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-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 distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [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 higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to star Clifford polygon sequences 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}} cl3pwiafe5snfrrnd03xu0rl9ggwox3 2812951 2812950 2026-06-05T20:50:01Z Dc.samizdat 2856930 /* The 600-cell */ 2812951 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 5<sup>2</sup> 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 5<sup>2</sup> (75) disjoint instances of itself in the 600-point (120-cell), which contains 3<sup>2</sup> times 5<sup>2</sup> (225) distinct instances of the 24-point (24-cell), and 3<sup>3</sup> times 5<sup>2</sup> (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> |} == Complementary chord pairs == The list of 30 chords 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}} 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 concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius <math>c_1</math> is a dodecahedron cell, and the largest, central section of radius <math>c_{15}</math> is a non-uniform 60-point [[w:Rhombicosidodecahedron|rhombicosidodecahedron.]] At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal dodecahedron cell at distance <math>c_{29}</math>. ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[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|B<sub>4</sub> 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 geodesic 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 invariant planes containing its edges, 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 isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular double helix which visits each vertex 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>. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. 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. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 30° turns. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F<sub>4</sub> 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.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of <math>\sqrt{2}</math> great square edges in the 24-cell.]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 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-cells. Three Clifford parallel octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular triple helix {24/9}=3{8/3} that visits each 24-cell vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of <math>\sqrt{3}</math> great hexagon diagonals in the 24-cell.]] We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. 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 Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of <math>r_5</math> chords. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular double helix {24/10}=2{12/5} that visits each 24-cell vertex once. Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math> is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an 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 a different 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 == 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 between the 24-cell's existing 24 vertices, in effect adding twenty-four more 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. [[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]] 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. The skew {30}-gons have these chords: :<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}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</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}</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. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 12° turns. [[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.]] [[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} of <math>r_7</math> edges.]] We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, 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, but it also intersects vertex positions of other 16-cells. Four Clifford parallel {30}-gon geodesic isoclines of circumference <math>6\pi</math> over <math>r_7</math> chords form a circular quadruple helix that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the <math>r_7</math> chord is the rotation of the 600-cell by the rotation characteristic of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges. [[File:Regular_star_figure_3(10,3).svg|thumb|left|150px|{30/9}=3{10/3} of <math>r_9</math> edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, 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, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular helix of ten twisted strands that visits each 600-cell vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.] Fontaine and Hurley's rotation over the <math>r_9</math> chord is the rotation of the 600-cell by the rotation characteristic of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges. [[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} of <math>r_4</math> edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of <math>r_4</math> chords. Successive <math>r_4</math> chords are edges of different 24-cells. The rotational curve over each <math>r_4</math> chord makes five 12° turns. Eight Clifford parallel pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>\sqrt{1}</math> chords form a circular helix of eight twisted strands that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the <math>r_4</math> star polygon {30/8}=2{15/4} which constructs <math>1/r_4</math> is the characteristic rotation of the 600-cell, the isoclinic rotation in invariant planes containing its edges. In the 600-cell an 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, and every great square to a Clifford parallel great square. 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 over 15 <math>\sqrt{1}</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 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. == 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. In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-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 distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [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 higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to star Clifford polygon sequences 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}} q2sya0m610jcg7wkoatyr9issi8cjnh Media Literacy and You 0 327555 2812994 2811813 2026-06-05T23:45:47Z DavidMCEddy 218607 add ch2-4 shouting fire ... 2812994 wikitext text/x-wiki [[File:Pharoah - James VI and I - Trump.png|thumb|Religious and media leaders from the time of the Pharaohs convinced common folk to give increasing shares of what they produced to elites.]] :''This book uses dates in [[:w:ISO 8601|ISO 8601]], YYYY-MM-DD, when convenient.'' == Invitation to edit this book == You, dear reader, are invited to contribute questions, ideas and citations to support or refute claims made in this book possibly adding chapters. Wikiversity like other Wikimedia Foundation Projects invites humans to [[w:Wikipedia:Be bold|“be bold but not reckless,”]] while writing from a [[Wikiversity:Disclosures|neutral point of view]], [[Wikiversity:Cite sources|citing credible sources]]. Others are invited to change or revert what you wrote. What stays tends to be written from a neutral point of view citing credible sources. If someone reverts your edit or you have a question, take it to the ''[[Wikiversity:FAQ|''''“Discuss”'''' page]]'' associated with the specific Wikiversity page most related to your concerns. Those who teach media literacy are encouraged to invite their students to debate and revise the contents of this book. Doing so would build on a tradition of [[:w:Wikipedia:Student assignments|instructors requiring students to edit wikipedia article(s).]] Editing [[:w:Wikipedia|Wikipedia]] and other [[:w:Wikimedia Foundation|Wikimedia Foundation]] projects like this book is itself an exercise in media literacy: :''Central tenets of media literacy might include writing from a neutral point of view citing credible sources and engaging others, some of whom may disagree, in civil, supportive conversations about what can and cannot be said based on a reasonable evaluation of the available evidence. Wikimedia rules invite contributors to do just that, encouraging them to “be bold but not reckless,” contributing revisions written from a neutral point of view, citing credible sources -- and raising other questions and concerns on the ''''“Discuss”'''' page associated with the specific Wikiversity page most related to your concerns, as mentioned above.''<ref>For more on this, see Graves (2024).</ref> == Text and self-help book and point of discuss == This book is intended both as a text and self-help book and as a point of discussion considering four levels of media literacy: :1. '''Think before you share''': [[Facebook whistleblower Frances Haugen says|Facebook whistleblower Frances Haugen said]], "The shortest path to a click is anger or hate." The social psychology behind this phenomenon exploited also by legacy media has contributed to [[Media Literacy and You/Media consolidation, social media, and political polarization|the dramatic increase in political polarization and violence worldwide]], especially since the end of the [[w:Fairness doctrine|Fairness doctrine]] in 1987. To counter this, DiResta (2024, p. 335) recommends, "Think before you share." :2. '''Look for information to contradict preconceptions''' (Disconfirmation bias): [[w:Information is a public good: Designing experiments to improve government#Previous research|Virtually everyone]] (a) thinks they know more than they do ([[w:Overconfidence effect|overconfidence effect]]), and (b) prefers information and sources consistent with preconceptions ([[w:Confirmation bias|confirmation bias]]). The major media everywhere exploit this to please those who control most of the money for the media. Humans can counter this by searching for sources to help us understand our designated enemies. If we cannot explain circumstances under which we could see ourselves doing what we see our designated enemies doing, we haven't looked hard enough. :3. '''Talk''': Push ourselves to have friendly supportive conversations with others with whom we may vehemently disagree with the goals of agreeing to disagree agreeably and building collaboration on areas of common concern.<ref>Graves and Bailey (2025).</ref> :4. '''Teach''': Humans who develop skills in the first three levels can leverage that knowledge in helping others acquire those skills. If each one teaches two<ref>"[[:w:Each one teach one|Each one teach one]]" is an African-American proverb from the time of legalized slavery. However, if each one teaches only one, the growth in literacy will only be linear. Having "each one teaching two", on average, unleashes the power of doubling and [[:w:exponential growth|exponential growth]], which has the potential of educating the entirety of humanity in a reasonable period of time -- namely after 33 doublings starting from one.</ref> in a certain period of time, that time period becomes a [[:w:Doubling time|doubling time]]. Ten doublings is a thousand -- actually 1,024 to be precise.<ref>2 time 2 = 4 times 2 = 8 times 2 = 16 times 2 = 32 times 2 = 64 times 2 = 128 times 2 = 256 times 2 = 512 times 2 = 1024: That's 10 doublings, as anyone with a modest understanding of modern digital [[:w:computer|computer]]s will tell you.</ref> Twenty doublings become a million. Thirty doublings become a billion. Three more doublings become 8 billion, the [[:w:World population|world population]] as of approximately 2022-11-15.<ref>This book uses dates in [[:w:ISO 8601|ISO 8601]], YYYY-MM-DD, when convenient.</ref> Many organizations, including several United Nations agencies, already have active [[w:media literacy|media literacy]] programs that have already trained many.<ref>''[[Wikibooks:Antiracist Activism for Teachers and Students]]'' includes a chapter on [[Wikibooks:Antiracist Activism for Teachers and Students/Points to Consider for Teaching Anti-racism/Media Literacy In Schools|Media Literacy In Schools]].</ref> This book is being written hoping to increase the effectiveness and accelerate the rate of growth in media literacy and thereby accelerate progress against many of the most pressing issues facing humanity today. Much of this book is a [[w:Monograph|research monograph]] summarizing research that seems to have been underreported by the major media to avoid offending people who control most of the money for the media. These research results seem to be central to major political divisions. Each chapter ends in exercises to help the reader practice media literacy skills and have fun doing it. Remember: :''I am entitled to my [[Wiktionary:cockamamie|cockamamie]] ideas, and you are entitled to yours.'' Humor is important but must be offered in a way that does not offend others. If others are offended, they may be less interested in dialogue. The term "cockamamie" is used here, hoping that this style of [[w:Self-deprecation|self-deprecation]] might be more inviting for dialogue. ''Never say, "You're wrong, and I'm right!" instead, ask, "May I offer a contrary perspective?" Or "May I share with you another view that I've heard?" '' Much of the information in this book seems to have been largely overlooked and perhaps suppressed, apparently because it would increase the cost of producing news, some of which would clearly offend people who control much of the money for the media; see the brief discussion of conflicts of interest by the major media in the next "Key claims" section. ==Key claims== * ''Primary drivers of every major conflict include differences between the media that the different parties find credible''. :-- This works, because everything we think we know is coded in systems of connections between neurons in our brains. These systems are more unique than fingerprints and evolve over time. The words we use do not mean the same to two different humans nor even to the same human at different points in time. In many cases these differences are inconsequential. ''Sometimes they are fatal.''<ref>Graves and Bailey (2026).</ref> :-- ''[[w:Social constructionism|Show me someone who knows the truth]], and I will show you someone who is dangerous'' -- especially during war or any other situation where humans may be moved to violence mandated by their belief system.<ref>[[w:Collateral damage|Collateral damage]] that "they" commit proves to "us" that "they" are subhuman or at best criminally misled and must be resisted by any means necessary. By contrast, collateral damage that "we" commit is unfortunate but necessary.</ref> * The major media everywhere have [[w:Conflict of interest|conflicts of interest ]] in honestly reporting on [[v:Information is a public good per communications prof Pickard|anything that might offend anyone who controls large portions of the money for the media]].<ref>Pickard and Graves (2025), accessed 2026-02-08; Pickard (2020).</ref> [[v:Media Reform Coalition challenges anti-democratic media bias in the UK|British journalist and media reform advocate Dan Hind]] said that the content produced by the [[w:BBC|BBC]] was frivolous, soap opera stuff, because leading media personalities know very little about issues of substance and believe "they might get in trouble if" they produced anything serious. Similar analyses seem to apply to the major media everywhere<ref>Hind and Graves (2025), accessed 2026-02-09.</ref> but may not apply to non-profit and local media, which seem more likely to produce [[w:Investigative journalism|investigative]] / [[v:Dean Starkman and the watchdog that didn't bark|accountability journalism]]:<ref>Usher and Kim-Leffingwell (2022); see also Starkman and Graves (2025), accessed 2026-02-09.</ref> [[w:Watchdog journalism|Watchdogs]] tend to protect the people who feed them. Argentine journalist [[w:Horacio Verbitsky|Horacio Verbitsky]] said, "Journalism is disseminating information that someone does not want known; the rest is [[w:propaganda|propaganda]]."<ref>p. 16 in Verbitsky (1997); English translation from [[Wikiquote:Horacio Verbitsky]], accessed 2026-02-09.</ref> * The major media everywhere create the stage upon which politicians read their lines. :- Their selection of acceptable topics for news and entertainment create and maintain the "[[w:Overton window|Overton window]]", which is the range of acceptable political discourse. For example, in early 1964, US President [[w:Lyndon B. Johnson|Lyndon Johnson]] understood that he could lose the 1964 presidential election that year if he were seen to be soft on communism. His response was to clandestinely provoke an attack on US naval vessels in the Gulf of Tonkin, which he could then denounce as "unprovoked". During a dark and stormy night 1964-08-04 the [[w:USS Maddox (DD-731)|USS ''Maddox'']] and [[w:USS Turner Joy|''Turner Joy'']] spent a couple of hours "defending themselves" against radar snow, then [[w:Gulf of Tonkin incident|reported that they had sunk two attacking North Vietnamese torpedo boats]]; subsequent investigations found no evidence of the reported attacks. That incident was used to justify the [[w:Gulf of Tonkin Resolution|Gulf of Tonkin Resolution]], with only two dissenting votes in the US Congress: Those two dissenters were defeated in their next reelection campaigns, illustrating the point that the major media create the environment in which many politicians cannot get elected without betraying the nation. === The value of noncommercial news outlets === Some of the problems with the media and their contributions to increasing political polarization and violence are documented in the research summary on "[[Information is a public good: Designing experiments to improve government]]" and in the podcast series available on Wikiversity under "[[:Category:Media reform to improve democracy]]" with leading experts discussing their recommendations. One of the most compelling of the references discussed in that podcast series is Usher and Kim-Leffingwell (2022), who tallied all the federal prosecutions for political corruption in each of the 94 [[w:United States federal judicial district|US federal court district]]s between 2003 and 2019. During that period, the number of journalists in the US fell by a factor of roughly 3 -- between 60 and 70 percent. They found no statistically significant impact on federal prosecutions for political corruption of that decline in the number of journalists. However, each member of the [[w:Institute for Nonprofit News|Institute for Nonprofit News]] (INN) in a federal court district in one year was associated with on average 1.4 additional prosecutions for political corruption the following year. This suggests that the major media outlets that had so dramatically reduced their staffs had not substantively reduced the amount of investigative journalism they did. If we assume that the people prosecuted for political corruption also control substantive advertising budgets, then the major media outlets have conflicts of interest in honestly reporting on such. They may report on it if some other organization like a member of INN does the research and they are threatened with a loss of audience from not reporting on it. :'''''Major point''''': You and I benefit, the vast majority of humans on earth benefit, from news reports presumably published by members if INN that contributed to those on average 1.4 additional prosecutions for political corruption estimated by Usher and Kim-Leffingwell (2022). We benefit even if we never heard about the news reports that contributed to those prosecutions. We benefit even if we have never heard of the news outlets that presumably did the investigative journalism behind those additional prosecutions. Why? Because on average those news reports likely deterred other incidents of political corruption, which likely contributed to broadly shared economic growth and the development of new technology that ultimately benefit the vast majority of humanity. Other aspects of this are documented in the research on the impact of [[w:news desert|news desert]]s, which we summarize next. === Costs increase in news deserts=== There's a growing body of research describing what happens when local newspapers die. Perhaps most important, a 2018 research report by Gao et al. reported that the death of a local newspaper was followed by … increases in local tax revenue, averaging $85 per human per year.<ref name = Gao2018>Gao et al. (2018).</ref> That $85 was roughly 13 hundredths of a percent of the 2019 US GDP. That's mentioned in the 2025-07-17 interview with [[Democratic delusions: Fix the media to fix democracy|Natalie Fenton about her new book, ''Democratic Delusions, How the Media Hollows out democracy and What We Can Do About It'']]. One of the most spectacular example of the cost of a news desert is the [[w:City of Bell scandal|Scandal of Bell, California]]. Their local newspaper died around 1999. Roughly a decade later the city was nearly bankrupt in spite of having property tax rates among the highest in the nation. An investigation by the ''[[w:Los Angeles Times|Los Angeles Times]]'' documented that the city manager had a compensation package worth $1.5 million a year, well over double that of the President of the United States. Other senior city officials were similarly well-remunerated. Some of the city officials went to jail over that. Did the city manager decide after 1999, "Wow: The watchdog is dead. Let's have a party"? Malfeasance also increases in business as pollution and workplace accidents increase as does the cost of capital, because investors know their money is not as secure without a local newspaper. That leads to a reduction in investments in new products, services and processes -- slowing economic growth. See "[[Local newspapers limit malfeasance]]", esp. Kim et al. (2021). And executive compensation in increases in nonprofits, so less of what people donate goes to the charitable purpose for which they donated, according to Felix et al. (2024). Also, voter participation and split-ticket voting decline, per Benton (2019) and other references discussed in "[[Information is a public good: Designing experiments to improve government]]". And the ultra-right does better, as noted in [[News from Germany 1900-1945 and implications for today]] and the section on "[[Information is a public good: Designing experiments to improve government#Previous research|Previous research]]" in the Wikiversity article on "[[Information is a public good: Designing experiments to improve government]]".<ref>Flößer (2024).</ref> The 0.13 percent of GDP savings estimated by Gao et al. (2018) is roughly $120 per human per year. With over 300 million humans in the U.S, that is roughly $40 billion nationwide. {| class="wikitable" |+ Table 1. Costs increase in news deserts |- ! Entity !! What !!Source |- | local government || costs incr. 0.13% of GDP || Gao et al. (2018) |- | local businesses || pollution & workplace accidents incr., innovation & econ growth decr. || Kim et al. (2021) |- | nonprofits || exec. compensation incr. || Felix et al. (2024) |- | rowspan=2 | elections | voter participation & split-ticket voting decl. || Benton (2019) |- | Ultra-right does better || Flößer (2024) |} === Government subsidies === John (1995) documented how in the first half of the nineteenth century the US had more independent newspaper publishers per million population than at any other time or place in human history.<ref>This is discussed in the 2025-06-08 [[Media concentration per Columbia History Professor Richard John|interview with him]], available on Wikiversity under [[:Category:Media reform to improve democracy]], accessed 2026-04-30.</ref> This encouraged literacy and limited political corruption, both of which helped [[The Great American Paradox|the early United States stay together and grow]] while contemporary [[w:New Spain|New Spain]] / [[w:Mexico|Mexico]], fractured, shrank, and stagnated economically. As documented with Figure 1 in the chapter below on [[/The impact of the media on political economy since the time of the Pharaohs/]], that growth catapulted the young United States into its current position of dominance in the international political economy, a position it has been losing since at least 1990 -- or since the Reagan Revolution began in 1981, according to the analysis in the chapter below on [[/Fox, the Great Depression, the Great Recession, and our future/]]. Other countries now have stronger democracies due in part to government subsidies for media in the range of 0.05 and 0.25 percent of GDP with a firewall that limits political interference in the content, according to Neff and Pickard (2024). Table 1 in "[[Information is a public good: Designing experiments to improve government]] compares media subsidies in various places with "other points of reference". McChesney and Nichols (2010, pp. 310-311, note 88) suggested that the relatively high rate of economic growth of the economy in the early US was due in part to postal subsidies under the US [[w:Postal Service Act|Postal Service Act]] of 1792.<ref>See also the Wikiversity article on "[[The Great American Paradox]]", accessed 2026-04-30.</ref> They estimated those subsidies at 0.21 percent of GDP. To improve the current political economy of the US, they recommended subsidies of 0.15 percent of GDP distributed to local news nonprofits on the basis of local elections.<ref>McChesney and Nichols (2021, 2022).</ref> The Wikipedia article on "[[Information is a public good: Designing experiments to improve government]]" documents how some jurisdictions can devote that much money to local news nonprofits by matching what they spend on accounting, advertising, and public relations.<ref>See the section on "[[Information is a public good: Designing experiments to improve government#Sampling units / experimental polities|Sampling units / experimental polities]]" in the Wikiversity article on "[[Information is a public good: Designing experiments to improve government]]", accessed 2026-04-30.</ref> Pickard (2023) describes three basic strategies for confronting concentrated commercial media power: (1) break them up, (2) regulate them, and (3) create non-commercial, public alternatives. A fourth possibility might be [[w:externality|a graduated tax on income and wealth]] in proportion to the threat that major corporations pose to democracy. One class of noncommercial alternatives that Packard mentions is local multi-media / Public Media Centers (PMCs) with management split between local journalists and boards, e.g., selected at random from registered voters. A key here is to have the boards selected in a way that cannot be influenced by people with power, whether business or political elites. Picard recommends considering '''six discrete layers''' when discussing PMCs, each of which, he says, must be radically democratised: # funding, # governance, # ascertainment (to determine a community’s ''critical information needs''), # infrastructure (including universal broadband service), # algorithmic (e.g., not allowing companies like Google and Facebook to suppress indexing information the might challenge their hegemony of those markets, [[w:Deep web|treating them like pedophilia and the Islamic State]]), # engagement, involving local communities in making their own news and in communicating their own stories; this is paramount to building trust and the grassroots-level support that this new local journalistic model requires. All this needs to be managed in ways that provide substantive support to news deserts and underserved communities that have long been subjected to various kinds of informational redlining. This might be done by including the proposed PMCs within local libraries staffed by professional journalists, who provide training in media literacy in local schools for children and supervise students producing school newspapers. Management of such PMCs might be split between journalists on staff and boards of, e.g., six members selected at random from voter registration rolls serving staggered terms of one year with a new member rotated in every 2 months. Another alternative that could be done in parallel with local PMCs calls for 200 journalists in each US Congressional district funded at $10 billion annually in 2022 dollars, which is just a little under 4 hundredths of one percent of GDP; if such allocations are expressed as fractions of a percent of GDP, they would grow naturally with the economy. (The nominal GDP for the US was roughly $26.1 trillion in 2022.<ref>Johnston and Williamson (2026).</ref> For 2026 it is estimated at $32.4 trillion.<ref>[[w:United States|United States]], accessed 2026-04-30.</ref>) A similar model is the [[w:BBC|BBC]]’s Local Democracy Reporting Service (LDRS), in which the BBC funds journalists to cover the work of local councils and other local public bodies, funded at £8 million per year, which is a little under 2 hundredths of a percent of the [[w:United Kingdom|UK]]'s GDP of £7.27 trillion.<ref>[[w:United Kingdom|United Kingdom]], accessed 2026-04-30.</ref> Pickard (2023) ended by saying, "Today we face a crossroads: technocracy and oligarchy from above or radical democracy and structural reform from below. ... [T]his is not just a journalism crisis: it is a democracy crisis." ==Table of Contents== *[[/Introduction/]] including an exercise, asking all to discuss perceptions of the settlement of ''[[w:Dominion Voting Systems v. Fox News Network|Dominion Voting Systems v. Fox News Network]]'' in a friendly supportive manner with humans with whom they may vehemently disagree, because the alternative could be killing humans over misunderstandings. ===Part I. The media and political economy=== # [[/The impact of the media on political economy since the time of the Pharaohs/]] describes how hierarchical societies prior to [[w:James VI and I|King James of the King James bible]] were divided between those who fought, prayed, and worked. It was the responsibility of those who prayed to convince those who worked to live in poverty while giving increasing shares of what they produced so those who fought and prayed could live lives of leisure and opulence. During the reign of King James, pamphlets and newspapers began to compete with the church for helping commoners understand their roles in society. This produced the Industrial Revolution and modern democracies. Media consolidation since World War II gradually slowed and then reversed this trend. # [[/Fox, the Great Depression, the Great Recession, and our future/]] describes the unprecedented performance of the US political economy during the presidency of Franklin Roosevelt (FDR), insisting that much of what FDR achieved can be replicated, giving a media system that supports honest discussion of the available evidence. # [[/Media consolidation, social media, and political polarization/]] (Combine from McChesney and Nichols discussing the [[w:Postal Service Act|US Postal Service Act]] of 1792 with [[Media concentration per Columbia History Professor Richard John]], the section on "[[v:Information is a public good: Designing experiments to improve government#Threats from social media|Threats from social media]]" in "[[Information is a public good: Designing experiments to improve government]], and the comments by [[v:Facebook whistleblower Frances Haugen says|Facebook whistleblower Frances Haugen that, "the shortest path to a click is anger or hate."]]. ===Part II. The media and war=== # [[/Deterrence without threat/]]: The historical record is clear: Nations that have prepared for war often got war, not peace. This happens for at least two reasons: First, some leaders cannot resist the temptation to use force inappropriately, sometimes clandestinely provoking others to do things that are then denounced as "unprovoked"; sometimes the media environment pushes them to do such. Alternatively, potential adversaries may believe -- or claim -- that you are actually preparing a first strike, and they must move preemptively or lose their ability to retaliate adequately. We can avoid these possibilities with three supportive policies: [a] Legislation that ''prohibits'' projecting force beyond our own borders. [b] Civilian-based defense training in nonviolent noncooperation like what helped Denmark survive Nazi occupation with minimal damage. And [c] a media system that penalizes rather than encourages a bellicose foreign policy. # [[/Responding to a nuclear attack/]] (draft in [[Responding to a nuclear attack]]. Add a discussion of Russia's Poseidon nuclear powered unmanned underwater vehicle, armed with nuclear weapons. With that, cite the record of "[[w:System accident|system accident]]s". Also add material from [[Nuclear weapons and effective defense]]). # [[/Threats from excessive government secrecy/]] (draft in [https://sanjosepeace.org/restrict-secrecy-more-than-data-collection/ "Restrict secrecy more than data collection"], adding material from [https://kkfi.org/program-episodes/does-us-government-secrecy-threaten-national-security/ Connelly (2023) ''The Declassification Engine: What History Reveals About America's Top Secrets''], [[Wikipedia:Moynihan Commission on Government Secrecy]] and [[1998 Embassy bombings and September 11]]. # [[/Shouting fire in a crowded theater/|Shouting ''fire'' in a crowded theater]]: Legal concerns about "[[w:Shouting fire in a crowded theater|Shouting ''fire'' in a crowded theater]]" date, at least in large part, from the [[w:Supreme Court of the United States|US Supreme Court]] decisions in ''[[w:Schenck v. United States|Schenck v. United States]]'' (1919) and ''[[w: Brandenburg v. Ohio| Brandenburg v. Ohio]]'' (1969). In ''Schenck'' the Court ruled that the government had a right to imprison Schenck and others, because their distribution of fliers encouraging draft resistance presented a [[w:clear and present danger|clear and present danger]] to the efficacy of ongoing military activities during [[w:World War I|World War I]], then in progress. The Court in ''Brandenburg'' held that the government cannot punish inflammatory speech ''unless that speech is "directed to inciting or producing imminent lawless action and is likely to incite or produce such action".'' Some could argue that many uses of military force by the US and Israel since 1948 have violated international law, encouraged by biases in the major US media "directed to inciting or producing imminent lawless action", though it may not be feasible to convince a court of that. Still, it might be useful to simulate such a case in a mock trial like the 1966 [[w:Russell Tribunal|Russell Tribunal]].<ref>Andersen (2006) provides such documentation for several such uses of force. Johnson (2026) ''How to Sell a Genocide: The Media's Complicity in the Destruction of Gaza'' organizes evidence supporting such claims for the current [[w:Gaza war|Gaza war]], which began with [[w:October 7 attacks|Palestinian attacks 2023-10-07]]. See also Andersen (2026). Might, e.g., Palestinians -- or at least Palestinian Americans -- be able to sue the [[w:Anti-Defamation League|Anti-Defamation League]] (ADL), the [[w:AIPAC|American Israel Public Affairs Committee]] (AIPAC), and all the major media outlets in the US for inciting genocide in the current [[w:Gaza war|Gaza war]]? That history includes routine suppression of coverage by the major media especially in the US of routine denial of equal protection of Israeli laws to non-Jews in Israel and under Israeli occupation, including suppression of Israeli violence against nonviolent protestors peaceably assembling and petitioning for a redress of grievances combined with over reporting of Palestinian violence and unquestioning coverage of fraudulent claims of Palestinian violence by Israel and supporters. The suppressions included underreporting of Palestinian nonviolence such as the [[w:2018–2019 Gaza border protests|(2018-2019) Great March of Return]], and suppression of the grievances inspiring such nonviolence such as indefinite detention without charges of thousands of Palestinians, including children, routine destruction of Palestinian property by settlers, confiscation of Palestinian property at gunpoint, closing [[w:Gaza Strip|Gaza]] to international trade, and maintaining Gaza on starvation rations. These routine biases in reporting have been encouraged by charges that more honest reporting would be "[[w:Antisemitism|antisemetic]], according to the ADL and AIPAC. This denial of coverage thereby encouraged Israel to increase the rate of such violations until the [[w:October 7 attacks|2023-10-07 attacks on Israel from Gaza]] unleashed Israeli "retaliations" way out of proportion to the alleged provocation. Sucharov (2022) reported that 69% of American Jews opposed privileging Jews over non-Jews in Israel. Their support of Israel in the current Gaza war is consistent with the media biases documented by Johnson (2026) and others including Andersen (2006, 2026). Regarding whether Israel could achieve anything positive from this war, Samuelson (2025) is skeptical. He summarized quantitative analyses of 60 previous insurgencies. The results including the observation that it is exceedingly difficult to defeat an insurgency without responding to the grievances that support it without force ratios far beyond Israel's resources.</ref> ===Part III. Climate, immigrants, education, public health, and criminal justice=== # [[/Global warming/]] [Summarize research especially on conflicts of interest of major media in honestly reporting on this issue and the research on global warming itself and activities of groups concerned about this issue. Decompose into global population times CO2 equivalents per human.] # [[/Immigrants/]] [Summarize research documenting that [[w:Sanctuary city|sanctuary cities tend to have higher median incomes and no more crime than non-sanctuary jurisdictions]], and some studies report less crime. Moreover economists have documented that immigrants tend to be more entrepreneurial, overrepresented in patent applications, and generally increasing the rate of economic growth. See, e.g., Aghion et al. (2022) ''The power of creative destruction''; Aghion shared the 2025 Nobel Memorial Prize in Economics with two others.] # [[/Education/]] (draft in [[Invest in children]].) # [[/Public health/]] [Draft in [[UN public health data]] to be revised to be consistent with Bezruchka (2023, 2025).] # [[/Substance abuse and addictive behavior/]] (Research in cited in "[[Wikipedia:War on drugs]]" insists that the US and the world would have fewer problems with substance abuse and addiction problems with 100 percent public funding for treatment programs and complete decriminalization of possession and use of retail quantities of addictive substances. We would also likely have fewer problems with immigrants, as that would make it harder for the US to intervene in the internal affairs of foreign countries funded off the books, as exposed in the [[w:Iran–Contra affair|Iran–Contra affair]].) # [[/Criminal justice/]] (The section on "[[w:United States incarceration rate#Editorial policies of major media|Editorial policies of major media]]" in "[[Wikipedia:United States incarceration rate]]" cites research claiming that within the range range of experience in the US political economy since 1925, the incarceration rate is uncorrelated with crime: It's a function of the public's perception of crime, and that's a function of the media. That suggest that the US would be safer and more prosperous if incarceration policies were driving more by research than by editorial policies of the media. For example, there is also research that says that incarcerees who receive visits are less likely to recidivate, but that evidence is overlooked when convicts are incarcerated substantial distance from their family and friends and when the cost of phone services is substantially higher for incarcerees than among the general pubic. Also, it's known that better educated incarcerees are less likely to recidivate, but it's difficult and maybe impossible for many incarcerees to obtain education in prison. # [[/Empower women and girls/]] [Cite research claiming that a primary restraint on population growth is empowering women and girls. Empowering women and girls is not just a matter of equity: It is also a means to reduce the threats of global warming, of increasing exposure to animal diseases and other problems that come with unrestrained population growth.] === Continuation === * [[/The evolving media literacy movement/]] to invite others to keep this book current with the evolving understanding of media literacy, how to encourage and promote it and the benefits of doing so. ==See also== * [[Wikibooks:Antiracist Activism for Teachers and Students/Points to Consider for Teaching Anti-racism/Media Literacy In Schools]] ==Notes== {{reflist}} ==Bibliography== * <!--Robin Andersen (2006) A century of media, a century of war-->{{cite Q|Q138795568}} * <!--Robin Andersen (2026-06-02) The Complicit Lens: US Media Coverage of Israel’s Genocide in Gaza-->{{cite Q|Q138796307}} * <!--Perry Bacon Jr. (2022-10-17) "America Should Spend Billions to Revive Local News"-->{{cite Q|Q139594786}} * <!-- Joshua Benton (9 April 2019). "When local newspapers shrink, fewer people bother to run for mayor". Nieman Foundation for Journalism -->{{cite Q|Q63127216}} * <!--Stephen Bezruchka (2023) Inequality Kills Us All-->{{cite Q|Q136047815}} * <!--Stephen Bezruchka (2025) ''Born Sick in the USA''-->{{cite Q|Q138749292}} * <!--Renée DiResta (2024) Invisible Rulers: The People Who Turn Lies into Reality-->{{cite Q|Q135107164}} * <!--Robert Felix, Joshua A. Khavis, and Mikhail Pevzner (2024) "The effects of local newspaper closures on nonprofits’ executive compensation"-->{{cite Q|Q132730972}} * <!--Maxim Flößer (2024-03-06) "Keine Lokalzeitung -- mehr AfD", Kontext-->{{cite Q|Q125287792}} * <!--Pengjie Gao, Chang Lee, and Dermot Murphy (2018) "Financing Dies in Darkness? The Impact of Newspaper Closures on Public Finance"-->{{cite Q|Q55670016}} * <!--Spencer Graves (2024) "Wikipedia: The most democratic force on earth-->{{cite Q|Q137796922}} * <!--Spencer Graves and Bryan Bailey (2025) "We have to talk", blog at PeaceWorksKC.org-->{{cite Q|Q136126262}} * [[d:Q138038060|Dan Hind and Spencer Graves (2025) "Media Reform Coalition challenges anti-democratic media bias in the UK" on Wikiversity]]. * <!--Richard R. John (1995) Spreading the News: The American Postal System from Franklin to Morse-->{{cite Q|Q54641943}} * <!--Adam H. Johnson (2026-04-21) How to Sell a Genocide: The Media's Complicity in the Destruction of Gaza-->{{cite Q|Q140073447}} * <!--Louis Johnston and Samuel H. Williamson, "What Was the U.S. GDP Then?" MeasuringWorth, 2026-->{{cite Q|Q56881105}} * <!-- Min Kim, Derrald Stice, Han Stice, and Roger M. White (2021) "Stop the presses! Or wait, we might need them: Firm responses to local newspaper closures and layoffs"-->{{cite Q|Q132459373}} * <!-- Robert W. McChesney; John Nichols (2010). The Death and Life of American Journalism (Bold Type Books) -->{{cite Q|Q104888067}}. * <!-- Robert W. McChesney; John Nichols (2021). "The Local Journalism Initiative: a proposal to protect and extend democracy". Columbia Journalism Review, 30 November 2021 -->{{cite Q|Q109978060}} * <!-- Robert W. McChesney; John Nichols (2022), To Protect and Extend Democracy, Recreate Local News Media (PDF), FreePress.net (updated 25 January 2022) -->{{cite Q|Q109978337|access-date=2024-06-23}} * <!-- Victor Pickard (2023-05-12) "Another Media System is Possible: Ripping Open the Overton Window, from Platforms to Public Broadcasting"-->{{cite Q|Q131398460}} * <!--Neff and Pickard (2024) "Funding Democracy: Public Media and Democratic Health in 33 Countries"-->{{cite Q|Q131468289}} * [[d:Q131398359|Victor Pickard (2020) ''Democracy without journalism? : confronting the misinformation society'' (Oxford U. Pr.)]]. * <!--Doug Samuelson (2025) Assessing Israel’s Approach in Gaza-->{{cite Q|Q138843324}} * [[d:Q138037937|Dean Starkman and Spencer Graves (2025) "Dean Starkman and the watchdog that didn't bark anglais" on Wikiversity]]. * <!--Mira Sucharov (2022) Do American Jews Really Know What 'Zionist' Means?-->{{cite Q|Q125903777}} * [[d:Q134715465|Nikki Usher and Sanghoon Kim-Leffingwell (2022) "How Loud Does the Watchdog Bark? A Reconsideration of Local Journalism, News Non-profits, and Political Corruption", ''SSRN Electronic Journal'']]. * [[d:Q61013892|Horacio Verbitsky (1997) ''Un mundo sin periodistas'' (in Spanish: A world without journalists; Editorial Sudamericana)]]. [[Category:Communication]] [[Category:Political science]] [[Category:Law]] [[Category:Psychology]] [[Category:Sociology]] [[Category:Education]] [[Category:Economics]] [[Category:Media Literacy and You]] [[Category:Freedom and abundance]] <!-- https://en.wikiversity.org/wiki/Category_Review --> p03gpl7xon693f6i68cymf0synv8ftq Eschatology of a Midday Quickie: The Endurance of "Afternoon Delight" 0 328005 2813004 2794049 2026-06-06T00:28:26Z Kirby - Electrotechnics 3074947 changed to 'category:Songs' instead of category:Music 2813004 wikitext text/x-wiki {{wikibox|type=white|shadow=|header=Attribution Statement|align=center|border color=black|This cultural criticism essay is a derivative of [https://doi.org/10.5281/zenodo.18779499 Eschatology of a Midday Quickie: The Endurance of "Afternoon Delight"] by C. Green published by Zenodo, February 2026, used under CC BY-SA 4.0. This derivative work is licensed under the same terms.}} '''Abstract''': How did a cheesy 1976 soft-rock anthem about a midday quickie manage to survive for fifty years as a pop-culture punchline? This essay dives into the bizarre history and zombie-like endurance of the Starland Vocal Band's hit single, "Afternoon Delight." From its absurd origins on a Georgetown restaurant appetizer menu to its accidental hijacking of the 1976 Bicentennial fireworks, and its infamous Grammy victory over the rock band Boston, the song has a history as weird as its lyrics. We explore exactly why this track refuses to die, examining how modern shows like ''Arrested Development'' and ''Anchorman'' continually mine it for cringe-comedy gold. Ultimately, the essay reveals how "Afternoon Delight" became a pop-culture Trojan horse with a deeply horny message disguised as a wholesome folksy singalong; and why we still use it today to lovingly mock the awkward, unironic earnestness of the 1970s. == Eschatology of a Midday Quickie: The Endurance of "Afternoon Delight" == This essay seeks to understand the peculiar cultural longevity of the Starland Vocal Bands 1976 hit "Afternoon Delight," a song that is, at its core, about the logistics and mechanics of intercourse occurring roughly between the hours of 1:00 and 4:00 PM. The origin story is that Bill Danoff, the primary architect of the Starland Vocal Band, was having lunch at Clydes, a heavily wood-paneled, faux-pub establishment in Georgetown, Washington D.C.<ref>Clyde’s of Georgetown is still operational as of this writing, though one assumes the "Afternoon Delights" section of the menu has been quietly retired.</ref> There, on the menu, describing an assortment of midday appetizers - spiced shrimp, hot Brie with almonds and perhaps a water chestnut - was a section titled "Afternoon Delights." From this culinary seed Danoff imagined an anthem of diurnal copulation. The song was released in the late spring of 1976 as the United States was preparing for the Bicentennial. And right at the chorus, "Afternoon Delight" deploys its killer earworm: "Skyrockets in flight / Afternoon delight." Every time a firework exploded in the summer of '76 - which is to say, constantly - the songs chorus was subconsciously triggered in the minds of millions of Americans. Danoff had essentially weaponized the Bicentennial, linking the celebration of the American republics founding to the friction of "rubbing sticks and stones together" in the middle of the day. And then, the Grammys. At the 19th Annual Grammy Awards in February 1977, the Starland Vocal Band improbably won Best New Artist, a decision that remains one of the Recording Academys more baffling judgments, in a category already known for its ridiculous positions. To grasp the staggering injustice, one must examine who they defeated. The nominees included Wild Cherry with the instantly recognizable "Play That Funky Music", and most egregiously, Boston. Tom Scholz, an MIT-educated electrical engineer, had essentially spent the better part of a decade locked in a basement inventing new acoustic hardware. He meticulously layered guitars and organs to create ''Boston'', an album that stands as one of the greatest technical and artistic achievements in the history of album rock.<ref>''Boston'' (the album) remains one of the best-selling debut albums of all time, eventually moving some 17 million copies in the US alone. Scholz achieved the signature "Boston guitar sound" not in a professional studio, but through obsessive tinkering with equalizers and tape loops in his basement. This Promethean effort was defeated by a song that features the lyric "my motto’s always been ' when it’s right, it’s right'".</ref> To pick one example, "Foreplay/Long Time" is a cathedral of sound, a meticulously engineered piece of emotional architecture with shades of baroque classical music. "Afternoon Delight," by comparison, sounds like a commercial for a feminine hygiene spray. The Recording Academy surveyed Scholzs transcendent, multi-platinum masterpiece, and then considered four adults harmonizing about taking a quick roll in the hay after a heavy lunch and absurdly chose the later Yet, the song refuses to die. It possesses a zombie-like afterlife in modern culture, "an animal with a life of its own," band member Taffy Danoff observed years later. If you examine the Wikipedia page for "Afternoon Delight" and specifically click on "What links here," you will find a lengthy list of back-references from modern television and film, assembled by Wikipedias apparently inexhaustible supply of "Afternoon Delight" documentarians. The joke, repeated across the pop culture spectrum (most sublimely in ''Arrested Development'', but also in ''Anchorman'', ''Good Will Hunting'', ''Glee'', and ''The Simpsons''), almost always hinges on the same exact comedic mechanism: Epistemological asymmetry. Specifically, an innocent or deeply naive character sings the song enthusiastically, entirely unaware that they are harmonizing about midday coitus, while the audience and the other characters writhe in sympathetic agony.<ref>The ''Arrested Development'' iteration of this joke (Season 2, Episode 6, "Afternoon Delight") is the apex of the form. Michael Bluth and his niece Maeby sing it as a karaoke duet at a company Christmas party, believing it to be a harmless, jaunty tune, before the realization of the lyrics dawns on them mid-verse.</ref> Why is this so reliably funny? Why do television writers return to this well with such alarming frequency? I submit that it has to do with the songs fundamental, jarring cognitive dissonance. "Afternoon Delight" sounds incredibly wholesome, the vocal harmonies are executed with the precision of a church choir, its acoustic guitar strumming along with the innocent cheer of a campfire singalong. To realize that it is actually a deeply horny song about abandoning your daily responsibilities to go have sex ("thinkin of you is workin up my appetite / looking forward to a little afternoon delight") creates a kind of cultural whiplash. When a naive character sings it, they are interacting only with the songs sonic signifiers (the cheery acoustic wholesomeness), completely blind to its semantic reality (the sex). We laugh because of the betrayal of form and content. It is a wolf in sheeps clothing, a Trojan horse of libido disguised as folk-pop innocence. But on a deeper, perhaps more eschatological level, the zombie-life of "Afternoon Delight" serves as a mechanism for modern culture to safely process the profoundly awkward adolescence of the sexual revolution. If one traces the sociological arc of pop-music horniness, the trajectory is pretty stark: in the 1950s, explicit sexuality was essentially nonexistent; in the '60s, it lurked mostly in the shadows of coy, heavily coded innuendo; and by the time we hit the late '90s and 2000s, it had mutated into something forthrightly, almost aggressively pornographic. Which is to say that today, when pop music is sexually explicit, the sonic form matches the semantic content. You get heavy, concussive bass, synthetic club-beats, and vocals that are either breathily sultry or overtly aggressive. The packaging aligns with the payload. The 1970s, conversely, represent this bizarre, transitional hiccup. Mainstream, casual sexuality had suddenly arrived for the American middle class, but musicians were still defaulting to the wholesome, acoustic-guitar templates of the preceding folk era to express it. The Starland Vocal Band were smiling, singing brightly about their libidos with the unbothered, sun-drenched earnestness of a family singalong. It is a horrifying lack of self-awareness. We keep the song alive as a memetic bludgeon because it allows us to feel superior to the era that produced it. We use it to remind ourselves that we are savvier, more detached, and infinitely more ironic than the people who once bought millions of copies of a song named after a plate of Georgetown appetizers. In laughing at "Afternoon Delight," we are not merely mocking a cheesy song about daytime coitus; we are exorcising the ghost of pre-ironic sexuality, keeping at bay the terrifying specter of earnest desire completely untainted by self-consciousness. ==Editions== *DOI edition: https://zenodo.org/records/18779499 *Flipbook edition (cover art): https://archive.org/details/afternoon-delight-eschatology-of-a-midday-quickie/mode/2up?view=theater ==Tags== * Pop culture, memetics, 1970s music, sexual revolution, Starland Vocal Band, cultural criticism, irony, television tropes ==References== {{Reflist}} ==Wikiversity status== {{Complete}} {{History}} {{Essay}} [[Category: Songs]] bchsj9i9gna2pd07tsr5anw2nrs4ied Digital Media Concepts/The Rise of Digital Music Therapy Apps and Teen Mental Health 0 328793 2813001 2802174 2026-06-06T00:24:05Z Kirby - Electrotechnics 3074947 changed to 'category:Music related pages' instead of category:Music 2813001 wikitext text/x-wiki == Khadija Safa == === introduction === In recent years, digital music therapy apps like Endel, Brain.fm, and Moodfit have become popular among teenagers who want to manage stress, anxiety, or mood swings. these apps offer personalized music experience that can help teens feel calmer and more focused. This article examines how these apps influence teen mental health, looking at both the advantages and possible challenges. ==== Background ==== Music therapy has long been recognized for its ability to reduce anxiety and improve mood, especially in young people. (American Music Therapy Association,2023: <ref>{{Cite web|url=https://www.musictherapy.org/|title=Research {{!}} American Music Therapy Association (AMTA)|website=www.musictherapy.org|language=en|access-date=2026-04-01}}</ref> Today, apps make it easy for teens to access music therapy on their smartphones. By analyzing mood patterns, these apps can suggest specific music or soundscapes to match how a teen is feeling, creating a personalized experience. For many teens, this is helpful because they deal with school stress, social pressure, and lack of sleep. <ref>{{Cite web|url=https://www.musictherapy.org/|title=Research {{!}} American Music Therapy Association (AMTA)|website=www.musictherapy.org|language=en|access-date=2026-04-01}}</ref> The impact of these apps depends on how they are used. Teens who include music therapy apps thoughtfully in their daily routine may gain emotional support, while those who rely too heavily on them could miss out on developing other coping strategies. ===== Positive Effects ===== Digital music therapy apps offer several clear benefits: Emotional Support: Research indicates that music therapy can help reduce stress and anxiety in adolescents (Maratos et al. 2008: Leubner & Hinterberger, 2017). <ref>{{Cite journal|last=Maratos|first=Anna|last2=Gold|first2=Christian|last3=Wang|first3=Xu|last4=Crawford|first4=Mike|date=2008|title=Music therapy for depression|url=https://www.cochranelibrary.com/cdsr/doi/10.1002/14651858.CD004517.pub2/full|journal=Cochrane Database of Systematic Reviews|language=en|issue=1|doi=10.1002/14651858.CD004517.pub2/full|issn=1465-1858}}</ref> Accessibility: These apps provide therapy like experiences anytime, without the need for in person sessions (Endel, 2026). personalization: Apps can adjust playlists or soundscapes based on mood data, which keeps teens engaged and attentive (Brain.fm,2026). <ref>{{Cite web|url=https://www.brain.fm/|title=Music to Focus Better - Brain.fm|website=BrainFM|language=en|access-date=2026-04-01}}</ref> Focus and sleep: Certain tracks can improve concentration during study and help with sleep routines (Chanda & Levitin, 2013). Overall, these benefits suggest that music therapy apps can be a useful tool for supporting teen mental health. <ref>{{Cite journal|last=Chanda|first=Mona Lisa|last2=Levitin|first2=Daniel J.|date=2013-04-01|title=The neurochemistry of music|url=https://www.cell.com/trends/cognitive-sciences/abstract/S1364-6613(13)00049-1|journal=Trends in Cognitive Sciences|language=English|volume=17|issue=4|pages=179–193|doi=10.1016/j.tics.2013.02.007|issn=1364-6613|pmid=23541122}}</ref> ====== Challenges and Considerations ====== Despite the advantages, there are some important points to keep in mind: Over Reliance, Teens may depend too much on apps instead of reaching out for professional help when needed (American Psychological Association, 2022). Limit Scope: Apps cannot replace human interaction or professional therapy for serious mental health issues. Privacy, Apps responsibly and with guidance from adults is key to getting the most benefit. <ref>{{Cite web|url=https://www.nimh.nih.gov/health/topics/child-and-adolescent-mental-health|title=Child and Adolescent Mental Health - National Institute of Mental Health (NIMH)|date=2024-12-12|website=www.nimh.nih.gov|language=en|access-date=2026-04-01}}</ref> ====== Impact on Teen Mental Health Habits ====== These apps are influencing how teens deal with stress. Many teens use them before exams, during homework, or before going to sleep. When used correctly, they can promote self awareness, relaxation, and healthy coping habits. However, relying solely on apps can limit the development of social and emotional skills. ====== Responsible Use of Digital Music Therapy Apps ====== To make the most of these apps, teens should: Use them along with support from parents, teachers, or mental health professionals. Balance app usage with offline activities and time ways from screens. Choose apps that are research based and credible. Discuss how they use the apps and any concerns with adults. <ref>{{Cite web|url=https://www.aap.org/en/patient-care/media-and-children/|title=Media and Children|website=www.aap.org|language=en|access-date=2026-04-01}}</ref> ====== Conclusion ====== Digital music therapy apps offer a modern way for teens to manage stress and support their mental health. They provide convenience, personalization, and engagement, but it is important to use them responsibly. When combined with traditional coping strategies and guidance from trusted adults, these apps can be a helpful part of a teen's mental health routine.(National Institute of Mental Health, 2023. <ref>{{Cite web|url=https://www.nimh.nih.gov/health/topics/child-and-adolescent-mental-health|title=Child and Adolescent Mental Health - National Institute of Mental Health (NIMH)|date=2024-12-12|website=www.nimh.nih.gov|language=en|access-date=2026-04-01}}</ref> == References == <references /> {{BookCat}} [[Category:Mental health]] [[Category:Music related pages]] 6pr94au1u638yoktfoz1v06mde2o19i Template:Support staff 10 329092 2812889 2812461 2026-06-05T17:19:26Z Codename Noreste 2969951 DannyS712 no longer has curator rights. 2812889 wikitext text/x-wiki <templatestyles src="Template:Support staff/styles.css"/> {|class="sortable" cellspacing="3" style="width:100%" |+'''Overview of the English Wikiversity support staff''' |- class="staff-row-cells" !scope="col"| User !scope="col"| Role 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{{dts|October 14 2018}} | | | [[Special:Log/Tegel|Tegel]] |}<noinclude> {{pp-template|small=yes}} </noinclude> [[Category:Wikiversity administration]] iycv7ybn1m5uipcoe751w2j2l98p4ep User:Atcovi/OGM & Suicide/The Paper 2 329353 2812840 2812550 2026-06-05T13:00:22Z Atcovi 276019 2812840 wikitext text/x-wiki {{Notice|MOVED}} '''Major objective''': OGM may function as a context-dependent cognitive vulnerability that contributes to suicidal escalation through mechanisms embedded within the IMV model, particularly in high-risk populations. Through mechanisms outlined in the IMV framework, OGM may contribute to increased suicidal ideation within high-risk populations. ==Introduction== '''[[w:Overgeneral_autobiographical_memory|Overgeneral autobiographical memory]]''' (OGM) describes a reduced ability to recall specific events in one's autobiographical memory. For example, one may remember attending a birthday party at some point in their life, but they could not uniquely recall a specific instance of attending a birthday party. OGM has been empirically associated with depression, with depressed individuals reporting higher levels of OGM than non-depressed individuals<ref name=":2">{{Cite journal|last=Sumner|first=Jennifer A.|last2=Griffith|first2=James W.|last3=Mineka|first3=Susan|date=2010-07|title=Overgeneral autobiographical memory as a predictor of the course of depression: a meta-analysis|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC2878838/|journal=Behaviour Research and Therapy|volume=48|issue=7|pages=614–625|doi=10.1016/j.brat.2010.03.013|issn=1873-622X|pmc=2878838|pmid=20399418}}</ref>. Given the association of depression and suicidal ideation<ref>{{Cite journal|last=Chachamovich|first=Eduardo|last2=Stefanello|first2=Sabrina|last3=Botega|first3=Neury|last4=Turecki|first4=Gustavo|date=2009-05|title=[Which are the recent clinical findings regarding the association between depression and suicide?]|url=https://pubmed.ncbi.nlm.nih.gov/19565147|journal=Revista Brasileira De Psiquiatria (Sao Paulo, Brazil: 1999)|volume=31 Suppl 1|pages=S18–25|doi=10.1590/s1516-44462009000500004|issn=1516-4446|pmid=19565147}}</ref>, utilizing the '''Integrated Motivational-Volitional (IMV) model''' provides a theoretical cognitive framework to argue that OGM intensifies vulnerability to suicidal escalation through moderators described in the model. The IMV model portrays suicidal behavior as an escalating, behavioral process divided into three phases: pre-motivational phase, motivational phase, and volitional phase. The motivational phase is characterized by suicidal ideation formation, where feelings of entrapment (described as a "proximal [predictor] of suicidal ideation"<ref>{{Cite journal|last=O'Connor|first=Rory C.|last2=Kirtley|first2=Olivia J.|date=2018-09-05|title=The integrated motivational-volitional model of suicidal behaviour|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC6053985/|journal=Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences|volume=373|issue=1754|pages=20170268|doi=10.1098/rstb.2017.0268|issn=1471-2970|pmc=6053985|pmid=30012735}}</ref>), poor problem-solving abilities, brooding, and interpersonal vulnerabilities (thwarted belongingness and perceived burdensomeness) may transition the individual to the volitional phase. When looking at the IMV model and assessing where OGM contributes to suicidal ideation, OGM appears to impair problem-solving capabilities and the ability to learn from the past through reduced retrieval of specific past experiences, leading to hopelessness<ref name=":3">{{Cite journal|last=Jiang|first=Wen|last2=Hu|first2=Guangtao|last3=Zhang|first3=Jingxuan|last4=Chen|first4=Ken|last5=Fan|first5=Dongni|last6=Feng|first6=Zhengzhi|date=2020-10-12|title=Distinct effects of over-general autobiographical memory on suicidal ideation among depressed and healthy people|url=https://doi.org/10.1186/s12888-020-02877-6|journal=BMC Psychiatry|language=en|volume=20|issue=1|pages=501|doi=10.1186/s12888-020-02877-6|issn=1471-244X|pmc=7549224|pmid=33046032}}</ref><ref>{{Cite book|url=https://doi.org/10.1017/9781316563205|title=The Neuroscience of Suicidal Behavior|last=van Heeringen|first=Kees|date=2018-08-23|publisher=Cambridge University Press|isbn=978-1-316-56320-5}}</ref>. The current literature aims to shed light on a neglected niche of suicide research: autobiographical memory. Despite the overwhelming research suggesting correlations between OGM and depression and suicidal ideation, research has not thoroughly explored OGM's exact role in a cognitive, theoretical framework of suicidal ideation (specifically within the IMV model). By conducting a narrative review and integrating research on OGM's role in suicidal ideation, this paper furthers understanding on OGM's role in suicidal ideation within the IMV framework in high-risk populations. ''[is this aligned with my conclusion?]'' == Methods == ''[how were the studies selected?]'' ==Mechanisms of OGM== To best understand OGM's contributions to the suicidal process, it is imperative to understand OGM and its influence on cognition. OGM is the hindered ability to retrieve specific memories from one's autobiographical memory. This may lead to inefficient problem-solving abilities, which can impact one's ability to deal with difficult situations as they lack past experiences to rely on<ref name=":2" /><ref name=":4">{{Cite journal|last=Arie|first=Miri|last2=Apter|first2=Alan|last3=Orbach|first3=Israel|last4=Yefet|first4=Yael|last5=Zalzman|first5=Gil|date=2008-01-01|title=Autobiographical memory, interpersonal problem solving, and suicidal behavior in adolescent inpatients|url=https://www.sciencedirect.com/science/article/pii/S0010440X07000922|journal=Comprehensive Psychiatry|volume=49|issue=1|pages=22–29|doi=10.1016/j.comppsych.2007.07.004|issn=0010-440X}}</ref>. Failing to deal with difficult situations can drive an individual to hopelessness. In 1986, an article by Mark J. Williams and Keith Broadbent found that individuals who recently attempted suicide had biased latencies in autobiographical memory retrieval and had reduced specificity in responses to especially positive cues<ref>{{Cite journal|last=Williams|first=J. Mark G.|last2=Dritschel|first2=Barbara H.|date=1988-07|title=Emotional Disturbance and the Specificity of Autobiographical Memory|url=https://www.tandfonline.com/doi/abs/10.1080/02699938808410925|journal=Cognition and Emotion|volume=2|issue=3|pages=221–234|doi=10.1080/02699938808410925|issn=0269-9931}}</ref>. Although the OGM may not have a direct effect on suicide, the 1986 findings suggest that OGM may intensify the risk of suicidal ideation through deteriorating cognitive functioning. Rumination involves maladaptive dwelling on one's past negative emotions and feelings<ref name=":0">{{Cite journal|last=Treynor|first=Wendy|last2=Gonzalez|first2=Richard|last3=Nolen-Hoeksema|first3=Susan|date=2003-06-01|title=Rumination Reconsidered: A Psychometric Analysis|url=https://doi.org/10.1023/A:1023910315561|journal=Cognitive Therapy and Research|language=en|volume=27|issue=3|pages=247–259|doi=10.1023/A:1023910315561|issn=1573-2819}}</ref>, and is associated with suicidal behavior/ideation<ref name=":1">{{Cite journal|last=O'Connor|first=Rory C.|last2=Kirtley|first2=Olivia J.|date=2018-09-05|title=The integrated motivational–volitional model of suicidal behaviour|url=https://royalsocietypublishing.org/doi/10.1098/rstb.2017.0268|journal=Philosophical Transactions of the Royal Society B: Biological Sciences|language=en|volume=373|issue=1754|pages=20170268|doi=10.1098/rstb.2017.0268|issn=0962-8436|pmc=6053985|pmid=30012735}}</ref>. One may reflect on their negative emotions and question such emotions in an abstract manner ("How did I get to feel this way?"<ref name=":0" />, "Why did this happen to me?"<ref name=":5">{{Cite journal|last=Sumner|first=Jennifer A.|date=2012-02|title=The mechanisms underlying overgeneral autobiographical memory: an evaluative review of evidence for the CaR-FA-X model|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC3246105/|journal=Clinical Psychology Review|volume=32|issue=1|pages=34–48|doi=10.1016/j.cpr.2011.10.003|issn=1873-7811|pmc=3246105|pmid=22142837}}</ref>), which causes one's memory retrieval to capture negative intermediate conceptual information (ex, "I'm a failure") instead of specific memories. Repeated rumination strengthens these negative self-beliefs, leading to frequent capture of negative conceptual themes and impeding memory retrieval. This association of rumination and general memory aligns with the CaR-FA-X model<ref name=":5" />. Altogether, I propose that OGM contributes to the vulnerability of suicidal ideation through entrapment (a perceived sense of being trapped by defeat/humilitation) by moderators highlighted in the IMV model. Within the IMV model, impaired autobiographical memory, repeated rumination, and hopelessness contribute to entrapment. Positive factors, such as motivation to live, positive future thinking, and belongingness can offset the transition of entrapment → suicidal ideation, though negative factors from '''Motivational Moderators''' (MM), such as thwarted belongingness, very little social support, and perceived burdensomeness, may increase the chance of entrapment converting into suicidal ideation<ref name=":1" />. ==OGM as a Vulnerability== After reviewing OGM's process and effect on cognition in relation to suicidal ideation, we are able to evaluate its association with psychopathological disorders. Evidence suggests that OGM is a cognitive vulnerability associated with depression and suicidal ideation, though its predictive relevance may vary depending on the population. A meta analysis performed by Sumner et. al (2010) found that OGM accounted for about 1-2% of the variance in depressive symptoms at follow-up<ref name=":2" />. A 2020 study found that OGM was associated with depressed patients' current suicidal ideation state and worse-point suicidal ideation, while OGM affected the healthy patients' worse-point suicidal ideation<ref name=":2" />. However, Crane et. al (2016) conducted a longitudinal study of n≈5800 adolescents from ages 13 to 16 and found that OGM was not significantly associated with depression and did not moderate the effect of life events, suggesting OGM may not be as generalizable to community samples vs. high-risk populations. OGM persists even past depression, as found in Hallford et. al (2022). A meta-analysis indicated that participants with remitted depression continue to experience small to moderate deficiencies in being able to recall "specific, event-level personal memories". This indicates that OGM isn't merely a symptom of depression, but may function as a risk factor for future depressive episodes<ref name=":2" />. Even though the findings indicated that OGM was not significantly associated with depression, the study highlights that OGM may function as a vulnerability within high-risk populations. Across the scientific literature, OGM is suggested to be a contributing factor in the development of suicidal ideation, but appears to be have a more prevalent predictive relevance in higher-risk populations. === OGM → Suicidal ideation === OGM's influence may not be just limited to clinical depression, but may further have a deleterious effect on suicidal ideation. Jiang et. al (2020) found in a study of 365 participants, with roughly 51% of the participants clinically depressed while the other roughly 49% of participants were classified as "healthy", that OGM had an increased presence in the depressed group vs. the "healthy" group. WSI (worst suicidal thoughts one has ever had [at a certain point]) and CSI (current point of suicidal ideation) were significantly affected by OGM in the depressed group. OGM was also found to be a mediator between CSI and childhood trauma in depressed patients. As OGM leads to negative memory biases and, therefore, the maintenance of a negative mental state, the researchers suggested that OGM may be a consistent contributor to suicidal ideation in depressed patients<ref name=":3" />. These findings are corroborated by a more recent (2025) study on depressed patients with varying levels of SI, where the researchers concluded that OGM may be "a maladaptive cognitive avoidance strategy" rather than simply a deterioration in memory. Zhu et. al (2025) further explain that individuals with OGM have a difficult time recalling positive memories, which reinforce negative recollections, spurring hopelessness and the transition to suicidal ideation<ref>{{Cite journal|last=Zhu|first=Ying|last2=Yin|first2=Qianlan|last3=Xu|first3=Huijing|last4=Xiao|first4=Fang|last5=Jiang|first5=Qian|last6=Liang|first6=Meng|last7=Cheng|first7=Qi|last8=Liu|first8=Taosheng|date=2025-11-24|title=Speech feature identification model for depressed individuals with suicidal ideation based on autobiographical memory|url=https://doi.org/10.1186/s12888-025-07635-0|journal=BMC Psychiatry|language=en|volume=25|issue=1|pages=1154|doi=10.1186/s12888-025-07635-0|issn=1471-244X|pmc=12713288|pmid=41286790}}</ref>. In a 2008 study on autobiographical memory, interpersonal problem-solving skills, and suicidal behaviour in adolescents and young adults, Arie et. al (2008) found that OGM was significantly associated with hopelessness and poor problem-solving abilities in adolescents. This suggests that being able to retrieve specific memories in one's autobiographical memory improves problem-solving skills, as they are able to draw back from past experiences to address challenging interpersonal situations<ref name=":4" />. Kaviani et. al (2011) found that depressed individuals with more severe suicidal ideation levels had more difficulty in retrieving specific thoughts in comparison to depressed individuals with less severe suicidal ideation<ref>{{Cite journal|last=Kaviani|first=H.|last2=Rahimi|first2=M.|last3=Rahimi-Darabad|first3=P.|last4=Naghavi|first4=K. Kamyar H.|date=2003|title=HOW AUTOBIOGRAPHICAL MEMORY DEFICITS AFFECT PROBLEM-SOLVING IN DEPRESSED PATIENTS|url=https://acta.tums.ac.ir/index.php/acta/article/view/2663|journal=Acta Medica Iranica|language=en-US|pages=194–198|issn=1735-9694}}</ref>. Accounting for the findings, they suggest that OGM may play a unique factor in contributing to suicidal ideation through maladaptive cognitive processing rather than being merely a symptom of depression. === Emphasis on High-Risk Population === Despite the importance of OGM and the findings indicating its potential amplification of suicidal ideation, OGM does not appear to be a consistent detriment in low-risk populations. Crane et. al (2016) conducted a longitudinal study of 5792 adolescents from ages 13 to 16 and found no significant findings that OGM played a direct or interactive role with depression, suicidal ideation, and self-harm (when accounted for confounding variables) in a general population<ref>{{Cite journal|last=Crane|first=Catherine|last2=Heron|first2=Jon|last3=Gunnell|first3=David|last4=Lewis|first4=Glyn|last5=Evans|first5=Jonathan|last6=Williams|first6=J. Mark G.|date=2016|title=Adolescent over-general memory, life events and mental health outcomes: Findings from a UK cohort study|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC4743605/|journal=Memory (Hove, England)|volume=24|issue=3|pages=348–363|doi=10.1080/09658211.2015.1008014|issn=1464-0686|pmc=4743605|pmid=25716137}}</ref>. The authors concluded that OGM appears to be more clinically meaningful in high-risk populations that are already cognitively vulnerable through depression and/or psychopathology. The OGM x Stress interaction theory is supported by another longitudinal study done on 174 Caucasian adolescents by Stange et. al (2012), where they found that OGM was found to be a vulnerability to adolescents with depression (especially emotional maltreatment)<ref>{{Cite journal|last=Stange|first=Jonathan P.|last2=Hamlat|first2=Elissa J.|last3=Hamilton|first3=Jessica L.|last4=Abramson|first4=Lyn Y.|last5=Alloy|first5=Lauren B.|date=2013-02|title=Overgeneral autobiographical memory, emotional maltreatment, and depressive symptoms in adolescence: evidence of a cognitive vulnerability-stress interaction|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC3530666/|journal=Journal of Adolescence|volume=36|issue=1|pages=201–208|doi=10.1016/j.adolescence.2012.11.001|issn=1095-9254|pmc=3530666|pmid=23186994}}</ref>. These findings, alongside with Jiang et. al (2020), suggest that OGM is context-dependent and may play a significant role in the development of depression and/or suicidal ideation if the individual is already susceptible for depression and/or mental disorders. This aligns with the diathesis-stress model, suggesting a set of factors interact with pre-existing vulnerabilities to produce a "disordered state"<ref>{{Cite book|url=https://doi.org/10.1017/9781316563205|title=The Neuroscience of Suicidal Behavior|last=van Heeringen|first=Kees|date=2018-08-23|publisher=Cambridge University Press|isbn=978-1-316-56320-5|pages=24-25}}</ref>. == Conceptual Figure == ''[tighten this]'' OGM ↓ Reduced memory specificity ↓ Rumination + impaired problem solving ↓ Hopelessness ↓ Entrapment ↓ Suicidal ideation (IMV) ↑ More predictive in high-risk populations ==Conclusion== '''Meat of the conclusion''': OGM has been found to be a cognitive vulnerability for suicidal ideation amongst clinically high-risk samples. Within the IMV model, OGM appears to contribute to entrapment through impaired retrieval of past experiences, repeated rumination, and hopelessness. From the literature, OGM appears to have a meaningful contribution to suicidal ideation for high-risk/depressed populations. This paper illustrates where OGM can contribute to suicidal ideation according to the IMV model to better inform researchers on identifying relevant risk factors. '''Limitations''': Limitations include the cross-sectional research method of the studies, the causal direction between OGM and suicidal ideation is still somewhat unclear, and the conclusion derived from this research cannot be generalized beyond high-risk populations. '''Future direction''': Future research should look into empirical testing of OGM and entrapment and to test if OGM moderates the transition from entrapment to suicidal ideation. Future research could also look into whether therapies targeting OGM, such as Memory Specificity Training (MEST), would be beneficial for high-risk populations in reducing the risk of suicidal ideation. == References == {{Reflist}} [[Category:Atcovi/OGM & Suicide Poster]] cgsonjaiuurijqgg1w25e2dmpe2cf4e Talk:WikiJournal Preprints/Using Wikidata to analyze the main historical trends in archaeological research on the Argentine continental Patagonian coast 1 329659 2812992 2812796 2026-06-05T22:34:56Z OhanaUnited 18921 /* Peer review 2 */ reviewer agrees to open identity 2812992 wikitext text/x-wiki == Editorial comments == {{editor's comments |date=28 May 2026 |Q=Q99632752 |text= Background: line 4 "the scientific contents" to "the scientific content". Conclusion: "Wikidata as a database as an analytical tool" to "Wikidata as a database and analytical tool" Acknowledgements: "funding of a grants since" to "funding of grants since". References: these need dates 2. (December 2023) 15. (2012) 24. (August 2002) In the Introduction line 1: "(in spanish," to "(in Spanish,", line 2: "a group of wikimedians" to "a group of Wikimedians" and paragraph 2 last line: "on the subject, Scholia for data analysis" to "on the subject, Scholia[5] for data analysis". In Methodology paragraph 3 line 3 Note 3 "about argentine academic journals" to "about Argentine academic journals". In Methodology paragraph 4 line 7 "Argentine Patagonian coast" (wich label" to "Argentine Patagonian coast" (which label" and line 32: "label in spanish is" to "label in Spanish is". Discussion paragraph 8 line 3 "in the last decades there are an increase" to "in the last decades there is an increase" and line 9 "one in wich these kind of publications are" to "one in which the kind of publications are". Final remarks first paragraph line 4 "wich regard" to "which regard". Discussion paragraph 8 line 9 change "one in which the kind of publications are" to "one in which the kinds of publications are". }} ==Peer review 1== Editor's note: The comment was provided in Spanish. Author (FMA) confirmed that they can read Spanish and able to accept reviewer comments in Spanish. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;"><sup>Talk page</sup></span></b>]] 23:24, 1 June 2026 (UTC) {{review |reviewer = Romina C. Vazquez |Q = Q106576509 |affiliation=Instituto de Investigación en Paleobiología y Geología, Universidad Nacional de Río Negro, Argentina |link = |date = 28 May 2026 |pdf = Wikidata Argentine Patagonian coast - Romina Vazquez peer review comments.pdf |text = }} ==Peer review 2== {{review |reviewer = Pablo Ambrústolo |Q = Q116889133 |affiliation=División Arqueología, Facultad de Ciencias Naturales y Museo, Universidad Nacional de La Plata |date = 2 June 2026 |text = This work offers a comprehensive approach to the main historical trends in archaeological research on the Argentine Patagonian coast. In this sense, the study is very thorough, considering virtually all publications on the subject within a historiographical perspective focused on the development of archaeology in Argentina. A progressive development of the discipline is observed in diachronic terms. By uploading the information to Wikidata, creating an open database, the researchers identified trends based on different aspects, such as the topics addressed, geographical variations, and the representation of authors. Within this framework, for example, from the year 2000 onward, they recognize a process of consolidation and maturation of research in the archaeology of the continental Patagonian coast, which they correctly correlate with the implementation of expansionary policies that favored the development of Argentine science at both the national and international levels. On the other hand, and in contrast, it is also important to note that these findings highlight a significant decrease in the number of publications related to this topic in recent years. This correlates with the current situation of marked reductions in science funding in Argentina, with all the repercussions this has for research groups. As they rightly point out, this is a scenario in which a deliberate policy is resulting in a kind of "scientific genocide" that impacts all levels of the scientific system. The manuscript can be published without modifications.--[[User:Paconicet|Paconicet]] ([[User talk:Paconicet|discuss]] • [[Special:Contributions/Paconicet|contribs]]) 17:19, 2 June 2026 (UTC) }} n6podi0w9pa8pr742o0ffz7fn6vx38q Wikiversity:Request custodian action/Archive/30 4 329678 2812894 2809730 2026-06-05T17:37:25Z Codename Noreste 2969951 /* ~2026-28792-52 */ archive from [[Wikiversity:Request custodian action]] ([[mw:c:Special:MyLanguage/User:JWBTH/CD|CD]]) 2812894 wikitext text/x-wiki {{Archive|Wikiversity:Request custodian action}} == Sidewide count.js == i would like something like: [[Template:User contrib count/count.js]]. i created [[Template:User contrib count]] and a user/common.js. {{User contrib count}}.<br><br> so a "count.js" would complete it. See [[User:Harold Foppele/common.js]]. If an Administrator could help please. Cheers [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 19:22, 18 January 2026 (UTC) == need to add my profile == im trying to add new profile content [[User:PAGURUMURTHY|PAGURUMURTHY]] ([[User talk:PAGURUMURTHY|discuss]] • [[Special:Contributions/PAGURUMURTHY|contribs]]) 18:03, 4 February 2026 (UTC) :You can edit it now. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 18:05, 4 February 2026 (UTC) ::where can create a new one [[User:PAGURUMURTHY|PAGURUMURTHY]] ([[User talk:PAGURUMURTHY|discuss]] • [[Special:Contributions/PAGURUMURTHY|contribs]]) 18:51, 4 February 2026 (UTC) :::i have created and its in sandbox. i would like to know when it will be approved [[User:PAGURUMURTHY|PAGURUMURTHY]] ([[User talk:PAGURUMURTHY|discuss]] • [[Special:Contributions/PAGURUMURTHY|contribs]]) 19:38, 4 February 2026 (UTC) ::::Please don’t create [[wv:spam|spam]] pages as it will be deleted. Please also read [[WV:Scope]] [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 04:01, 5 February 2026 (UTC) == Im trying to add new profile while add content its shows not alowed == This action has been automatically identified as potentially harmful, and therefore disallowed. If you believe your action was constructive, please [[Wikiversity:Request custodian action|inform an administrator]] of what you were trying to do. A brief description of the abuse rule which your action matched is: New User Exceeded New Page Limit This action has been automatically identified as potentially harmful, and therefore disallowed. If you believe your action was constructive, please [[Wikiversity:Request custodian action|inform an administrator]] of what you were trying to do. A brief description of the abuse rule which your action matched is: New User Created Page with External Link [[User:PAGURUMURTHY|PAGURUMURTHY]] ([[User talk:PAGURUMURTHY|discuss]] • [[Special:Contributions/PAGURUMURTHY|contribs]]) 18:51, 4 February 2026 (UTC) == New User: cannot create talk page == Hi, I am a new user of Wikiversity and I wanted to create a talk page for the article [[ChatGPT's Essay on Kohlberg's Theory: AI's Use in Academic Writing]]. As a new user, I was barred from performing this action. The text that I wanted to add to the talk page is: <blockquote> I have doubts as to to the reliability of this essay. Take for rexample the sentence: <blockquote> Due to its efficiency, AI has saved 380,000-403,000 lives per year in European healthcare as reported in a recent Deloitte and MedTech Europe report<ref>Dantas, C., Mackiewicz, K., Tageo, V., Jacucci, G., Guardado, D., Ortet, S., Varlamis, I., Maniadakis, M., De Lera, E., Quintas, J., Kocsis, O., & Vassiliou, C. (2021). Benefits and hurdles of AI in the workplace – what comes next? ''International Journal of Artificial Intelligence and Expert Systems, 10'', 9-17. https://www.researchgate.net/publication/351993615_Benefits_and_Hurdles_of_AI_In_The_Workplace_-What_Comes_Next</ref>. </blockquote> Reading the reference (freely available on ResearchGate), one notes: # that the reference is from 2021 (predating the widespread use of LLMs such as ChatGPT and the associated 'AI' boom), and # that the reference factually contradicts this essay. Quoting from the reference: <blockquote> There are enormous benefits of applying AI-based solutions to monitor workers’ health and prevent accidents or, currently, COVID-19 infections, and those benefits are reported with enormous potential. According to the recent Deloitte and MedTech Europe report [11], implementing AI in European healthcare systems could save up 380,000 to 403,000 lives annually or €170.9 to 212.4 billion per year. </blockquote> Not that the reference says ''could save'', not ''saves'' as in the essay. This calls into question the reliability of the essay. </blockquote> Could an administrator make this addition for me? Thank you! {{reflist}} [[User:Æolus|Æolus]] ([[User talk:Æolus|discuss]] • [[Special:Contributions/Æolus|contribs]]) 06:53, 5 February 2026 (UTC) :@[[User:Æolus|Æolus]] I have added it for you, you can change the header and sign it now. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 08:05, 5 February 2026 (UTC) ::Thank you! [[User:Æolus|Æolus]] ([[User talk:Æolus|discuss]] • [[Special:Contributions/Æolus|contribs]]) 12:43, 5 February 2026 (UTC) == Disallowed to add a page on a course == I'm trying to populate a newly created course on Wikiversity, but it blocks me from creating more pages with "New User Exceeded New Page Limit". Could this be lifted please? [[User:Berkeleywho|Berkeleywho]] ([[User talk:Berkeleywho|discuss]] • [[Special:Contributions/Berkeleywho|contribs]]) 13:21, 15 February 2026 (UTC) :Sorry! Never mind. I was trying to create a new article instead of a new page. All good now. [[User:Berkeleywho|Berkeleywho]] ([[User talk:Berkeleywho|discuss]] • [[Special:Contributions/Berkeleywho|contribs]]) 14:03, 15 February 2026 (UTC) == Harold Foppele adding LLM-generated nonsense and personal fiction == I became aware of [[User:Harold Foppele]]'s editing after I deleted some of his uploads on Commons. He appears to be adding a large amount of text and images that are some combination of personal fiction and LLM-generated nonsense. This includes: *[[Quantum Ultra fast lasers#Future thought experiment|Personal speculative fiction]] in an otherwise "nonfiction" article *Uploading nonsense LLM-created [[:File:Rontosecond pulse laser (Schematic).jpg|diagrams]] and [[:File:Rontosecond pulse laser (Futuristic).jpg|renders]] for nonexistent lab equipment, with fake source (on Commons, he indicated these files as having been created by him using an LLM) *Uploading nonsense LLM-created images of equations with obvious artifacts. These images, such as [[:File:Redfield equation (non-Markovian).png]] and [[:File:Lindblad equation (Markovian).png]], don't even match the text he puts them with. Much of his writing is also of extremely poor quality, to the point where it's not clear whether it's written by him or an LLM. I'm not an active editor on this project, so I'm not as familiar with the standards here, but I believe this is worth custodian attention. [[User:Pi.1415926535|Pi.1415926535]] ([[User talk:Pi.1415926535|discuss]] • [[Special:Contributions/Pi.1415926535|contribs]]) 03:06, 23 February 2026 (UTC) :Fake source ''and'' contradictory copyright info, claiming both public domain and CC license. Moreover, if they are indeed nearly-direct LLM output, depending on jurisdiction they may not even be eligible for copyright. :I've put speedy deletion marks for the equations, because they're obviously not coherent mathematical equations (the parentheses don't match, the symbols merge into each other the way text in image models often do, etc) [[User:Sesquilinear|Sesquilinear]] ([[User talk:Sesquilinear|discuss]] • [[Special:Contributions/Sesquilinear|contribs]]) 21:50, 7 March 2026 (UTC) == Repeated removal of RFD notices by Harold Foppele == {{User|Harold Foppele }} This editor is appearing in multiple noticeboards for behaviour which is contentious. Ther latest adventure is the repeated removal of tye RFD notice at [[Quantum/Henry C. Kapteyn]]. You will see from their contributions record the number of times. I have warned Tham on their user tag page that this is tantaomunt to volunteering to be blocked here. They have a track record of achieving blocks on enWiki and Commons already. They have all the appearance of shooting not to understand when given direct information about their behaviour, whichever project they are editing, and are fast becoming a time sink. Their behaviour across multiple WMF sites may well lead to a Global Lock, but I do not believe they have quite reached the threshold for that. I believe that what is required is a preventative block to seek to ensure thatchy understand the seriousness of their behaviour, and the need to be collegial. 🇵🇸&zwj;🇺🇦&nbsp;[[User:Timtrent|Timtrent]]&nbsp;🇺🇦&nbsp;[[User talk:Timtrent|talk to me]]&nbsp;🇺🇦&zwj;🇵🇸 23:03, 4 March 2026 (UTC) : {{Done}} [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 11:45, 8 March 2026 (UTC) == Blocks for sockpuppet == Please block [[User:Harold Foppele]] and [[User:Johnwilliamsiii]] for sockpuppetry based on [https://en.wikipedia.org/wiki/Wikipedia:Sockpuppet_investigations/Harold_Foppele en wiki] CU and [https://commons.wikimedia.org/w/index.php?diff=1177465640 commons] CU investigations. The user has also violated copyright, see the above discussion. A block is necessary to prevent further abuse. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 11:30, 8 March 2026 (UTC) :<small>@[[User:MathXplore|MathXplore]]</small> [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 11:31, 8 March 2026 (UTC) :: {{Done}} [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 11:44, 8 March 2026 (UTC) :CC. @[[User:Timtrent|Timtrent]], @[[User:Sesquilinear|Sesquilinear]], @[[User:Pi.1415926535|Pi.1415926535]] [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 11:33, 8 March 2026 (UTC) ::Thank you for the ping. I concur based on [[w:en:WP:DUCK|behaviour]]. CUs appear divided. 🇵🇸&zwj;🇺🇦&nbsp;[[User:Timtrent|Timtrent]]&nbsp;🇺🇦&nbsp;[[User talk:Timtrent|talk to me]]&nbsp;🇺🇦&zwj;🇵🇸 11:41, 8 March 2026 (UTC) == Problem when trying to start a discussion with authors of the Plurilingual education portal == The authors I wanted to discuss with are called "Project PEP" and my name is Franch Chandler. How can I be allowed to do so ? [[User:French Chandler|French Chandler]] ([[User talk:French Chandler|discuss]] • [[Special:Contributions/French Chandler|contribs]]) 18:25, 16 March 2026 (UTC) :@[[User:French Chandler|French Chandler]] place your qestion [https://en.wikiversity.org/w/index.php?title=User_talk:Projet_PEP&action=edit into the dialog box] on this link and hit Publish page. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 20:22, 16 March 2026 (UTC) == Please publish my post == My post is about "Every child grows and develops at their own pace, but some may experience challenges that affect their ability to perform everyday tasks. These challenges can include difficulties with fine motor skills, sensory processing, handwriting, feeding, and self-regulation. When these issues are not addressed early, they can impact a child’s confidence, academic performance, and independence. With the rise of digital healthcare services, '''online physical therapy''' has emerged as a powerful and accessible solution for parents seeking support for their children. This modern approach provides structured, personalized therapy programs that can be accessed from the comfort of home, making it easier for families to ensure consistent care." [[User:Skyabovetherapy|Skyabovetherapy]] ([[User talk:Skyabovetherapy|discuss]] • [[Special:Contributions/Skyabovetherapy|contribs]]) 12:28, 28 March 2026 (UTC) :@[[User:Skyabovetherapy|Skyabovetherapy]] Well, you can publish it yourself, Wikiversity is a free environement, where everybody can create educational resources. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 14:11, 29 March 2026 (UTC) ::They actually triggered some abuse filters. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:24, 29 March 2026 (UTC) :I looked at your attempts to add this text and I see a link to one website repeated many times, which reminds me of the misuse of Wikiversity for self-promotion or to increase the importance of the website. It is necessary to remind you here that Wikiversity is not a place for promotion, but a place for education. So if you want to educate, it will not be a problem to create the page without external links with a clearly defined procedure for how people should use it and what to expect from it. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:07, 1 April 2026 (UTC) == New user limit == Hi, I am creating an AIPA Method learning resource page. I am the author of the linked research, and I hit the “new user limit” and “new page with external link” filters while publishing. Here is the link to the page in creation: [https://en.wikiversity.org/w/index.php?title=AIPA_Method&veaction=edit] Thank you for your help. Best regards, Senad Dizdarević [[User:Senad Dizdarević|Senad Dizdarević]] ([[User talk:Senad Dizdarević|discuss]] • [[Special:Contributions/Senad Dizdarević|contribs]]) 07:19, 30 March 2026 (UTC) :@[[User:Senad Dizdarević|Senad Dizdarević]] I should admit I dont know, what is "new user limit", but if filter blocks your page because of certain external link, you may force to save anyway and sometimes it works. It should not work, when the website is blacklisted. As of now, I am not seeing you to save page in main namespace, so try to save it without external links first. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 07:30, 30 March 2026 (UTC) ::Thank you, you are very kind. ::I will wait a day, and try again (without links, too). ::Today, I already created About Me info page, and maybe that is enough for the filters for one day. [[User:Senad Dizdarević|Senad Dizdarević]] ([[User talk:Senad Dizdarević|discuss]] • [[Special:Contributions/Senad Dizdarević|contribs]]) 07:53, 30 March 2026 (UTC) :::Well, I have analyzed your contribution to Wikiversity and I should point out here, that this project is not a place for advertising, so there is no way of promoting your books and authority this way. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:56, 1 April 2026 (UTC) ::::Hi, my About Me page is just an info page with the neutral as possible presentation of my work. ::::There is a big difference between informing and advertising. Informing is neutrally stating that something exists and requiring no action, while advertising is a special communication form with intent to cause certain action from readers. For example, click here, click there, order this, buy that. ::::There is no such intention, form, or terms on my info page. Just neutral information. I don't hide and I am not ashamed that I am write and author, and that is a part of the usual bio, including works. I checked your user page: "I graduated from the Czech University of Life Sciences in Prague and studied information science at the Faculty of Arts of Charles University." I think that if you had written a book on Life Science, you would have mentioned that as well. ::::Most of the Info page is about my research and AIPA Method which is a valid contribution to psychology, consciousness studies, identity theory, and personality development. Actually, my paper '''AIPA Method: A Cognitive-Phenomenological Model for Identity Reconstruction and Stabilization in Pure Awareness''' is now in the peer review procedure at Journal of Consciousness Studies. ::::Here is a part from the Wikiversity AIPA Method page in creation (waiting for the end of the time limit for new users): [[User:Senad Dizdarević|Senad Dizdarević]] ([[User talk:Senad Dizdarević|discuss]] • [[Special:Contributions/Senad Dizdarević|contribs]]) 06:47, 2 April 2026 (UTC) :::::For the unknown reasons, the form didn't publish my second part of the message: :::::I believe this is a valid contribution to Wikiversity. :::::Best Regards, :::::Senad [[User:Senad Dizdarević|Senad Dizdarević]] ([[User talk:Senad Dizdarević|discuss]] • [[Special:Contributions/Senad Dizdarević|contribs]]) 06:52, 2 April 2026 (UTC) ::::::And the third try: :::::: == Introduction == ::::::The AIPA Method addresses a gap in contemporary personal development and consciousness science: most evidence‑based approaches (CBT, MBSR, MBCT, standard meditation) operate at the level of mental content—reframing thoughts, observing them, or reducing their impact—rather than at the level of identity structure. In contrast, AIPA targets the structural relationship between the self and the mind, aiming at durable identity reconstruction rooted in Pure Awareness rather than symptom management. ::::::The central research question of the primary AIPA preprint is whether a structured, sequentially staged method can produce permanent identity reconstruction rooted in Pure Awareness, and how such a method compares to established approaches in scope, mechanism, and outcome. :::::: == Theoretical foundations == ::::::The AIPA framework is grounded in the cognitive‑phenomenological tradition (e.g., McAdams, Varela, Metzinger, Erikson), contemporary consciousness science on minimal phenomenal experience, and qualitative methods advocacy in psychology. It builds directly on: ::::::* Empirical work on pure awareness and Minimal Phenomenal Experience (MPE), especially Gamma & Metzinger’s large‑scale study of content‑reduced awareness states. ::::::* Metzinger’s proposal of minimal phenomenal experience as an entry point for a minimal unifying model of consciousness. ::::::* Narrative identity and partial‑self models within personality and identity theory. ::::::Within this backdrop, AIPA proposes Pure Awareness as a distinct, operationally specified state that can become a structural ground of identity rather than a transient meditative experience. :::::: == Experiential empiricism == ::::::The empirical foundation of the AIPA Method is explicitly first‑person and experiential, combining: ::::::* A 22‑year longitudinal autoethnographic self‑study (2003–2025) documenting partial personality episodes, protocol use, and outcomes. ::::::* A 13‑year prospective verification period with zero self‑reported recurrence of targeted harmful behaviors after a dated stabilization point (1 January 2006). ::::::* A high‑ecological‑validity “stress test” during acute bereavement, used to examine whether non‑reactive awareness remains stable under maximal provocation. ::::::* Two independent practitioner cases (an Amazon‑verified report and a structured questionnaire case) providing preliminary convergent signals across cognitive, emotional, behavioral, and identity dimensions. ::::::All central constructs (Pure Awareness, partial personalities, the Switch, identity stabilization) are operationalized with explicit phenomenological and behavioral criteria intended to enable replication and future third‑person measurement. ::::::I believe this is a valid contribution to Wikiversity. ::::::Best regards, ::::::Senad [[User:Senad Dizdarević|Senad Dizdarević]] ([[User talk:Senad Dizdarević|discuss]] • [[Special:Contributions/Senad Dizdarević|contribs]]) 06:54, 2 April 2026 (UTC) == Unable to publish pages == Whenever I try to publish a page with linked sources it gets flagged and says I'm a new user attempting to publish content with outside links. Those outside links are my sources. [[User:Soboyed|Soboyed]] ([[User talk:Soboyed|discuss]] • [[Special:Contributions/Soboyed|contribs]]) 04:52, 2 April 2026 (UTC) :This restriction is automatically lifted after you have edited for a certain time (I don't recall that time off-hand, but it is not long). This is designed to stop spam. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 04:53, 2 April 2026 (UTC) == Showing error to publish a Post == My action was constructive, not destructive, please allow to publish it. [[Special:Contributions/&#126;2026-20906-18|&#126;2026-20906-18]] ([[User talk:&#126;2026-20906-18|talk]]) 08:06, 4 April 2026 (UTC) :Maybe you got caught in a filter. Consider [[Special:CreateAccount|creating an account]]. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 09:06, 4 April 2026 (UTC) :Your edits, [https://en.wikiversity.org/w/index.php?title=Special:AbuseLog&wpSearchUser=%7E2026-20906-18 these ones], seems to have tripped a filter when you tried to create a page on [[Create]] which external links. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 23:58, 4 April 2026 (UTC) :Have you read my [https://en.wikiversity.org/w/index.php?title=Wikiversity:Request_custodian_action&diff=prev&oldid=2802219 previous reply] to you @[[User:~2026-20906-18|~2026-20906-18]]? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:02, 6 April 2026 (UTC) == Abuse filters which should be deleted == Hi, there are some abuse filters which should probably be deleted. * [[Special:AbuseFilter/1]] (not needed anymore) * [[Special:AbuseFilter/2]] (no hits since 2018) * [[Special:AbuseFilter/3]] (not needed since there are global filters that disallow this specific type of spam filter 3 would have catched) * [[Special:AbuseFilter/4]] (looking at the logs, there are too many false positives) * [[Special:AbuseFilter/5]] (no hits since 2023) * Abuse filters 7, 8, 9, 10, 11, 12 (these filters are not needed anymore) * [[Special:AbuseFilter/17]] (no hits since 2022) * [[Special:AbuseFilter/19]] (no hits since 2019) * [[Special:AbuseFilter/21]] (false positives, vandal currently inactive) Thanks. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 03:51, 5 April 2026 (UTC) :Why do these need to be deleted rather than inactivated? Do inactive abuse filters cause a server strain? —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:39, 5 April 2026 (UTC) :: Deleted filters do not cause strain to the servers. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 14:28, 5 April 2026 (UTC) :These sounds like sensible suggestions but, yes, would inactivation perhaps make more sense than deletion for at least some filters? -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 09:35, 5 April 2026 (UTC) :I would keep them @[[User:Codename Noreste|Codename Noreste]]. Alternatively, I would turn off the ones that haven't caught anything for a long time, but I would leave them enabled in case they need to be turned on or improved. If someone has already written the code and we don't have hundreds of free man-hours of programmers on Wikiversity, the server load seems secondary to me, and is negligible compared to other things. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:11, 6 April 2026 (UTC) :: I know how to write abuse filter code and regex, but I would recommend disabling filters that have never caught anything in a long time ''and'' those who made lots of false positives. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:09, 6 April 2026 (UTC) :::Of course @[[User:Codename Noreste|Codename Noreste]], there are people here today who are capable of changing the code. But the question is what it will be like in a few years, the question is what will happen if those two are busy for a long time, etc. That's why I would leave it so that those who don't know much about code can be inspired by it and will need to do something with it someday - plus, more code for different types of filtering is actually great educational material on how those filters work. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 11:41, 17 April 2026 (UTC) Here's the updated list of abuse filters under review with actions I've taken (several disabled, one basic code improvement, and some actions changed) - none have been deleted so they can all be edited and reactivayed - please suggest any further changes: * [[Special:AbuseFilter/1]] (not needed anymore) - One time account spam bot - 4 hits over 10 years ago - Disabled in 2024 - May be useful in future * [[Special:AbuseFilter/2]] (no hits since 2018) - Userspace spamming - 778 hits; none since 2018 likely due to global filters - Now disabled * [[Special:AbuseFilter/3]] (not needed since there are global filters that disallow this specific type of spam filter 3 would have catched) - Specific user page spam - 1,101 hits most recent 7 March 2026 - Still active - Kept enabled * [[Special:AbuseFilter/4]] (looking at the logs, there are too many false positives) - Questionable Language (profanity) - 6,055 hits including very recently - However it was logging hits without taking any actions - Edited to reduce likelihood of false positives by only applying filter to users with low (< 20) edit count and applied weak actions to tag and warn but not prevent publishing the content * [[Special:AbuseFilter/5]] (no hits since 2023) - Blocked Solicitation Links - 95 hits; none since 2023 - blocks specific historical spam sites - Non-active - Now disabled * Abuse filters 7, 8, 9, 10, 11, 12 (these filters are not needed anymore) - Not reviewed - They are currently disabled * [[Special:AbuseFilter/17]] (no hits since 2022) - Fundamental Physics Edits - 347 hits; none since 2022 - Non-active and very specific for a historical issue - Now disabled * [[Special:AbuseFilter/19]] (no hits since 2019) - Page Creation - 20 hits; none since 2019 - Retained for historical reference and possible future updates - Now disabled * [[Special:AbuseFilter/21]] (false positives, vandal currently inactive) - Globally Banned Editor (renamed to Low-edit Spam Monitor) - 2,829 hits including very recent - Only applies to users with less than 5 edits and takes no actions / monitoring only - Reviewing the details of the hits I don't see many false positives and have strengthened its actions to add a tag and warning -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:57, 9 April 2026 (UTC) == Block request == Please block ~2026-20985-80/~2026-21079-90/~2026-21223-88. Reason: Vandalism. [[User:Àncilu|Àncilu]] ([[User talk:Àncilu|discuss]] • [[Special:Contributions/Àncilu|contribs]]) 23:24, 5 April 2026 (UTC) :All edits should be deleted and the first is blocked by Atcovi. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 00:33, 6 April 2026 (UTC) == Antispam - Filter 12 == {{ping|Codename Noreste}} Thanks for contacting me with a suggested [[Special:AbuseFilter|abuse filter]] for the coupon spam we've been getting. A very much appreciated time saver. Per your suggestion, abuse filter 12 has been reactivated with your updated regex. It should tag and prevent page creation actions for coupon promo etc. Sincerely, James -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 10:33, 15 April 2026 (UTC) : @[[User:Jtneill|Jtneill]] Please do the following to filter 12: :# Enable the block action (temporary for unregistered users, indefinite for registered users) :# Disable both the disallow and tag actions, since that filter will be set to block : Thank you. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 13:42, 5 May 2026 (UTC) ::Excellent. Done: ::* Block duration for non-registered users = 1 week ::* Block duration for registered users = indefinite ::* Disallow and tagging disabled ::Thankyou. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 10:24, 9 May 2026 (UTC) == Urgent! error message This action has been automatically identified as potentially harmful, and therefore disallowed == While submitting the post this error was coming "This action has been automatically identified as potentially harmful, and therefore disallowed. If you believe your action was constructive, please inform an administrator of what you were trying to do. A brief description of the abuse rule which your action matched is: New User Created Page with External Link" How to resolve it? Here is the content: {{note|marketing material removed}} [[User:EasyshikshaMarketing|EasyshikshaMarketing]] ([[User talk:EasyshikshaMarketing|discuss]] • [[Special:Contributions/EasyshikshaMarketing|contribs]]) 05:14, 17 April 2026 (UTC) : That's because Wikiversity doesn't accept advertising. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 05:20, 17 April 2026 (UTC) ::So this is fine ::'''Online Internship and Digital Learning for Students''' ::Online learning has become an important part of modern education. With the help of the internet and digital tools, students can now study, practice, and gain experience without being physically present in a classroom. One key part of this system is the online internship, which helps students learn real-world skills along with their studies. ::An online internship allows students to work on tasks and projects through digital platforms. This makes learning more practical and useful, especially for those who want to understand how real work environments function. ::'''Background''' ::The concept of online learning developed from distance education, where students learned from remote locations. Over time, with the growth of digital technology, learning has become more interactive and flexible. ::The introduction of the online internship has added another important layer to digital education. It combines theoretical learning with practical experience, helping students prepare for future careers. ::During global events such as the COVID-19 pandemic, online education and online internship programs became essential. They helped students continue learning and gaining experience despite restrictions on physical movement. ::'''Importance of Online Internship''' ::An online internship plays an important role in student development. It helps bridge the gap between academic knowledge and practical skills. ::'''Some key points include:''' ::Students understand how real work is done ::They develop basic professional skills ::It supports career readiness ::It allows learning without location limits ::By participating in an online internship, students can improve their confidence and gain early exposure to different fields. ::'''Features of Online Internship-Based Learning''' ::Modern education platforms often include online internship opportunities as part of their learning system. These usually offer: ::Flexible schedules for students ::Access to learning from home ::Beginner-friendly tasks and projects ::A combination of theory and practice ::Such features make online internship programs suitable for a wide range of learners, including beginners. ::'''Learning Tasks''' ::Explore how online internship programs support student learning in digital environments. ::Identify how students can gain practical experience through an online internship ::Analyze the role of flexible learning in improving student engagement ::Understand how online education and internships work together ::'''References''' ::[https://ijpsl.in/wp-content/uploads/2020/09/E-Learning_Hanaaya-Navaneeth.pdf The Past, Present and Future of E-Learning: Hanaaya Varyani and Navaneeth M S] ::[https://ieeexplore.ieee.org/document/9788102 Current Trends and Future Perspectives of e-Learning in India] ::'''See Also''' ::[https://easyshiksha.com/online_courses/internship Online Internship] ::[https://easyshiksha.com/online_courses/ Online Courses] ::[https://easyshiksha.com/online_courses/kids-learning Kids Learning] ::[https://easyshiksha.com/career_helper/ Career Guidance] [[User:EasyshikshaMarketing|EasyshikshaMarketing]] ([[User talk:EasyshikshaMarketing|discuss]] • [[Special:Contributions/EasyshikshaMarketing|contribs]]) 05:27, 17 April 2026 (UTC) :::@[[User:EasyshikshaMarketing|EasyshikshaMarketing]] Wikiversity is a resource for education, or a space for education. However, your intention to link to another website is obvious, and such content does not belong here, as it contradicts the purpose of Wikiversity. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 11:38, 17 April 2026 (UTC) ==AI-generated images== Seeking your advice. Myself and students use some AI-generated images in the [[Motivation and emotion]] project. [[User:Dronebogus]] has been removing some of these images from Wikiversity pages and nominating them for deletion at Commons. As a result, some have been deleted and some have been kept. Dronebogus has made some useful edits and image suggestions for [[Motivation and emotion]] which I've appreciated and incorporated. However, there are other edits to remove an AI image by Dronebogus that I've reverted where I think the image is more educational than no image or an alternative image suggested by Dronebogus. There are a couple of pages where Dronebogus has reverted my reversion, so we are at risk of edit warring. We have briefly discussed and warned each other on our user talk pages, but it seems to come down to a difference in perception about the educational usefulness of the AI images. So, I'm asking here for others to please review the recent edit histories for these pages: * [[Motivation and emotion/Lectures/Brain and physiological needs]] * [[Motivation and emotion/Book/2025/Stockholm syndrome emotion]] and let us know what you think about the AI image suitability vs. using no image or alternative images suggested by Dronebogus. Sincerely, James -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:45, 26 April 2026 (UTC) :I think Jtneill needs to try harder to find non-AI alternatives both on Commons and the web. I’m not reiterating the well known problems with generative AI— you can read about those on Wikipedia and the broader Internet. Needless to say it’s kind of just inherently toxic. If you use it, it should be the last resort of last resorts. Just my stance, which I consider perfectly valid and reasonable. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 04:10, 27 April 2026 (UTC) == Request for Page Creation Approval - Research on Dr. Ian Stevenson == '''Hello Administrators,''' I am a digital archivist and researcher attempting to create an educational page regarding Dr. Ian Stevenson's scientific work, specifically titled "Where Reincarnation and Biology Intersect" and its official Vietnamese edition. My edits are purely constructive, aimed at integrating local research context with global structured data (including '''Wikidata Q139548587'''). The automated filter is currently preventing publication due to "new user external link limits." I have ensured that all citations, references, and images (sourced from Wikimedia Commons) are strictly academic and non-promotional. Could you please review my contributions and consider whitelisting my account or approving this specific page creation? Thank you for your assistance in preserving this research '''legacy''' within the open knowledge ecosystem. '''Best regards,''' [[User:Ian Stevenson777|Ian Stevenson777]] ([[User talk:Ian Stevenson777|discuss]] • [[Special:Contributions/Ian Stevenson777|contribs]]) 07:26, 28 April 2026 (UTC) : The page title should probably be in English, as this is English Wikiversity. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 14:19, 28 April 2026 (UTC) :@[[User:Ian Stevenson777|Ian Stevenson777]] I looked at the text and it seems to me that it is informative, but the user cannot learn anything from it because they would have to buy the book. In this regard, it seems to me that it is enough [[:w:Where Reincarnation and Biology Intersect|to have that article on Wikipedia]] and there is no need to duplicate it here on Wikiversity. If you are particularly interested with its Vietnamese translation, I would focus on Vietnamese Wikipedia, but it seems to me, that the article there [https://vi.wikipedia.org/w/index.php?title=N%C6%A1i_lu%C3%A2n_h%E1%BB%93i_v%C3%A0_sinh_h%E1%BB%8Dc_giao_nhau&oldid=75029095 have some problems at the moment]. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 12:51, 5 May 2026 (UTC) :This request is AI-generated and was made by a lock evading LTA. Please ignore it. [[User:NguoiDungKhongDinhDanh|NguoiDungKhongDinhDanh]] ([[User talk:NguoiDungKhongDinhDanh|discuss]] • [[Special:Contributions/NguoiDungKhongDinhDanh|contribs]]) 23:58, 13 May 2026 (UTC) == Call for custodians and bureaucrats == <div class="cd-moveMark">''Moved to [[Wikiversity:Notices for custodians#Call for custodians and bureaucrats]]. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 18:46, 12 May 2026 (UTC)''</div> == Interface administrator flag request == @[[User:Jtneill|Jtneill]], @[[User:Mu301|Mu301]] I am requesting IA flag to delete [[MediaWiki:Gadget-WikiSign.js]]. It was requested for deletion by [[User:Sophivorus|a trusted user]] and I think it can be deleted. Once I delete it, I will notify you and you can remove the flag. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:24, 10 May 2026 (UTC) :Can you enable two-factor authentication? :Policy: [[Wikiversity:Interface administrators]] -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 10:27, 10 May 2026 (UTC) ::Done @[[User:Jtneill|Jtneill]]. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:20, 11 May 2026 (UTC) :::You are an interface administrator for 24 hours. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 09:57, 11 May 2026 (UTC) ::::Thx @[[User:Jtneill|Jtneill]], I have just deleted the page. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 09:59, 11 May 2026 (UTC) == Block request == Hi. Please block @[[User:Skibidi Titan 5.0|Skibidi Titan 5.0]]. Reason : Vandalism. Thanks, [[User:Locpac|Locpac]] ([[User talk:Locpac|discuss]] • [[Special:Contributions/Locpac|contribs]]) 15:10, 10 May 2026 (UTC) :Thankyou. Done. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 15:25, 10 May 2026 (UTC) == ~2026-28496-53 == Please block [[Special:Contribs/~2026-28496-53]], vandalism. [[User:Tenshi Hinanawi|Tenshi Hinanawi]] ([[User talk:Tenshi Hinanawi|トーク]] • [[Special:Contributions/Tenshi Hinanawi|投稿記録]]) 17:02, 11 May 2026 (UTC) :{{done}} Thank you! —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 17:14, 11 May 2026 (UTC) == ~2026-28792-52 == Please block [[Special:Contribs/~2026-28792-52]], vandalism. Appears to be same user as above. [[User:Tenshi Hinanawi|Tenshi Hinanawi]] ([[User talk:Tenshi Hinanawi|トーク]] • [[Special:Contributions/Tenshi Hinanawi|投稿記録]]) 17:38, 12 May 2026 (UTC) : Blocked locally by Barras. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 17:46, 12 May 2026 (UTC) fwf7wlqbojlhrkpsg8ypsbtp0e8m9kl 2812896 2812894 2026-06-05T17:38:28Z Codename Noreste 2969951 /* New User Exceeded New Page Limit */ archive from [[Wikiversity:Request custodian action]] ([[mw:c:Special:MyLanguage/User:JWBTH/CD|CD]]) 2812896 wikitext text/x-wiki {{Archive|Wikiversity:Request custodian action}} == Sidewide count.js == i would like something like: [[Template:User contrib count/count.js]]. i created [[Template:User contrib count]] and a user/common.js. {{User contrib count}}.<br><br> so a "count.js" would complete it. See [[User:Harold Foppele/common.js]]. If an Administrator could help please. Cheers [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 19:22, 18 January 2026 (UTC) == need to add my profile == im trying to add new profile content [[User:PAGURUMURTHY|PAGURUMURTHY]] ([[User talk:PAGURUMURTHY|discuss]] • [[Special:Contributions/PAGURUMURTHY|contribs]]) 18:03, 4 February 2026 (UTC) :You can edit it now. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 18:05, 4 February 2026 (UTC) ::where can create a new one [[User:PAGURUMURTHY|PAGURUMURTHY]] ([[User talk:PAGURUMURTHY|discuss]] • [[Special:Contributions/PAGURUMURTHY|contribs]]) 18:51, 4 February 2026 (UTC) :::i have created and its in sandbox. i would like to know when it will be approved [[User:PAGURUMURTHY|PAGURUMURTHY]] ([[User talk:PAGURUMURTHY|discuss]] • [[Special:Contributions/PAGURUMURTHY|contribs]]) 19:38, 4 February 2026 (UTC) ::::Please don’t create [[wv:spam|spam]] pages as it will be deleted. Please also read [[WV:Scope]] [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 04:01, 5 February 2026 (UTC) == Im trying to add new profile while add content its shows not alowed == This action has been automatically identified as potentially harmful, and therefore disallowed. If you believe your action was constructive, please [[Wikiversity:Request custodian action|inform an administrator]] of what you were trying to do. A brief description of the abuse rule which your action matched is: New User Exceeded New Page Limit This action has been automatically identified as potentially harmful, and therefore disallowed. If you believe your action was constructive, please [[Wikiversity:Request custodian action|inform an administrator]] of what you were trying to do. A brief description of the abuse rule which your action matched is: New User Created Page with External Link [[User:PAGURUMURTHY|PAGURUMURTHY]] ([[User talk:PAGURUMURTHY|discuss]] • [[Special:Contributions/PAGURUMURTHY|contribs]]) 18:51, 4 February 2026 (UTC) == New User: cannot create talk page == Hi, I am a new user of Wikiversity and I wanted to create a talk page for the article [[ChatGPT's Essay on Kohlberg's Theory: AI's Use in Academic Writing]]. As a new user, I was barred from performing this action. The text that I wanted to add to the talk page is: <blockquote> I have doubts as to to the reliability of this essay. Take for rexample the sentence: <blockquote> Due to its efficiency, AI has saved 380,000-403,000 lives per year in European healthcare as reported in a recent Deloitte and MedTech Europe report<ref>Dantas, C., Mackiewicz, K., Tageo, V., Jacucci, G., Guardado, D., Ortet, S., Varlamis, I., Maniadakis, M., De Lera, E., Quintas, J., Kocsis, O., & Vassiliou, C. (2021). Benefits and hurdles of AI in the workplace – what comes next? ''International Journal of Artificial Intelligence and Expert Systems, 10'', 9-17. https://www.researchgate.net/publication/351993615_Benefits_and_Hurdles_of_AI_In_The_Workplace_-What_Comes_Next</ref>. </blockquote> Reading the reference (freely available on ResearchGate), one notes: # that the reference is from 2021 (predating the widespread use of LLMs such as ChatGPT and the associated 'AI' boom), and # that the reference factually contradicts this essay. Quoting from the reference: <blockquote> There are enormous benefits of applying AI-based solutions to monitor workers’ health and prevent accidents or, currently, COVID-19 infections, and those benefits are reported with enormous potential. According to the recent Deloitte and MedTech Europe report [11], implementing AI in European healthcare systems could save up 380,000 to 403,000 lives annually or €170.9 to 212.4 billion per year. </blockquote> Not that the reference says ''could save'', not ''saves'' as in the essay. This calls into question the reliability of the essay. </blockquote> Could an administrator make this addition for me? Thank you! {{reflist}} [[User:Æolus|Æolus]] ([[User talk:Æolus|discuss]] • [[Special:Contributions/Æolus|contribs]]) 06:53, 5 February 2026 (UTC) :@[[User:Æolus|Æolus]] I have added it for you, you can change the header and sign it now. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 08:05, 5 February 2026 (UTC) ::Thank you! [[User:Æolus|Æolus]] ([[User talk:Æolus|discuss]] • [[Special:Contributions/Æolus|contribs]]) 12:43, 5 February 2026 (UTC) == Disallowed to add a page on a course == I'm trying to populate a newly created course on Wikiversity, but it blocks me from creating more pages with "New User Exceeded New Page Limit". Could this be lifted please? [[User:Berkeleywho|Berkeleywho]] ([[User talk:Berkeleywho|discuss]] • [[Special:Contributions/Berkeleywho|contribs]]) 13:21, 15 February 2026 (UTC) :Sorry! Never mind. I was trying to create a new article instead of a new page. All good now. [[User:Berkeleywho|Berkeleywho]] ([[User talk:Berkeleywho|discuss]] • [[Special:Contributions/Berkeleywho|contribs]]) 14:03, 15 February 2026 (UTC) == Harold Foppele adding LLM-generated nonsense and personal fiction == I became aware of [[User:Harold Foppele]]'s editing after I deleted some of his uploads on Commons. He appears to be adding a large amount of text and images that are some combination of personal fiction and LLM-generated nonsense. This includes: *[[Quantum Ultra fast lasers#Future thought experiment|Personal speculative fiction]] in an otherwise "nonfiction" article *Uploading nonsense LLM-created [[:File:Rontosecond pulse laser (Schematic).jpg|diagrams]] and [[:File:Rontosecond pulse laser (Futuristic).jpg|renders]] for nonexistent lab equipment, with fake source (on Commons, he indicated these files as having been created by him using an LLM) *Uploading nonsense LLM-created images of equations with obvious artifacts. These images, such as [[:File:Redfield equation (non-Markovian).png]] and [[:File:Lindblad equation (Markovian).png]], don't even match the text he puts them with. Much of his writing is also of extremely poor quality, to the point where it's not clear whether it's written by him or an LLM. I'm not an active editor on this project, so I'm not as familiar with the standards here, but I believe this is worth custodian attention. [[User:Pi.1415926535|Pi.1415926535]] ([[User talk:Pi.1415926535|discuss]] • [[Special:Contributions/Pi.1415926535|contribs]]) 03:06, 23 February 2026 (UTC) :Fake source ''and'' contradictory copyright info, claiming both public domain and CC license. Moreover, if they are indeed nearly-direct LLM output, depending on jurisdiction they may not even be eligible for copyright. :I've put speedy deletion marks for the equations, because they're obviously not coherent mathematical equations (the parentheses don't match, the symbols merge into each other the way text in image models often do, etc) [[User:Sesquilinear|Sesquilinear]] ([[User talk:Sesquilinear|discuss]] • [[Special:Contributions/Sesquilinear|contribs]]) 21:50, 7 March 2026 (UTC) == Repeated removal of RFD notices by Harold Foppele == {{User|Harold Foppele }} This editor is appearing in multiple noticeboards for behaviour which is contentious. Ther latest adventure is the repeated removal of tye RFD notice at [[Quantum/Henry C. Kapteyn]]. You will see from their contributions record the number of times. I have warned Tham on their user tag page that this is tantaomunt to volunteering to be blocked here. They have a track record of achieving blocks on enWiki and Commons already. They have all the appearance of shooting not to understand when given direct information about their behaviour, whichever project they are editing, and are fast becoming a time sink. Their behaviour across multiple WMF sites may well lead to a Global Lock, but I do not believe they have quite reached the threshold for that. I believe that what is required is a preventative block to seek to ensure thatchy understand the seriousness of their behaviour, and the need to be collegial. 🇵🇸&zwj;🇺🇦&nbsp;[[User:Timtrent|Timtrent]]&nbsp;🇺🇦&nbsp;[[User talk:Timtrent|talk to me]]&nbsp;🇺🇦&zwj;🇵🇸 23:03, 4 March 2026 (UTC) : {{Done}} [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 11:45, 8 March 2026 (UTC) == Blocks for sockpuppet == Please block [[User:Harold Foppele]] and [[User:Johnwilliamsiii]] for sockpuppetry based on [https://en.wikipedia.org/wiki/Wikipedia:Sockpuppet_investigations/Harold_Foppele en wiki] CU and [https://commons.wikimedia.org/w/index.php?diff=1177465640 commons] CU investigations. The user has also violated copyright, see the above discussion. A block is necessary to prevent further abuse. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 11:30, 8 March 2026 (UTC) :<small>@[[User:MathXplore|MathXplore]]</small> [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 11:31, 8 March 2026 (UTC) :: {{Done}} [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 11:44, 8 March 2026 (UTC) :CC. @[[User:Timtrent|Timtrent]], @[[User:Sesquilinear|Sesquilinear]], @[[User:Pi.1415926535|Pi.1415926535]] [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 11:33, 8 March 2026 (UTC) ::Thank you for the ping. I concur based on [[w:en:WP:DUCK|behaviour]]. CUs appear divided. 🇵🇸&zwj;🇺🇦&nbsp;[[User:Timtrent|Timtrent]]&nbsp;🇺🇦&nbsp;[[User talk:Timtrent|talk to me]]&nbsp;🇺🇦&zwj;🇵🇸 11:41, 8 March 2026 (UTC) == Problem when trying to start a discussion with authors of the Plurilingual education portal == The authors I wanted to discuss with are called "Project PEP" and my name is Franch Chandler. How can I be allowed to do so ? [[User:French Chandler|French Chandler]] ([[User talk:French Chandler|discuss]] • [[Special:Contributions/French Chandler|contribs]]) 18:25, 16 March 2026 (UTC) :@[[User:French Chandler|French Chandler]] place your qestion [https://en.wikiversity.org/w/index.php?title=User_talk:Projet_PEP&action=edit into the dialog box] on this link and hit Publish page. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 20:22, 16 March 2026 (UTC) == Please publish my post == My post is about "Every child grows and develops at their own pace, but some may experience challenges that affect their ability to perform everyday tasks. These challenges can include difficulties with fine motor skills, sensory processing, handwriting, feeding, and self-regulation. When these issues are not addressed early, they can impact a child’s confidence, academic performance, and independence. With the rise of digital healthcare services, '''online physical therapy''' has emerged as a powerful and accessible solution for parents seeking support for their children. This modern approach provides structured, personalized therapy programs that can be accessed from the comfort of home, making it easier for families to ensure consistent care." [[User:Skyabovetherapy|Skyabovetherapy]] ([[User talk:Skyabovetherapy|discuss]] • [[Special:Contributions/Skyabovetherapy|contribs]]) 12:28, 28 March 2026 (UTC) :@[[User:Skyabovetherapy|Skyabovetherapy]] Well, you can publish it yourself, Wikiversity is a free environement, where everybody can create educational resources. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 14:11, 29 March 2026 (UTC) ::They actually triggered some abuse filters. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:24, 29 March 2026 (UTC) :I looked at your attempts to add this text and I see a link to one website repeated many times, which reminds me of the misuse of Wikiversity for self-promotion or to increase the importance of the website. It is necessary to remind you here that Wikiversity is not a place for promotion, but a place for education. So if you want to educate, it will not be a problem to create the page without external links with a clearly defined procedure for how people should use it and what to expect from it. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:07, 1 April 2026 (UTC) == New user limit == Hi, I am creating an AIPA Method learning resource page. I am the author of the linked research, and I hit the “new user limit” and “new page with external link” filters while publishing. Here is the link to the page in creation: [https://en.wikiversity.org/w/index.php?title=AIPA_Method&veaction=edit] Thank you for your help. Best regards, Senad Dizdarević [[User:Senad Dizdarević|Senad Dizdarević]] ([[User talk:Senad Dizdarević|discuss]] • [[Special:Contributions/Senad Dizdarević|contribs]]) 07:19, 30 March 2026 (UTC) :@[[User:Senad Dizdarević|Senad Dizdarević]] I should admit I dont know, what is "new user limit", but if filter blocks your page because of certain external link, you may force to save anyway and sometimes it works. It should not work, when the website is blacklisted. As of now, I am not seeing you to save page in main namespace, so try to save it without external links first. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 07:30, 30 March 2026 (UTC) ::Thank you, you are very kind. ::I will wait a day, and try again (without links, too). ::Today, I already created About Me info page, and maybe that is enough for the filters for one day. [[User:Senad Dizdarević|Senad Dizdarević]] ([[User talk:Senad Dizdarević|discuss]] • [[Special:Contributions/Senad Dizdarević|contribs]]) 07:53, 30 March 2026 (UTC) :::Well, I have analyzed your contribution to Wikiversity and I should point out here, that this project is not a place for advertising, so there is no way of promoting your books and authority this way. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:56, 1 April 2026 (UTC) ::::Hi, my About Me page is just an info page with the neutral as possible presentation of my work. ::::There is a big difference between informing and advertising. Informing is neutrally stating that something exists and requiring no action, while advertising is a special communication form with intent to cause certain action from readers. For example, click here, click there, order this, buy that. ::::There is no such intention, form, or terms on my info page. Just neutral information. I don't hide and I am not ashamed that I am write and author, and that is a part of the usual bio, including works. I checked your user page: "I graduated from the Czech University of Life Sciences in Prague and studied information science at the Faculty of Arts of Charles University." I think that if you had written a book on Life Science, you would have mentioned that as well. ::::Most of the Info page is about my research and AIPA Method which is a valid contribution to psychology, consciousness studies, identity theory, and personality development. Actually, my paper '''AIPA Method: A Cognitive-Phenomenological Model for Identity Reconstruction and Stabilization in Pure Awareness''' is now in the peer review procedure at Journal of Consciousness Studies. ::::Here is a part from the Wikiversity AIPA Method page in creation (waiting for the end of the time limit for new users): [[User:Senad Dizdarević|Senad Dizdarević]] ([[User talk:Senad Dizdarević|discuss]] • [[Special:Contributions/Senad Dizdarević|contribs]]) 06:47, 2 April 2026 (UTC) :::::For the unknown reasons, the form didn't publish my second part of the message: :::::I believe this is a valid contribution to Wikiversity. :::::Best Regards, :::::Senad [[User:Senad Dizdarević|Senad Dizdarević]] ([[User talk:Senad Dizdarević|discuss]] • [[Special:Contributions/Senad Dizdarević|contribs]]) 06:52, 2 April 2026 (UTC) ::::::And the third try: :::::: == Introduction == ::::::The AIPA Method addresses a gap in contemporary personal development and consciousness science: most evidence‑based approaches (CBT, MBSR, MBCT, standard meditation) operate at the level of mental content—reframing thoughts, observing them, or reducing their impact—rather than at the level of identity structure. In contrast, AIPA targets the structural relationship between the self and the mind, aiming at durable identity reconstruction rooted in Pure Awareness rather than symptom management. ::::::The central research question of the primary AIPA preprint is whether a structured, sequentially staged method can produce permanent identity reconstruction rooted in Pure Awareness, and how such a method compares to established approaches in scope, mechanism, and outcome. :::::: == Theoretical foundations == ::::::The AIPA framework is grounded in the cognitive‑phenomenological tradition (e.g., McAdams, Varela, Metzinger, Erikson), contemporary consciousness science on minimal phenomenal experience, and qualitative methods advocacy in psychology. It builds directly on: ::::::* Empirical work on pure awareness and Minimal Phenomenal Experience (MPE), especially Gamma & Metzinger’s large‑scale study of content‑reduced awareness states. ::::::* Metzinger’s proposal of minimal phenomenal experience as an entry point for a minimal unifying model of consciousness. ::::::* Narrative identity and partial‑self models within personality and identity theory. ::::::Within this backdrop, AIPA proposes Pure Awareness as a distinct, operationally specified state that can become a structural ground of identity rather than a transient meditative experience. :::::: == Experiential empiricism == ::::::The empirical foundation of the AIPA Method is explicitly first‑person and experiential, combining: ::::::* A 22‑year longitudinal autoethnographic self‑study (2003–2025) documenting partial personality episodes, protocol use, and outcomes. ::::::* A 13‑year prospective verification period with zero self‑reported recurrence of targeted harmful behaviors after a dated stabilization point (1 January 2006). ::::::* A high‑ecological‑validity “stress test” during acute bereavement, used to examine whether non‑reactive awareness remains stable under maximal provocation. ::::::* Two independent practitioner cases (an Amazon‑verified report and a structured questionnaire case) providing preliminary convergent signals across cognitive, emotional, behavioral, and identity dimensions. ::::::All central constructs (Pure Awareness, partial personalities, the Switch, identity stabilization) are operationalized with explicit phenomenological and behavioral criteria intended to enable replication and future third‑person measurement. ::::::I believe this is a valid contribution to Wikiversity. ::::::Best regards, ::::::Senad [[User:Senad Dizdarević|Senad Dizdarević]] ([[User talk:Senad Dizdarević|discuss]] • [[Special:Contributions/Senad Dizdarević|contribs]]) 06:54, 2 April 2026 (UTC) == Unable to publish pages == Whenever I try to publish a page with linked sources it gets flagged and says I'm a new user attempting to publish content with outside links. Those outside links are my sources. [[User:Soboyed|Soboyed]] ([[User talk:Soboyed|discuss]] • [[Special:Contributions/Soboyed|contribs]]) 04:52, 2 April 2026 (UTC) :This restriction is automatically lifted after you have edited for a certain time (I don't recall that time off-hand, but it is not long). This is designed to stop spam. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 04:53, 2 April 2026 (UTC) == Showing error to publish a Post == My action was constructive, not destructive, please allow to publish it. [[Special:Contributions/&#126;2026-20906-18|&#126;2026-20906-18]] ([[User talk:&#126;2026-20906-18|talk]]) 08:06, 4 April 2026 (UTC) :Maybe you got caught in a filter. Consider [[Special:CreateAccount|creating an account]]. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 09:06, 4 April 2026 (UTC) :Your edits, [https://en.wikiversity.org/w/index.php?title=Special:AbuseLog&wpSearchUser=%7E2026-20906-18 these ones], seems to have tripped a filter when you tried to create a page on [[Create]] which external links. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 23:58, 4 April 2026 (UTC) :Have you read my [https://en.wikiversity.org/w/index.php?title=Wikiversity:Request_custodian_action&diff=prev&oldid=2802219 previous reply] to you @[[User:~2026-20906-18|~2026-20906-18]]? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:02, 6 April 2026 (UTC) == Abuse filters which should be deleted == Hi, there are some abuse filters which should probably be deleted. * [[Special:AbuseFilter/1]] (not needed anymore) * [[Special:AbuseFilter/2]] (no hits since 2018) * [[Special:AbuseFilter/3]] (not needed since there are global filters that disallow this specific type of spam filter 3 would have catched) * [[Special:AbuseFilter/4]] (looking at the logs, there are too many false positives) * [[Special:AbuseFilter/5]] (no hits since 2023) * Abuse filters 7, 8, 9, 10, 11, 12 (these filters are not needed anymore) * [[Special:AbuseFilter/17]] (no hits since 2022) * [[Special:AbuseFilter/19]] (no hits since 2019) * [[Special:AbuseFilter/21]] (false positives, vandal currently inactive) Thanks. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 03:51, 5 April 2026 (UTC) :Why do these need to be deleted rather than inactivated? Do inactive abuse filters cause a server strain? —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:39, 5 April 2026 (UTC) :: Deleted filters do not cause strain to the servers. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 14:28, 5 April 2026 (UTC) :These sounds like sensible suggestions but, yes, would inactivation perhaps make more sense than deletion for at least some filters? -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 09:35, 5 April 2026 (UTC) :I would keep them @[[User:Codename Noreste|Codename Noreste]]. Alternatively, I would turn off the ones that haven't caught anything for a long time, but I would leave them enabled in case they need to be turned on or improved. If someone has already written the code and we don't have hundreds of free man-hours of programmers on Wikiversity, the server load seems secondary to me, and is negligible compared to other things. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:11, 6 April 2026 (UTC) :: I know how to write abuse filter code and regex, but I would recommend disabling filters that have never caught anything in a long time ''and'' those who made lots of false positives. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:09, 6 April 2026 (UTC) :::Of course @[[User:Codename Noreste|Codename Noreste]], there are people here today who are capable of changing the code. But the question is what it will be like in a few years, the question is what will happen if those two are busy for a long time, etc. That's why I would leave it so that those who don't know much about code can be inspired by it and will need to do something with it someday - plus, more code for different types of filtering is actually great educational material on how those filters work. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 11:41, 17 April 2026 (UTC) Here's the updated list of abuse filters under review with actions I've taken (several disabled, one basic code improvement, and some actions changed) - none have been deleted so they can all be edited and reactivayed - please suggest any further changes: * [[Special:AbuseFilter/1]] (not needed anymore) - One time account spam bot - 4 hits over 10 years ago - Disabled in 2024 - May be useful in future * [[Special:AbuseFilter/2]] (no hits since 2018) - Userspace spamming - 778 hits; none since 2018 likely due to global filters - Now disabled * [[Special:AbuseFilter/3]] (not needed since there are global filters that disallow this specific type of spam filter 3 would have catched) - Specific user page spam - 1,101 hits most recent 7 March 2026 - Still active - Kept enabled * [[Special:AbuseFilter/4]] (looking at the logs, there are too many false positives) - Questionable Language (profanity) - 6,055 hits including very recently - However it was logging hits without taking any actions - Edited to reduce likelihood of false positives by only applying filter to users with low (< 20) edit count and applied weak actions to tag and warn but not prevent publishing the content * [[Special:AbuseFilter/5]] (no hits since 2023) - Blocked Solicitation Links - 95 hits; none since 2023 - blocks specific historical spam sites - Non-active - Now disabled * Abuse filters 7, 8, 9, 10, 11, 12 (these filters are not needed anymore) - Not reviewed - They are currently disabled * [[Special:AbuseFilter/17]] (no hits since 2022) - Fundamental Physics Edits - 347 hits; none since 2022 - Non-active and very specific for a historical issue - Now disabled * [[Special:AbuseFilter/19]] (no hits since 2019) - Page Creation - 20 hits; none since 2019 - Retained for historical reference and possible future updates - Now disabled * [[Special:AbuseFilter/21]] (false positives, vandal currently inactive) - Globally Banned Editor (renamed to Low-edit Spam Monitor) - 2,829 hits including very recent - Only applies to users with less than 5 edits and takes no actions / monitoring only - Reviewing the details of the hits I don't see many false positives and have strengthened its actions to add a tag and warning -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:57, 9 April 2026 (UTC) == Block request == Please block ~2026-20985-80/~2026-21079-90/~2026-21223-88. Reason: Vandalism. [[User:Àncilu|Àncilu]] ([[User talk:Àncilu|discuss]] • [[Special:Contributions/Àncilu|contribs]]) 23:24, 5 April 2026 (UTC) :All edits should be deleted and the first is blocked by Atcovi. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 00:33, 6 April 2026 (UTC) == Antispam - Filter 12 == {{ping|Codename Noreste}} Thanks for contacting me with a suggested [[Special:AbuseFilter|abuse filter]] for the coupon spam we've been getting. A very much appreciated time saver. Per your suggestion, abuse filter 12 has been reactivated with your updated regex. It should tag and prevent page creation actions for coupon promo etc. Sincerely, James -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 10:33, 15 April 2026 (UTC) : @[[User:Jtneill|Jtneill]] Please do the following to filter 12: :# Enable the block action (temporary for unregistered users, indefinite for registered users) :# Disable both the disallow and tag actions, since that filter will be set to block : Thank you. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 13:42, 5 May 2026 (UTC) ::Excellent. Done: ::* Block duration for non-registered users = 1 week ::* Block duration for registered users = indefinite ::* Disallow and tagging disabled ::Thankyou. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 10:24, 9 May 2026 (UTC) == Urgent! error message This action has been automatically identified as potentially harmful, and therefore disallowed == While submitting the post this error was coming "This action has been automatically identified as potentially harmful, and therefore disallowed. If you believe your action was constructive, please inform an administrator of what you were trying to do. A brief description of the abuse rule which your action matched is: New User Created Page with External Link" How to resolve it? Here is the content: {{note|marketing material removed}} [[User:EasyshikshaMarketing|EasyshikshaMarketing]] ([[User talk:EasyshikshaMarketing|discuss]] • [[Special:Contributions/EasyshikshaMarketing|contribs]]) 05:14, 17 April 2026 (UTC) : That's because Wikiversity doesn't accept advertising. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 05:20, 17 April 2026 (UTC) ::So this is fine ::'''Online Internship and Digital Learning for Students''' ::Online learning has become an important part of modern education. With the help of the internet and digital tools, students can now study, practice, and gain experience without being physically present in a classroom. One key part of this system is the online internship, which helps students learn real-world skills along with their studies. ::An online internship allows students to work on tasks and projects through digital platforms. This makes learning more practical and useful, especially for those who want to understand how real work environments function. ::'''Background''' ::The concept of online learning developed from distance education, where students learned from remote locations. Over time, with the growth of digital technology, learning has become more interactive and flexible. ::The introduction of the online internship has added another important layer to digital education. It combines theoretical learning with practical experience, helping students prepare for future careers. ::During global events such as the COVID-19 pandemic, online education and online internship programs became essential. They helped students continue learning and gaining experience despite restrictions on physical movement. ::'''Importance of Online Internship''' ::An online internship plays an important role in student development. It helps bridge the gap between academic knowledge and practical skills. ::'''Some key points include:''' ::Students understand how real work is done ::They develop basic professional skills ::It supports career readiness ::It allows learning without location limits ::By participating in an online internship, students can improve their confidence and gain early exposure to different fields. ::'''Features of Online Internship-Based Learning''' ::Modern education platforms often include online internship opportunities as part of their learning system. These usually offer: ::Flexible schedules for students ::Access to learning from home ::Beginner-friendly tasks and projects ::A combination of theory and practice ::Such features make online internship programs suitable for a wide range of learners, including beginners. ::'''Learning Tasks''' ::Explore how online internship programs support student learning in digital environments. ::Identify how students can gain practical experience through an online internship ::Analyze the role of flexible learning in improving student engagement ::Understand how online education and internships work together ::'''References''' ::[https://ijpsl.in/wp-content/uploads/2020/09/E-Learning_Hanaaya-Navaneeth.pdf The Past, Present and Future of E-Learning: Hanaaya Varyani and Navaneeth M S] ::[https://ieeexplore.ieee.org/document/9788102 Current Trends and Future Perspectives of e-Learning in India] ::'''See Also''' ::[https://easyshiksha.com/online_courses/internship Online Internship] ::[https://easyshiksha.com/online_courses/ Online Courses] ::[https://easyshiksha.com/online_courses/kids-learning Kids Learning] ::[https://easyshiksha.com/career_helper/ Career Guidance] [[User:EasyshikshaMarketing|EasyshikshaMarketing]] ([[User talk:EasyshikshaMarketing|discuss]] • [[Special:Contributions/EasyshikshaMarketing|contribs]]) 05:27, 17 April 2026 (UTC) :::@[[User:EasyshikshaMarketing|EasyshikshaMarketing]] Wikiversity is a resource for education, or a space for education. However, your intention to link to another website is obvious, and such content does not belong here, as it contradicts the purpose of Wikiversity. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 11:38, 17 April 2026 (UTC) ==AI-generated images== Seeking your advice. Myself and students use some AI-generated images in the [[Motivation and emotion]] project. [[User:Dronebogus]] has been removing some of these images from Wikiversity pages and nominating them for deletion at Commons. As a result, some have been deleted and some have been kept. Dronebogus has made some useful edits and image suggestions for [[Motivation and emotion]] which I've appreciated and incorporated. However, there are other edits to remove an AI image by Dronebogus that I've reverted where I think the image is more educational than no image or an alternative image suggested by Dronebogus. There are a couple of pages where Dronebogus has reverted my reversion, so we are at risk of edit warring. We have briefly discussed and warned each other on our user talk pages, but it seems to come down to a difference in perception about the educational usefulness of the AI images. So, I'm asking here for others to please review the recent edit histories for these pages: * [[Motivation and emotion/Lectures/Brain and physiological needs]] * [[Motivation and emotion/Book/2025/Stockholm syndrome emotion]] and let us know what you think about the AI image suitability vs. using no image or alternative images suggested by Dronebogus. Sincerely, James -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:45, 26 April 2026 (UTC) :I think Jtneill needs to try harder to find non-AI alternatives both on Commons and the web. I’m not reiterating the well known problems with generative AI— you can read about those on Wikipedia and the broader Internet. Needless to say it’s kind of just inherently toxic. If you use it, it should be the last resort of last resorts. Just my stance, which I consider perfectly valid and reasonable. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 04:10, 27 April 2026 (UTC) == Request for Page Creation Approval - Research on Dr. Ian Stevenson == '''Hello Administrators,''' I am a digital archivist and researcher attempting to create an educational page regarding Dr. Ian Stevenson's scientific work, specifically titled "Where Reincarnation and Biology Intersect" and its official Vietnamese edition. My edits are purely constructive, aimed at integrating local research context with global structured data (including '''Wikidata Q139548587'''). The automated filter is currently preventing publication due to "new user external link limits." I have ensured that all citations, references, and images (sourced from Wikimedia Commons) are strictly academic and non-promotional. Could you please review my contributions and consider whitelisting my account or approving this specific page creation? Thank you for your assistance in preserving this research '''legacy''' within the open knowledge ecosystem. '''Best regards,''' [[User:Ian Stevenson777|Ian Stevenson777]] ([[User talk:Ian Stevenson777|discuss]] • [[Special:Contributions/Ian Stevenson777|contribs]]) 07:26, 28 April 2026 (UTC) : The page title should probably be in English, as this is English Wikiversity. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 14:19, 28 April 2026 (UTC) :@[[User:Ian Stevenson777|Ian Stevenson777]] I looked at the text and it seems to me that it is informative, but the user cannot learn anything from it because they would have to buy the book. In this regard, it seems to me that it is enough [[:w:Where Reincarnation and Biology Intersect|to have that article on Wikipedia]] and there is no need to duplicate it here on Wikiversity. If you are particularly interested with its Vietnamese translation, I would focus on Vietnamese Wikipedia, but it seems to me, that the article there [https://vi.wikipedia.org/w/index.php?title=N%C6%A1i_lu%C3%A2n_h%E1%BB%93i_v%C3%A0_sinh_h%E1%BB%8Dc_giao_nhau&oldid=75029095 have some problems at the moment]. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 12:51, 5 May 2026 (UTC) :This request is AI-generated and was made by a lock evading LTA. Please ignore it. [[User:NguoiDungKhongDinhDanh|NguoiDungKhongDinhDanh]] ([[User talk:NguoiDungKhongDinhDanh|discuss]] • [[Special:Contributions/NguoiDungKhongDinhDanh|contribs]]) 23:58, 13 May 2026 (UTC) == Call for custodians and bureaucrats == <div class="cd-moveMark">''Moved to [[Wikiversity:Notices for custodians#Call for custodians and bureaucrats]]. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 18:46, 12 May 2026 (UTC)''</div> == Interface administrator flag request == @[[User:Jtneill|Jtneill]], @[[User:Mu301|Mu301]] I am requesting IA flag to delete [[MediaWiki:Gadget-WikiSign.js]]. It was requested for deletion by [[User:Sophivorus|a trusted user]] and I think it can be deleted. Once I delete it, I will notify you and you can remove the flag. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:24, 10 May 2026 (UTC) :Can you enable two-factor authentication? :Policy: [[Wikiversity:Interface administrators]] -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 10:27, 10 May 2026 (UTC) ::Done @[[User:Jtneill|Jtneill]]. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:20, 11 May 2026 (UTC) :::You are an interface administrator for 24 hours. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 09:57, 11 May 2026 (UTC) ::::Thx @[[User:Jtneill|Jtneill]], I have just deleted the page. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 09:59, 11 May 2026 (UTC) == Block request == Hi. Please block @[[User:Skibidi Titan 5.0|Skibidi Titan 5.0]]. Reason : Vandalism. Thanks, [[User:Locpac|Locpac]] ([[User talk:Locpac|discuss]] • [[Special:Contributions/Locpac|contribs]]) 15:10, 10 May 2026 (UTC) :Thankyou. Done. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 15:25, 10 May 2026 (UTC) == ~2026-28496-53 == Please block [[Special:Contribs/~2026-28496-53]], vandalism. [[User:Tenshi Hinanawi|Tenshi Hinanawi]] ([[User talk:Tenshi Hinanawi|トーク]] • [[Special:Contributions/Tenshi Hinanawi|投稿記録]]) 17:02, 11 May 2026 (UTC) :{{done}} Thank you! —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 17:14, 11 May 2026 (UTC) == ~2026-28792-52 == Please block [[Special:Contribs/~2026-28792-52]], vandalism. Appears to be same user as above. [[User:Tenshi Hinanawi|Tenshi Hinanawi]] ([[User talk:Tenshi Hinanawi|トーク]] • [[Special:Contributions/Tenshi Hinanawi|投稿記録]]) 17:38, 12 May 2026 (UTC) : Blocked locally by Barras. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 17:46, 12 May 2026 (UTC) == New User Exceeded New Page Limit == Hello, my action of creating a Portal for Banjo learning (Portal:Banjo) was denied as I am a user whose had my account for a while but not written before today. My actions are constructive, but if you would rather me wait and let the system work as it intends to then that's okay too. --[[User:Kirby - Electrotechnics|Kirby - Electrotechnics]] ([[User talk:Kirby - Electrotechnics|discuss]] • [[Special:Contributions/Kirby - Electrotechnics|contribs]]) 01:13, 17 May 2026 (UTC) :That is standard and the rate limit will fall off as you stay and edit; it has some pretty easy barriers to cross. I can create a blank [[Portal:Banjo]] if you want, but to be clear, portal pages are usually much broader topics that can help orient you to specific pages. Do you think there will be that many pages about the topic of banjo playing? If not, then you can just continue editing [[Banjo]]. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 03:36, 17 May 2026 (UTC) dky4azc0z9db4jb4zpnk3m7z3itoxb4 2812898 2812896 2026-06-05T17:38:39Z Codename Noreste 2969951 /* Disable Special:AbuseFilter/3 */ archive from [[Wikiversity:Request custodian action]] ([[mw:c:Special:MyLanguage/User:JWBTH/CD|CD]]) 2812898 wikitext text/x-wiki {{Archive|Wikiversity:Request custodian action}} == Sidewide count.js == i would like something like: [[Template:User contrib count/count.js]]. i created [[Template:User contrib count]] and a user/common.js. {{User contrib count}}.<br><br> so a "count.js" would complete it. See [[User:Harold Foppele/common.js]]. If an Administrator could help please. Cheers [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 19:22, 18 January 2026 (UTC) == need to add my profile == im trying to add new profile content [[User:PAGURUMURTHY|PAGURUMURTHY]] ([[User talk:PAGURUMURTHY|discuss]] • [[Special:Contributions/PAGURUMURTHY|contribs]]) 18:03, 4 February 2026 (UTC) :You can edit it now. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 18:05, 4 February 2026 (UTC) ::where can create a new one [[User:PAGURUMURTHY|PAGURUMURTHY]] ([[User talk:PAGURUMURTHY|discuss]] • [[Special:Contributions/PAGURUMURTHY|contribs]]) 18:51, 4 February 2026 (UTC) :::i have created and its in sandbox. i would like to know when it will be approved [[User:PAGURUMURTHY|PAGURUMURTHY]] ([[User talk:PAGURUMURTHY|discuss]] • [[Special:Contributions/PAGURUMURTHY|contribs]]) 19:38, 4 February 2026 (UTC) ::::Please don’t create [[wv:spam|spam]] pages as it will be deleted. Please also read [[WV:Scope]] [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 04:01, 5 February 2026 (UTC) == Im trying to add new profile while add content its shows not alowed == This action has been automatically identified as potentially harmful, and therefore disallowed. If you believe your action was constructive, please [[Wikiversity:Request custodian action|inform an administrator]] of what you were trying to do. A brief description of the abuse rule which your action matched is: New User Exceeded New Page Limit This action has been automatically identified as potentially harmful, and therefore disallowed. If you believe your action was constructive, please [[Wikiversity:Request custodian action|inform an administrator]] of what you were trying to do. A brief description of the abuse rule which your action matched is: New User Created Page with External Link [[User:PAGURUMURTHY|PAGURUMURTHY]] ([[User talk:PAGURUMURTHY|discuss]] • [[Special:Contributions/PAGURUMURTHY|contribs]]) 18:51, 4 February 2026 (UTC) == New User: cannot create talk page == Hi, I am a new user of Wikiversity and I wanted to create a talk page for the article [[ChatGPT's Essay on Kohlberg's Theory: AI's Use in Academic Writing]]. As a new user, I was barred from performing this action. The text that I wanted to add to the talk page is: <blockquote> I have doubts as to to the reliability of this essay. Take for rexample the sentence: <blockquote> Due to its efficiency, AI has saved 380,000-403,000 lives per year in European healthcare as reported in a recent Deloitte and MedTech Europe report<ref>Dantas, C., Mackiewicz, K., Tageo, V., Jacucci, G., Guardado, D., Ortet, S., Varlamis, I., Maniadakis, M., De Lera, E., Quintas, J., Kocsis, O., & Vassiliou, C. (2021). Benefits and hurdles of AI in the workplace – what comes next? ''International Journal of Artificial Intelligence and Expert Systems, 10'', 9-17. https://www.researchgate.net/publication/351993615_Benefits_and_Hurdles_of_AI_In_The_Workplace_-What_Comes_Next</ref>. </blockquote> Reading the reference (freely available on ResearchGate), one notes: # that the reference is from 2021 (predating the widespread use of LLMs such as ChatGPT and the associated 'AI' boom), and # that the reference factually contradicts this essay. Quoting from the reference: <blockquote> There are enormous benefits of applying AI-based solutions to monitor workers’ health and prevent accidents or, currently, COVID-19 infections, and those benefits are reported with enormous potential. According to the recent Deloitte and MedTech Europe report [11], implementing AI in European healthcare systems could save up 380,000 to 403,000 lives annually or €170.9 to 212.4 billion per year. </blockquote> Not that the reference says ''could save'', not ''saves'' as in the essay. This calls into question the reliability of the essay. </blockquote> Could an administrator make this addition for me? Thank you! {{reflist}} [[User:Æolus|Æolus]] ([[User talk:Æolus|discuss]] • [[Special:Contributions/Æolus|contribs]]) 06:53, 5 February 2026 (UTC) :@[[User:Æolus|Æolus]] I have added it for you, you can change the header and sign it now. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 08:05, 5 February 2026 (UTC) ::Thank you! [[User:Æolus|Æolus]] ([[User talk:Æolus|discuss]] • [[Special:Contributions/Æolus|contribs]]) 12:43, 5 February 2026 (UTC) == Disallowed to add a page on a course == I'm trying to populate a newly created course on Wikiversity, but it blocks me from creating more pages with "New User Exceeded New Page Limit". Could this be lifted please? [[User:Berkeleywho|Berkeleywho]] ([[User talk:Berkeleywho|discuss]] • [[Special:Contributions/Berkeleywho|contribs]]) 13:21, 15 February 2026 (UTC) :Sorry! Never mind. I was trying to create a new article instead of a new page. All good now. [[User:Berkeleywho|Berkeleywho]] ([[User talk:Berkeleywho|discuss]] • [[Special:Contributions/Berkeleywho|contribs]]) 14:03, 15 February 2026 (UTC) == Harold Foppele adding LLM-generated nonsense and personal fiction == I became aware of [[User:Harold Foppele]]'s editing after I deleted some of his uploads on Commons. He appears to be adding a large amount of text and images that are some combination of personal fiction and LLM-generated nonsense. This includes: *[[Quantum Ultra fast lasers#Future thought experiment|Personal speculative fiction]] in an otherwise "nonfiction" article *Uploading nonsense LLM-created [[:File:Rontosecond pulse laser (Schematic).jpg|diagrams]] and [[:File:Rontosecond pulse laser (Futuristic).jpg|renders]] for nonexistent lab equipment, with fake source (on Commons, he indicated these files as having been created by him using an LLM) *Uploading nonsense LLM-created images of equations with obvious artifacts. These images, such as [[:File:Redfield equation (non-Markovian).png]] and [[:File:Lindblad equation (Markovian).png]], don't even match the text he puts them with. Much of his writing is also of extremely poor quality, to the point where it's not clear whether it's written by him or an LLM. I'm not an active editor on this project, so I'm not as familiar with the standards here, but I believe this is worth custodian attention. [[User:Pi.1415926535|Pi.1415926535]] ([[User talk:Pi.1415926535|discuss]] • [[Special:Contributions/Pi.1415926535|contribs]]) 03:06, 23 February 2026 (UTC) :Fake source ''and'' contradictory copyright info, claiming both public domain and CC license. Moreover, if they are indeed nearly-direct LLM output, depending on jurisdiction they may not even be eligible for copyright. :I've put speedy deletion marks for the equations, because they're obviously not coherent mathematical equations (the parentheses don't match, the symbols merge into each other the way text in image models often do, etc) [[User:Sesquilinear|Sesquilinear]] ([[User talk:Sesquilinear|discuss]] • [[Special:Contributions/Sesquilinear|contribs]]) 21:50, 7 March 2026 (UTC) == Repeated removal of RFD notices by Harold Foppele == {{User|Harold Foppele }} This editor is appearing in multiple noticeboards for behaviour which is contentious. Ther latest adventure is the repeated removal of tye RFD notice at [[Quantum/Henry C. Kapteyn]]. You will see from their contributions record the number of times. I have warned Tham on their user tag page that this is tantaomunt to volunteering to be blocked here. They have a track record of achieving blocks on enWiki and Commons already. They have all the appearance of shooting not to understand when given direct information about their behaviour, whichever project they are editing, and are fast becoming a time sink. Their behaviour across multiple WMF sites may well lead to a Global Lock, but I do not believe they have quite reached the threshold for that. I believe that what is required is a preventative block to seek to ensure thatchy understand the seriousness of their behaviour, and the need to be collegial. 🇵🇸&zwj;🇺🇦&nbsp;[[User:Timtrent|Timtrent]]&nbsp;🇺🇦&nbsp;[[User talk:Timtrent|talk to me]]&nbsp;🇺🇦&zwj;🇵🇸 23:03, 4 March 2026 (UTC) : {{Done}} [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 11:45, 8 March 2026 (UTC) == Blocks for sockpuppet == Please block [[User:Harold Foppele]] and [[User:Johnwilliamsiii]] for sockpuppetry based on [https://en.wikipedia.org/wiki/Wikipedia:Sockpuppet_investigations/Harold_Foppele en wiki] CU and [https://commons.wikimedia.org/w/index.php?diff=1177465640 commons] CU investigations. The user has also violated copyright, see the above discussion. A block is necessary to prevent further abuse. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 11:30, 8 March 2026 (UTC) :<small>@[[User:MathXplore|MathXplore]]</small> [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 11:31, 8 March 2026 (UTC) :: {{Done}} [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 11:44, 8 March 2026 (UTC) :CC. @[[User:Timtrent|Timtrent]], @[[User:Sesquilinear|Sesquilinear]], @[[User:Pi.1415926535|Pi.1415926535]] [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 11:33, 8 March 2026 (UTC) ::Thank you for the ping. I concur based on [[w:en:WP:DUCK|behaviour]]. CUs appear divided. 🇵🇸&zwj;🇺🇦&nbsp;[[User:Timtrent|Timtrent]]&nbsp;🇺🇦&nbsp;[[User talk:Timtrent|talk to me]]&nbsp;🇺🇦&zwj;🇵🇸 11:41, 8 March 2026 (UTC) == Problem when trying to start a discussion with authors of the Plurilingual education portal == The authors I wanted to discuss with are called "Project PEP" and my name is Franch Chandler. How can I be allowed to do so ? [[User:French Chandler|French Chandler]] ([[User talk:French Chandler|discuss]] • [[Special:Contributions/French Chandler|contribs]]) 18:25, 16 March 2026 (UTC) :@[[User:French Chandler|French Chandler]] place your qestion [https://en.wikiversity.org/w/index.php?title=User_talk:Projet_PEP&action=edit into the dialog box] on this link and hit Publish page. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 20:22, 16 March 2026 (UTC) == Please publish my post == My post is about "Every child grows and develops at their own pace, but some may experience challenges that affect their ability to perform everyday tasks. These challenges can include difficulties with fine motor skills, sensory processing, handwriting, feeding, and self-regulation. When these issues are not addressed early, they can impact a child’s confidence, academic performance, and independence. With the rise of digital healthcare services, '''online physical therapy''' has emerged as a powerful and accessible solution for parents seeking support for their children. This modern approach provides structured, personalized therapy programs that can be accessed from the comfort of home, making it easier for families to ensure consistent care." [[User:Skyabovetherapy|Skyabovetherapy]] ([[User talk:Skyabovetherapy|discuss]] • [[Special:Contributions/Skyabovetherapy|contribs]]) 12:28, 28 March 2026 (UTC) :@[[User:Skyabovetherapy|Skyabovetherapy]] Well, you can publish it yourself, Wikiversity is a free environement, where everybody can create educational resources. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 14:11, 29 March 2026 (UTC) ::They actually triggered some abuse filters. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:24, 29 March 2026 (UTC) :I looked at your attempts to add this text and I see a link to one website repeated many times, which reminds me of the misuse of Wikiversity for self-promotion or to increase the importance of the website. It is necessary to remind you here that Wikiversity is not a place for promotion, but a place for education. So if you want to educate, it will not be a problem to create the page without external links with a clearly defined procedure for how people should use it and what to expect from it. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:07, 1 April 2026 (UTC) == New user limit == Hi, I am creating an AIPA Method learning resource page. I am the author of the linked research, and I hit the “new user limit” and “new page with external link” filters while publishing. Here is the link to the page in creation: [https://en.wikiversity.org/w/index.php?title=AIPA_Method&veaction=edit] Thank you for your help. Best regards, Senad Dizdarević [[User:Senad Dizdarević|Senad Dizdarević]] ([[User talk:Senad Dizdarević|discuss]] • [[Special:Contributions/Senad Dizdarević|contribs]]) 07:19, 30 March 2026 (UTC) :@[[User:Senad Dizdarević|Senad Dizdarević]] I should admit I dont know, what is "new user limit", but if filter blocks your page because of certain external link, you may force to save anyway and sometimes it works. It should not work, when the website is blacklisted. As of now, I am not seeing you to save page in main namespace, so try to save it without external links first. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 07:30, 30 March 2026 (UTC) ::Thank you, you are very kind. ::I will wait a day, and try again (without links, too). ::Today, I already created About Me info page, and maybe that is enough for the filters for one day. [[User:Senad Dizdarević|Senad Dizdarević]] ([[User talk:Senad Dizdarević|discuss]] • [[Special:Contributions/Senad Dizdarević|contribs]]) 07:53, 30 March 2026 (UTC) :::Well, I have analyzed your contribution to Wikiversity and I should point out here, that this project is not a place for advertising, so there is no way of promoting your books and authority this way. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:56, 1 April 2026 (UTC) ::::Hi, my About Me page is just an info page with the neutral as possible presentation of my work. ::::There is a big difference between informing and advertising. Informing is neutrally stating that something exists and requiring no action, while advertising is a special communication form with intent to cause certain action from readers. For example, click here, click there, order this, buy that. ::::There is no such intention, form, or terms on my info page. Just neutral information. I don't hide and I am not ashamed that I am write and author, and that is a part of the usual bio, including works. I checked your user page: "I graduated from the Czech University of Life Sciences in Prague and studied information science at the Faculty of Arts of Charles University." I think that if you had written a book on Life Science, you would have mentioned that as well. ::::Most of the Info page is about my research and AIPA Method which is a valid contribution to psychology, consciousness studies, identity theory, and personality development. Actually, my paper '''AIPA Method: A Cognitive-Phenomenological Model for Identity Reconstruction and Stabilization in Pure Awareness''' is now in the peer review procedure at Journal of Consciousness Studies. ::::Here is a part from the Wikiversity AIPA Method page in creation (waiting for the end of the time limit for new users): [[User:Senad Dizdarević|Senad Dizdarević]] ([[User talk:Senad Dizdarević|discuss]] • [[Special:Contributions/Senad Dizdarević|contribs]]) 06:47, 2 April 2026 (UTC) :::::For the unknown reasons, the form didn't publish my second part of the message: :::::I believe this is a valid contribution to Wikiversity. :::::Best Regards, :::::Senad [[User:Senad Dizdarević|Senad Dizdarević]] ([[User talk:Senad Dizdarević|discuss]] • [[Special:Contributions/Senad Dizdarević|contribs]]) 06:52, 2 April 2026 (UTC) ::::::And the third try: :::::: == Introduction == ::::::The AIPA Method addresses a gap in contemporary personal development and consciousness science: most evidence‑based approaches (CBT, MBSR, MBCT, standard meditation) operate at the level of mental content—reframing thoughts, observing them, or reducing their impact—rather than at the level of identity structure. In contrast, AIPA targets the structural relationship between the self and the mind, aiming at durable identity reconstruction rooted in Pure Awareness rather than symptom management. ::::::The central research question of the primary AIPA preprint is whether a structured, sequentially staged method can produce permanent identity reconstruction rooted in Pure Awareness, and how such a method compares to established approaches in scope, mechanism, and outcome. :::::: == Theoretical foundations == ::::::The AIPA framework is grounded in the cognitive‑phenomenological tradition (e.g., McAdams, Varela, Metzinger, Erikson), contemporary consciousness science on minimal phenomenal experience, and qualitative methods advocacy in psychology. It builds directly on: ::::::* Empirical work on pure awareness and Minimal Phenomenal Experience (MPE), especially Gamma & Metzinger’s large‑scale study of content‑reduced awareness states. ::::::* Metzinger’s proposal of minimal phenomenal experience as an entry point for a minimal unifying model of consciousness. ::::::* Narrative identity and partial‑self models within personality and identity theory. ::::::Within this backdrop, AIPA proposes Pure Awareness as a distinct, operationally specified state that can become a structural ground of identity rather than a transient meditative experience. :::::: == Experiential empiricism == ::::::The empirical foundation of the AIPA Method is explicitly first‑person and experiential, combining: ::::::* A 22‑year longitudinal autoethnographic self‑study (2003–2025) documenting partial personality episodes, protocol use, and outcomes. ::::::* A 13‑year prospective verification period with zero self‑reported recurrence of targeted harmful behaviors after a dated stabilization point (1 January 2006). ::::::* A high‑ecological‑validity “stress test” during acute bereavement, used to examine whether non‑reactive awareness remains stable under maximal provocation. ::::::* Two independent practitioner cases (an Amazon‑verified report and a structured questionnaire case) providing preliminary convergent signals across cognitive, emotional, behavioral, and identity dimensions. ::::::All central constructs (Pure Awareness, partial personalities, the Switch, identity stabilization) are operationalized with explicit phenomenological and behavioral criteria intended to enable replication and future third‑person measurement. ::::::I believe this is a valid contribution to Wikiversity. ::::::Best regards, ::::::Senad [[User:Senad Dizdarević|Senad Dizdarević]] ([[User talk:Senad Dizdarević|discuss]] • [[Special:Contributions/Senad Dizdarević|contribs]]) 06:54, 2 April 2026 (UTC) == Unable to publish pages == Whenever I try to publish a page with linked sources it gets flagged and says I'm a new user attempting to publish content with outside links. Those outside links are my sources. [[User:Soboyed|Soboyed]] ([[User talk:Soboyed|discuss]] • [[Special:Contributions/Soboyed|contribs]]) 04:52, 2 April 2026 (UTC) :This restriction is automatically lifted after you have edited for a certain time (I don't recall that time off-hand, but it is not long). This is designed to stop spam. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 04:53, 2 April 2026 (UTC) == Showing error to publish a Post == My action was constructive, not destructive, please allow to publish it. [[Special:Contributions/&#126;2026-20906-18|&#126;2026-20906-18]] ([[User talk:&#126;2026-20906-18|talk]]) 08:06, 4 April 2026 (UTC) :Maybe you got caught in a filter. Consider [[Special:CreateAccount|creating an account]]. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 09:06, 4 April 2026 (UTC) :Your edits, [https://en.wikiversity.org/w/index.php?title=Special:AbuseLog&wpSearchUser=%7E2026-20906-18 these ones], seems to have tripped a filter when you tried to create a page on [[Create]] which external links. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 23:58, 4 April 2026 (UTC) :Have you read my [https://en.wikiversity.org/w/index.php?title=Wikiversity:Request_custodian_action&diff=prev&oldid=2802219 previous reply] to you @[[User:~2026-20906-18|~2026-20906-18]]? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:02, 6 April 2026 (UTC) == Abuse filters which should be deleted == Hi, there are some abuse filters which should probably be deleted. * [[Special:AbuseFilter/1]] (not needed anymore) * [[Special:AbuseFilter/2]] (no hits since 2018) * [[Special:AbuseFilter/3]] (not needed since there are global filters that disallow this specific type of spam filter 3 would have catched) * [[Special:AbuseFilter/4]] (looking at the logs, there are too many false positives) * [[Special:AbuseFilter/5]] (no hits since 2023) * Abuse filters 7, 8, 9, 10, 11, 12 (these filters are not needed anymore) * [[Special:AbuseFilter/17]] (no hits since 2022) * [[Special:AbuseFilter/19]] (no hits since 2019) * [[Special:AbuseFilter/21]] (false positives, vandal currently inactive) Thanks. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 03:51, 5 April 2026 (UTC) :Why do these need to be deleted rather than inactivated? Do inactive abuse filters cause a server strain? —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:39, 5 April 2026 (UTC) :: Deleted filters do not cause strain to the servers. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 14:28, 5 April 2026 (UTC) :These sounds like sensible suggestions but, yes, would inactivation perhaps make more sense than deletion for at least some filters? -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 09:35, 5 April 2026 (UTC) :I would keep them @[[User:Codename Noreste|Codename Noreste]]. Alternatively, I would turn off the ones that haven't caught anything for a long time, but I would leave them enabled in case they need to be turned on or improved. If someone has already written the code and we don't have hundreds of free man-hours of programmers on Wikiversity, the server load seems secondary to me, and is negligible compared to other things. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:11, 6 April 2026 (UTC) :: I know how to write abuse filter code and regex, but I would recommend disabling filters that have never caught anything in a long time ''and'' those who made lots of false positives. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:09, 6 April 2026 (UTC) :::Of course @[[User:Codename Noreste|Codename Noreste]], there are people here today who are capable of changing the code. But the question is what it will be like in a few years, the question is what will happen if those two are busy for a long time, etc. That's why I would leave it so that those who don't know much about code can be inspired by it and will need to do something with it someday - plus, more code for different types of filtering is actually great educational material on how those filters work. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 11:41, 17 April 2026 (UTC) Here's the updated list of abuse filters under review with actions I've taken (several disabled, one basic code improvement, and some actions changed) - none have been deleted so they can all be edited and reactivayed - please suggest any further changes: * [[Special:AbuseFilter/1]] (not needed anymore) - One time account spam bot - 4 hits over 10 years ago - Disabled in 2024 - May be useful in future * [[Special:AbuseFilter/2]] (no hits since 2018) - Userspace spamming - 778 hits; none since 2018 likely due to global filters - Now disabled * [[Special:AbuseFilter/3]] (not needed since there are global filters that disallow this specific type of spam filter 3 would have catched) - Specific user page spam - 1,101 hits most recent 7 March 2026 - Still active - Kept enabled * [[Special:AbuseFilter/4]] (looking at the logs, there are too many false positives) - Questionable Language (profanity) - 6,055 hits including very recently - However it was logging hits without taking any actions - Edited to reduce likelihood of false positives by only applying filter to users with low (< 20) edit count and applied weak actions to tag and warn but not prevent publishing the content * [[Special:AbuseFilter/5]] (no hits since 2023) - Blocked Solicitation Links - 95 hits; none since 2023 - blocks specific historical spam sites - Non-active - Now disabled * Abuse filters 7, 8, 9, 10, 11, 12 (these filters are not needed anymore) - Not reviewed - They are currently disabled * [[Special:AbuseFilter/17]] (no hits since 2022) - Fundamental Physics Edits - 347 hits; none since 2022 - Non-active and very specific for a historical issue - Now disabled * [[Special:AbuseFilter/19]] (no hits since 2019) - Page Creation - 20 hits; none since 2019 - Retained for historical reference and possible future updates - Now disabled * [[Special:AbuseFilter/21]] (false positives, vandal currently inactive) - Globally Banned Editor (renamed to Low-edit Spam Monitor) - 2,829 hits including very recent - Only applies to users with less than 5 edits and takes no actions / monitoring only - Reviewing the details of the hits I don't see many false positives and have strengthened its actions to add a tag and warning -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:57, 9 April 2026 (UTC) == Block request == Please block ~2026-20985-80/~2026-21079-90/~2026-21223-88. Reason: Vandalism. [[User:Àncilu|Àncilu]] ([[User talk:Àncilu|discuss]] • [[Special:Contributions/Àncilu|contribs]]) 23:24, 5 April 2026 (UTC) :All edits should be deleted and the first is blocked by Atcovi. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 00:33, 6 April 2026 (UTC) == Antispam - Filter 12 == {{ping|Codename Noreste}} Thanks for contacting me with a suggested [[Special:AbuseFilter|abuse filter]] for the coupon spam we've been getting. A very much appreciated time saver. Per your suggestion, abuse filter 12 has been reactivated with your updated regex. It should tag and prevent page creation actions for coupon promo etc. Sincerely, James -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 10:33, 15 April 2026 (UTC) : @[[User:Jtneill|Jtneill]] Please do the following to filter 12: :# Enable the block action (temporary for unregistered users, indefinite for registered users) :# Disable both the disallow and tag actions, since that filter will be set to block : Thank you. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 13:42, 5 May 2026 (UTC) ::Excellent. Done: ::* Block duration for non-registered users = 1 week ::* Block duration for registered users = indefinite ::* Disallow and tagging disabled ::Thankyou. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 10:24, 9 May 2026 (UTC) == Urgent! error message This action has been automatically identified as potentially harmful, and therefore disallowed == While submitting the post this error was coming "This action has been automatically identified as potentially harmful, and therefore disallowed. If you believe your action was constructive, please inform an administrator of what you were trying to do. A brief description of the abuse rule which your action matched is: New User Created Page with External Link" How to resolve it? Here is the content: {{note|marketing material removed}} [[User:EasyshikshaMarketing|EasyshikshaMarketing]] ([[User talk:EasyshikshaMarketing|discuss]] • [[Special:Contributions/EasyshikshaMarketing|contribs]]) 05:14, 17 April 2026 (UTC) : That's because Wikiversity doesn't accept advertising. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 05:20, 17 April 2026 (UTC) ::So this is fine ::'''Online Internship and Digital Learning for Students''' ::Online learning has become an important part of modern education. With the help of the internet and digital tools, students can now study, practice, and gain experience without being physically present in a classroom. One key part of this system is the online internship, which helps students learn real-world skills along with their studies. ::An online internship allows students to work on tasks and projects through digital platforms. This makes learning more practical and useful, especially for those who want to understand how real work environments function. ::'''Background''' ::The concept of online learning developed from distance education, where students learned from remote locations. Over time, with the growth of digital technology, learning has become more interactive and flexible. ::The introduction of the online internship has added another important layer to digital education. It combines theoretical learning with practical experience, helping students prepare for future careers. ::During global events such as the COVID-19 pandemic, online education and online internship programs became essential. They helped students continue learning and gaining experience despite restrictions on physical movement. ::'''Importance of Online Internship''' ::An online internship plays an important role in student development. It helps bridge the gap between academic knowledge and practical skills. ::'''Some key points include:''' ::Students understand how real work is done ::They develop basic professional skills ::It supports career readiness ::It allows learning without location limits ::By participating in an online internship, students can improve their confidence and gain early exposure to different fields. ::'''Features of Online Internship-Based Learning''' ::Modern education platforms often include online internship opportunities as part of their learning system. These usually offer: ::Flexible schedules for students ::Access to learning from home ::Beginner-friendly tasks and projects ::A combination of theory and practice ::Such features make online internship programs suitable for a wide range of learners, including beginners. ::'''Learning Tasks''' ::Explore how online internship programs support student learning in digital environments. ::Identify how students can gain practical experience through an online internship ::Analyze the role of flexible learning in improving student engagement ::Understand how online education and internships work together ::'''References''' ::[https://ijpsl.in/wp-content/uploads/2020/09/E-Learning_Hanaaya-Navaneeth.pdf The Past, Present and Future of E-Learning: Hanaaya Varyani and Navaneeth M S] ::[https://ieeexplore.ieee.org/document/9788102 Current Trends and Future Perspectives of e-Learning in India] ::'''See Also''' ::[https://easyshiksha.com/online_courses/internship Online Internship] ::[https://easyshiksha.com/online_courses/ Online Courses] ::[https://easyshiksha.com/online_courses/kids-learning Kids Learning] ::[https://easyshiksha.com/career_helper/ Career Guidance] [[User:EasyshikshaMarketing|EasyshikshaMarketing]] ([[User talk:EasyshikshaMarketing|discuss]] • [[Special:Contributions/EasyshikshaMarketing|contribs]]) 05:27, 17 April 2026 (UTC) :::@[[User:EasyshikshaMarketing|EasyshikshaMarketing]] Wikiversity is a resource for education, or a space for education. However, your intention to link to another website is obvious, and such content does not belong here, as it contradicts the purpose of Wikiversity. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 11:38, 17 April 2026 (UTC) ==AI-generated images== Seeking your advice. Myself and students use some AI-generated images in the [[Motivation and emotion]] project. [[User:Dronebogus]] has been removing some of these images from Wikiversity pages and nominating them for deletion at Commons. As a result, some have been deleted and some have been kept. Dronebogus has made some useful edits and image suggestions for [[Motivation and emotion]] which I've appreciated and incorporated. However, there are other edits to remove an AI image by Dronebogus that I've reverted where I think the image is more educational than no image or an alternative image suggested by Dronebogus. There are a couple of pages where Dronebogus has reverted my reversion, so we are at risk of edit warring. We have briefly discussed and warned each other on our user talk pages, but it seems to come down to a difference in perception about the educational usefulness of the AI images. So, I'm asking here for others to please review the recent edit histories for these pages: * [[Motivation and emotion/Lectures/Brain and physiological needs]] * [[Motivation and emotion/Book/2025/Stockholm syndrome emotion]] and let us know what you think about the AI image suitability vs. using no image or alternative images suggested by Dronebogus. Sincerely, James -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:45, 26 April 2026 (UTC) :I think Jtneill needs to try harder to find non-AI alternatives both on Commons and the web. I’m not reiterating the well known problems with generative AI— you can read about those on Wikipedia and the broader Internet. Needless to say it’s kind of just inherently toxic. If you use it, it should be the last resort of last resorts. Just my stance, which I consider perfectly valid and reasonable. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 04:10, 27 April 2026 (UTC) == Request for Page Creation Approval - Research on Dr. Ian Stevenson == '''Hello Administrators,''' I am a digital archivist and researcher attempting to create an educational page regarding Dr. Ian Stevenson's scientific work, specifically titled "Where Reincarnation and Biology Intersect" and its official Vietnamese edition. My edits are purely constructive, aimed at integrating local research context with global structured data (including '''Wikidata Q139548587'''). The automated filter is currently preventing publication due to "new user external link limits." I have ensured that all citations, references, and images (sourced from Wikimedia Commons) are strictly academic and non-promotional. Could you please review my contributions and consider whitelisting my account or approving this specific page creation? Thank you for your assistance in preserving this research '''legacy''' within the open knowledge ecosystem. '''Best regards,''' [[User:Ian Stevenson777|Ian Stevenson777]] ([[User talk:Ian Stevenson777|discuss]] • [[Special:Contributions/Ian Stevenson777|contribs]]) 07:26, 28 April 2026 (UTC) : The page title should probably be in English, as this is English Wikiversity. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 14:19, 28 April 2026 (UTC) :@[[User:Ian Stevenson777|Ian Stevenson777]] I looked at the text and it seems to me that it is informative, but the user cannot learn anything from it because they would have to buy the book. In this regard, it seems to me that it is enough [[:w:Where Reincarnation and Biology Intersect|to have that article on Wikipedia]] and there is no need to duplicate it here on Wikiversity. If you are particularly interested with its Vietnamese translation, I would focus on Vietnamese Wikipedia, but it seems to me, that the article there [https://vi.wikipedia.org/w/index.php?title=N%C6%A1i_lu%C3%A2n_h%E1%BB%93i_v%C3%A0_sinh_h%E1%BB%8Dc_giao_nhau&oldid=75029095 have some problems at the moment]. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 12:51, 5 May 2026 (UTC) :This request is AI-generated and was made by a lock evading LTA. Please ignore it. [[User:NguoiDungKhongDinhDanh|NguoiDungKhongDinhDanh]] ([[User talk:NguoiDungKhongDinhDanh|discuss]] • [[Special:Contributions/NguoiDungKhongDinhDanh|contribs]]) 23:58, 13 May 2026 (UTC) == Call for custodians and bureaucrats == <div class="cd-moveMark">''Moved to [[Wikiversity:Notices for custodians#Call for custodians and bureaucrats]]. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 18:46, 12 May 2026 (UTC)''</div> == Interface administrator flag request == @[[User:Jtneill|Jtneill]], @[[User:Mu301|Mu301]] I am requesting IA flag to delete [[MediaWiki:Gadget-WikiSign.js]]. It was requested for deletion by [[User:Sophivorus|a trusted user]] and I think it can be deleted. Once I delete it, I will notify you and you can remove the flag. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:24, 10 May 2026 (UTC) :Can you enable two-factor authentication? :Policy: [[Wikiversity:Interface administrators]] -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 10:27, 10 May 2026 (UTC) ::Done @[[User:Jtneill|Jtneill]]. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:20, 11 May 2026 (UTC) :::You are an interface administrator for 24 hours. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 09:57, 11 May 2026 (UTC) ::::Thx @[[User:Jtneill|Jtneill]], I have just deleted the page. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 09:59, 11 May 2026 (UTC) == Block request == Hi. Please block @[[User:Skibidi Titan 5.0|Skibidi Titan 5.0]]. Reason : Vandalism. Thanks, [[User:Locpac|Locpac]] ([[User talk:Locpac|discuss]] • [[Special:Contributions/Locpac|contribs]]) 15:10, 10 May 2026 (UTC) :Thankyou. Done. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 15:25, 10 May 2026 (UTC) == ~2026-28496-53 == Please block [[Special:Contribs/~2026-28496-53]], vandalism. [[User:Tenshi Hinanawi|Tenshi Hinanawi]] ([[User talk:Tenshi Hinanawi|トーク]] • [[Special:Contributions/Tenshi Hinanawi|投稿記録]]) 17:02, 11 May 2026 (UTC) :{{done}} Thank you! —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 17:14, 11 May 2026 (UTC) == ~2026-28792-52 == Please block [[Special:Contribs/~2026-28792-52]], vandalism. Appears to be same user as above. [[User:Tenshi Hinanawi|Tenshi Hinanawi]] ([[User talk:Tenshi Hinanawi|トーク]] • [[Special:Contributions/Tenshi Hinanawi|投稿記録]]) 17:38, 12 May 2026 (UTC) : Blocked locally by Barras. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 17:46, 12 May 2026 (UTC) == New User Exceeded New Page Limit == Hello, my action of creating a Portal for Banjo learning (Portal:Banjo) was denied as I am a user whose had my account for a while but not written before today. My actions are constructive, but if you would rather me wait and let the system work as it intends to then that's okay too. --[[User:Kirby - Electrotechnics|Kirby - Electrotechnics]] ([[User talk:Kirby - Electrotechnics|discuss]] • [[Special:Contributions/Kirby - Electrotechnics|contribs]]) 01:13, 17 May 2026 (UTC) :That is standard and the rate limit will fall off as you stay and edit; it has some pretty easy barriers to cross. I can create a blank [[Portal:Banjo]] if you want, but to be clear, portal pages are usually much broader topics that can help orient you to specific pages. Do you think there will be that many pages about the topic of banjo playing? If not, then you can just continue editing [[Banjo]]. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 03:36, 17 May 2026 (UTC) == Disable [[Special:AbuseFilter/3]] == Please disable [[Special:AbuseFilter/3]] because it is redundant with [[m:Special:AbuseFilter/104|a global filter]] that disallows the same type of spam that filter 3 would have caught. Thank you. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 01:06, 19 May 2026 (UTC) : [[Special:AbuseFilter/history/3/diff/prev/454|Done]] -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 02:15, 19 May 2026 (UTC) 9lz918vwwr9yf35omfn3o5iih2sm1gj 2812908 2812898 2026-06-05T18:13:39Z Codename Noreste 2969951 /* One man's look at concept */ archive from [[Wikiversity:Request custodian action]] ([[mw:c:Special:MyLanguage/User:JWBTH/CD|CD]]) 2812908 wikitext text/x-wiki {{Archive|Wikiversity:Request custodian action}} == Sidewide count.js == i would like something like: [[Template:User contrib count/count.js]]. i created [[Template:User contrib count]] and a user/common.js. {{User contrib count}}.<br><br> so a "count.js" would complete it. See [[User:Harold Foppele/common.js]]. If an Administrator could help please. Cheers [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 19:22, 18 January 2026 (UTC) == need to add my profile == im trying to add new profile content [[User:PAGURUMURTHY|PAGURUMURTHY]] ([[User talk:PAGURUMURTHY|discuss]] • [[Special:Contributions/PAGURUMURTHY|contribs]]) 18:03, 4 February 2026 (UTC) :You can edit it now. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 18:05, 4 February 2026 (UTC) ::where can create a new one [[User:PAGURUMURTHY|PAGURUMURTHY]] ([[User talk:PAGURUMURTHY|discuss]] • [[Special:Contributions/PAGURUMURTHY|contribs]]) 18:51, 4 February 2026 (UTC) :::i have created and its in sandbox. i would like to know when it will be approved [[User:PAGURUMURTHY|PAGURUMURTHY]] ([[User talk:PAGURUMURTHY|discuss]] • [[Special:Contributions/PAGURUMURTHY|contribs]]) 19:38, 4 February 2026 (UTC) ::::Please don’t create [[wv:spam|spam]] pages as it will be deleted. Please also read [[WV:Scope]] [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 04:01, 5 February 2026 (UTC) == Im trying to add new profile while add content its shows not alowed == This action has been automatically identified as potentially harmful, and therefore disallowed. If you believe your action was constructive, please [[Wikiversity:Request custodian action|inform an administrator]] of what you were trying to do. A brief description of the abuse rule which your action matched is: New User Exceeded New Page Limit This action has been automatically identified as potentially harmful, and therefore disallowed. If you believe your action was constructive, please [[Wikiversity:Request custodian action|inform an administrator]] of what you were trying to do. A brief description of the abuse rule which your action matched is: New User Created Page with External Link [[User:PAGURUMURTHY|PAGURUMURTHY]] ([[User talk:PAGURUMURTHY|discuss]] • [[Special:Contributions/PAGURUMURTHY|contribs]]) 18:51, 4 February 2026 (UTC) == New User: cannot create talk page == Hi, I am a new user of Wikiversity and I wanted to create a talk page for the article [[ChatGPT's Essay on Kohlberg's Theory: AI's Use in Academic Writing]]. As a new user, I was barred from performing this action. The text that I wanted to add to the talk page is: <blockquote> I have doubts as to to the reliability of this essay. Take for rexample the sentence: <blockquote> Due to its efficiency, AI has saved 380,000-403,000 lives per year in European healthcare as reported in a recent Deloitte and MedTech Europe report<ref>Dantas, C., Mackiewicz, K., Tageo, V., Jacucci, G., Guardado, D., Ortet, S., Varlamis, I., Maniadakis, M., De Lera, E., Quintas, J., Kocsis, O., & Vassiliou, C. (2021). Benefits and hurdles of AI in the workplace – what comes next? ''International Journal of Artificial Intelligence and Expert Systems, 10'', 9-17. https://www.researchgate.net/publication/351993615_Benefits_and_Hurdles_of_AI_In_The_Workplace_-What_Comes_Next</ref>. </blockquote> Reading the reference (freely available on ResearchGate), one notes: # that the reference is from 2021 (predating the widespread use of LLMs such as ChatGPT and the associated 'AI' boom), and # that the reference factually contradicts this essay. Quoting from the reference: <blockquote> There are enormous benefits of applying AI-based solutions to monitor workers’ health and prevent accidents or, currently, COVID-19 infections, and those benefits are reported with enormous potential. According to the recent Deloitte and MedTech Europe report [11], implementing AI in European healthcare systems could save up 380,000 to 403,000 lives annually or €170.9 to 212.4 billion per year. </blockquote> Not that the reference says ''could save'', not ''saves'' as in the essay. This calls into question the reliability of the essay. </blockquote> Could an administrator make this addition for me? Thank you! {{reflist}} [[User:Æolus|Æolus]] ([[User talk:Æolus|discuss]] • [[Special:Contributions/Æolus|contribs]]) 06:53, 5 February 2026 (UTC) :@[[User:Æolus|Æolus]] I have added it for you, you can change the header and sign it now. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 08:05, 5 February 2026 (UTC) ::Thank you! [[User:Æolus|Æolus]] ([[User talk:Æolus|discuss]] • [[Special:Contributions/Æolus|contribs]]) 12:43, 5 February 2026 (UTC) == Disallowed to add a page on a course == I'm trying to populate a newly created course on Wikiversity, but it blocks me from creating more pages with "New User Exceeded New Page Limit". Could this be lifted please? [[User:Berkeleywho|Berkeleywho]] ([[User talk:Berkeleywho|discuss]] • [[Special:Contributions/Berkeleywho|contribs]]) 13:21, 15 February 2026 (UTC) :Sorry! Never mind. I was trying to create a new article instead of a new page. All good now. [[User:Berkeleywho|Berkeleywho]] ([[User talk:Berkeleywho|discuss]] • [[Special:Contributions/Berkeleywho|contribs]]) 14:03, 15 February 2026 (UTC) == Harold Foppele adding LLM-generated nonsense and personal fiction == I became aware of [[User:Harold Foppele]]'s editing after I deleted some of his uploads on Commons. He appears to be adding a large amount of text and images that are some combination of personal fiction and LLM-generated nonsense. This includes: *[[Quantum Ultra fast lasers#Future thought experiment|Personal speculative fiction]] in an otherwise "nonfiction" article *Uploading nonsense LLM-created [[:File:Rontosecond pulse laser (Schematic).jpg|diagrams]] and [[:File:Rontosecond pulse laser (Futuristic).jpg|renders]] for nonexistent lab equipment, with fake source (on Commons, he indicated these files as having been created by him using an LLM) *Uploading nonsense LLM-created images of equations with obvious artifacts. These images, such as [[:File:Redfield equation (non-Markovian).png]] and [[:File:Lindblad equation (Markovian).png]], don't even match the text he puts them with. Much of his writing is also of extremely poor quality, to the point where it's not clear whether it's written by him or an LLM. I'm not an active editor on this project, so I'm not as familiar with the standards here, but I believe this is worth custodian attention. [[User:Pi.1415926535|Pi.1415926535]] ([[User talk:Pi.1415926535|discuss]] • [[Special:Contributions/Pi.1415926535|contribs]]) 03:06, 23 February 2026 (UTC) :Fake source ''and'' contradictory copyright info, claiming both public domain and CC license. Moreover, if they are indeed nearly-direct LLM output, depending on jurisdiction they may not even be eligible for copyright. :I've put speedy deletion marks for the equations, because they're obviously not coherent mathematical equations (the parentheses don't match, the symbols merge into each other the way text in image models often do, etc) [[User:Sesquilinear|Sesquilinear]] ([[User talk:Sesquilinear|discuss]] • [[Special:Contributions/Sesquilinear|contribs]]) 21:50, 7 March 2026 (UTC) == Repeated removal of RFD notices by Harold Foppele == {{User|Harold Foppele }} This editor is appearing in multiple noticeboards for behaviour which is contentious. Ther latest adventure is the repeated removal of tye RFD notice at [[Quantum/Henry C. Kapteyn]]. You will see from their contributions record the number of times. I have warned Tham on their user tag page that this is tantaomunt to volunteering to be blocked here. They have a track record of achieving blocks on enWiki and Commons already. They have all the appearance of shooting not to understand when given direct information about their behaviour, whichever project they are editing, and are fast becoming a time sink. Their behaviour across multiple WMF sites may well lead to a Global Lock, but I do not believe they have quite reached the threshold for that. I believe that what is required is a preventative block to seek to ensure thatchy understand the seriousness of their behaviour, and the need to be collegial. 🇵🇸&zwj;🇺🇦&nbsp;[[User:Timtrent|Timtrent]]&nbsp;🇺🇦&nbsp;[[User talk:Timtrent|talk to me]]&nbsp;🇺🇦&zwj;🇵🇸 23:03, 4 March 2026 (UTC) : {{Done}} [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 11:45, 8 March 2026 (UTC) == Blocks for sockpuppet == Please block [[User:Harold Foppele]] and [[User:Johnwilliamsiii]] for sockpuppetry based on [https://en.wikipedia.org/wiki/Wikipedia:Sockpuppet_investigations/Harold_Foppele en wiki] CU and [https://commons.wikimedia.org/w/index.php?diff=1177465640 commons] CU investigations. The user has also violated copyright, see the above discussion. A block is necessary to prevent further abuse. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 11:30, 8 March 2026 (UTC) :<small>@[[User:MathXplore|MathXplore]]</small> [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 11:31, 8 March 2026 (UTC) :: {{Done}} [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 11:44, 8 March 2026 (UTC) :CC. @[[User:Timtrent|Timtrent]], @[[User:Sesquilinear|Sesquilinear]], @[[User:Pi.1415926535|Pi.1415926535]] [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 11:33, 8 March 2026 (UTC) ::Thank you for the ping. I concur based on [[w:en:WP:DUCK|behaviour]]. CUs appear divided. 🇵🇸&zwj;🇺🇦&nbsp;[[User:Timtrent|Timtrent]]&nbsp;🇺🇦&nbsp;[[User talk:Timtrent|talk to me]]&nbsp;🇺🇦&zwj;🇵🇸 11:41, 8 March 2026 (UTC) == Problem when trying to start a discussion with authors of the Plurilingual education portal == The authors I wanted to discuss with are called "Project PEP" and my name is Franch Chandler. How can I be allowed to do so ? [[User:French Chandler|French Chandler]] ([[User talk:French Chandler|discuss]] • [[Special:Contributions/French Chandler|contribs]]) 18:25, 16 March 2026 (UTC) :@[[User:French Chandler|French Chandler]] place your qestion [https://en.wikiversity.org/w/index.php?title=User_talk:Projet_PEP&action=edit into the dialog box] on this link and hit Publish page. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 20:22, 16 March 2026 (UTC) == Please publish my post == My post is about "Every child grows and develops at their own pace, but some may experience challenges that affect their ability to perform everyday tasks. These challenges can include difficulties with fine motor skills, sensory processing, handwriting, feeding, and self-regulation. When these issues are not addressed early, they can impact a child’s confidence, academic performance, and independence. With the rise of digital healthcare services, '''online physical therapy''' has emerged as a powerful and accessible solution for parents seeking support for their children. This modern approach provides structured, personalized therapy programs that can be accessed from the comfort of home, making it easier for families to ensure consistent care." [[User:Skyabovetherapy|Skyabovetherapy]] ([[User talk:Skyabovetherapy|discuss]] • [[Special:Contributions/Skyabovetherapy|contribs]]) 12:28, 28 March 2026 (UTC) :@[[User:Skyabovetherapy|Skyabovetherapy]] Well, you can publish it yourself, Wikiversity is a free environement, where everybody can create educational resources. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 14:11, 29 March 2026 (UTC) ::They actually triggered some abuse filters. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:24, 29 March 2026 (UTC) :I looked at your attempts to add this text and I see a link to one website repeated many times, which reminds me of the misuse of Wikiversity for self-promotion or to increase the importance of the website. It is necessary to remind you here that Wikiversity is not a place for promotion, but a place for education. So if you want to educate, it will not be a problem to create the page without external links with a clearly defined procedure for how people should use it and what to expect from it. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:07, 1 April 2026 (UTC) == New user limit == Hi, I am creating an AIPA Method learning resource page. I am the author of the linked research, and I hit the “new user limit” and “new page with external link” filters while publishing. Here is the link to the page in creation: [https://en.wikiversity.org/w/index.php?title=AIPA_Method&veaction=edit] Thank you for your help. Best regards, Senad Dizdarević [[User:Senad Dizdarević|Senad Dizdarević]] ([[User talk:Senad Dizdarević|discuss]] • [[Special:Contributions/Senad Dizdarević|contribs]]) 07:19, 30 March 2026 (UTC) :@[[User:Senad Dizdarević|Senad Dizdarević]] I should admit I dont know, what is "new user limit", but if filter blocks your page because of certain external link, you may force to save anyway and sometimes it works. It should not work, when the website is blacklisted. As of now, I am not seeing you to save page in main namespace, so try to save it without external links first. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 07:30, 30 March 2026 (UTC) ::Thank you, you are very kind. ::I will wait a day, and try again (without links, too). ::Today, I already created About Me info page, and maybe that is enough for the filters for one day. [[User:Senad Dizdarević|Senad Dizdarević]] ([[User talk:Senad Dizdarević|discuss]] • [[Special:Contributions/Senad Dizdarević|contribs]]) 07:53, 30 March 2026 (UTC) :::Well, I have analyzed your contribution to Wikiversity and I should point out here, that this project is not a place for advertising, so there is no way of promoting your books and authority this way. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:56, 1 April 2026 (UTC) ::::Hi, my About Me page is just an info page with the neutral as possible presentation of my work. ::::There is a big difference between informing and advertising. Informing is neutrally stating that something exists and requiring no action, while advertising is a special communication form with intent to cause certain action from readers. For example, click here, click there, order this, buy that. ::::There is no such intention, form, or terms on my info page. Just neutral information. I don't hide and I am not ashamed that I am write and author, and that is a part of the usual bio, including works. I checked your user page: "I graduated from the Czech University of Life Sciences in Prague and studied information science at the Faculty of Arts of Charles University." I think that if you had written a book on Life Science, you would have mentioned that as well. ::::Most of the Info page is about my research and AIPA Method which is a valid contribution to psychology, consciousness studies, identity theory, and personality development. Actually, my paper '''AIPA Method: A Cognitive-Phenomenological Model for Identity Reconstruction and Stabilization in Pure Awareness''' is now in the peer review procedure at Journal of Consciousness Studies. ::::Here is a part from the Wikiversity AIPA Method page in creation (waiting for the end of the time limit for new users): [[User:Senad Dizdarević|Senad Dizdarević]] ([[User talk:Senad Dizdarević|discuss]] • [[Special:Contributions/Senad Dizdarević|contribs]]) 06:47, 2 April 2026 (UTC) :::::For the unknown reasons, the form didn't publish my second part of the message: :::::I believe this is a valid contribution to Wikiversity. :::::Best Regards, :::::Senad [[User:Senad Dizdarević|Senad Dizdarević]] ([[User talk:Senad Dizdarević|discuss]] • [[Special:Contributions/Senad Dizdarević|contribs]]) 06:52, 2 April 2026 (UTC) ::::::And the third try: :::::: == Introduction == ::::::The AIPA Method addresses a gap in contemporary personal development and consciousness science: most evidence‑based approaches (CBT, MBSR, MBCT, standard meditation) operate at the level of mental content—reframing thoughts, observing them, or reducing their impact—rather than at the level of identity structure. In contrast, AIPA targets the structural relationship between the self and the mind, aiming at durable identity reconstruction rooted in Pure Awareness rather than symptom management. ::::::The central research question of the primary AIPA preprint is whether a structured, sequentially staged method can produce permanent identity reconstruction rooted in Pure Awareness, and how such a method compares to established approaches in scope, mechanism, and outcome. :::::: == Theoretical foundations == ::::::The AIPA framework is grounded in the cognitive‑phenomenological tradition (e.g., McAdams, Varela, Metzinger, Erikson), contemporary consciousness science on minimal phenomenal experience, and qualitative methods advocacy in psychology. It builds directly on: ::::::* Empirical work on pure awareness and Minimal Phenomenal Experience (MPE), especially Gamma & Metzinger’s large‑scale study of content‑reduced awareness states. ::::::* Metzinger’s proposal of minimal phenomenal experience as an entry point for a minimal unifying model of consciousness. ::::::* Narrative identity and partial‑self models within personality and identity theory. ::::::Within this backdrop, AIPA proposes Pure Awareness as a distinct, operationally specified state that can become a structural ground of identity rather than a transient meditative experience. :::::: == Experiential empiricism == ::::::The empirical foundation of the AIPA Method is explicitly first‑person and experiential, combining: ::::::* A 22‑year longitudinal autoethnographic self‑study (2003–2025) documenting partial personality episodes, protocol use, and outcomes. ::::::* A 13‑year prospective verification period with zero self‑reported recurrence of targeted harmful behaviors after a dated stabilization point (1 January 2006). ::::::* A high‑ecological‑validity “stress test” during acute bereavement, used to examine whether non‑reactive awareness remains stable under maximal provocation. ::::::* Two independent practitioner cases (an Amazon‑verified report and a structured questionnaire case) providing preliminary convergent signals across cognitive, emotional, behavioral, and identity dimensions. ::::::All central constructs (Pure Awareness, partial personalities, the Switch, identity stabilization) are operationalized with explicit phenomenological and behavioral criteria intended to enable replication and future third‑person measurement. ::::::I believe this is a valid contribution to Wikiversity. ::::::Best regards, ::::::Senad [[User:Senad Dizdarević|Senad Dizdarević]] ([[User talk:Senad Dizdarević|discuss]] • [[Special:Contributions/Senad Dizdarević|contribs]]) 06:54, 2 April 2026 (UTC) == Unable to publish pages == Whenever I try to publish a page with linked sources it gets flagged and says I'm a new user attempting to publish content with outside links. Those outside links are my sources. [[User:Soboyed|Soboyed]] ([[User talk:Soboyed|discuss]] • [[Special:Contributions/Soboyed|contribs]]) 04:52, 2 April 2026 (UTC) :This restriction is automatically lifted after you have edited for a certain time (I don't recall that time off-hand, but it is not long). This is designed to stop spam. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 04:53, 2 April 2026 (UTC) == Showing error to publish a Post == My action was constructive, not destructive, please allow to publish it. [[Special:Contributions/&#126;2026-20906-18|&#126;2026-20906-18]] ([[User talk:&#126;2026-20906-18|talk]]) 08:06, 4 April 2026 (UTC) :Maybe you got caught in a filter. Consider [[Special:CreateAccount|creating an account]]. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 09:06, 4 April 2026 (UTC) :Your edits, [https://en.wikiversity.org/w/index.php?title=Special:AbuseLog&wpSearchUser=%7E2026-20906-18 these ones], seems to have tripped a filter when you tried to create a page on [[Create]] which external links. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 23:58, 4 April 2026 (UTC) :Have you read my [https://en.wikiversity.org/w/index.php?title=Wikiversity:Request_custodian_action&diff=prev&oldid=2802219 previous reply] to you @[[User:~2026-20906-18|~2026-20906-18]]? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:02, 6 April 2026 (UTC) == Abuse filters which should be deleted == Hi, there are some abuse filters which should probably be deleted. * [[Special:AbuseFilter/1]] (not needed anymore) * [[Special:AbuseFilter/2]] (no hits since 2018) * [[Special:AbuseFilter/3]] (not needed since there are global filters that disallow this specific type of spam filter 3 would have catched) * [[Special:AbuseFilter/4]] (looking at the logs, there are too many false positives) * [[Special:AbuseFilter/5]] (no hits since 2023) * Abuse filters 7, 8, 9, 10, 11, 12 (these filters are not needed anymore) * [[Special:AbuseFilter/17]] (no hits since 2022) * [[Special:AbuseFilter/19]] (no hits since 2019) * [[Special:AbuseFilter/21]] (false positives, vandal currently inactive) Thanks. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 03:51, 5 April 2026 (UTC) :Why do these need to be deleted rather than inactivated? Do inactive abuse filters cause a server strain? —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:39, 5 April 2026 (UTC) :: Deleted filters do not cause strain to the servers. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 14:28, 5 April 2026 (UTC) :These sounds like sensible suggestions but, yes, would inactivation perhaps make more sense than deletion for at least some filters? -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 09:35, 5 April 2026 (UTC) :I would keep them @[[User:Codename Noreste|Codename Noreste]]. Alternatively, I would turn off the ones that haven't caught anything for a long time, but I would leave them enabled in case they need to be turned on or improved. If someone has already written the code and we don't have hundreds of free man-hours of programmers on Wikiversity, the server load seems secondary to me, and is negligible compared to other things. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:11, 6 April 2026 (UTC) :: I know how to write abuse filter code and regex, but I would recommend disabling filters that have never caught anything in a long time ''and'' those who made lots of false positives. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:09, 6 April 2026 (UTC) :::Of course @[[User:Codename Noreste|Codename Noreste]], there are people here today who are capable of changing the code. But the question is what it will be like in a few years, the question is what will happen if those two are busy for a long time, etc. That's why I would leave it so that those who don't know much about code can be inspired by it and will need to do something with it someday - plus, more code for different types of filtering is actually great educational material on how those filters work. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 11:41, 17 April 2026 (UTC) Here's the updated list of abuse filters under review with actions I've taken (several disabled, one basic code improvement, and some actions changed) - none have been deleted so they can all be edited and reactivayed - please suggest any further changes: * [[Special:AbuseFilter/1]] (not needed anymore) - One time account spam bot - 4 hits over 10 years ago - Disabled in 2024 - May be useful in future * [[Special:AbuseFilter/2]] (no hits since 2018) - Userspace spamming - 778 hits; none since 2018 likely due to global filters - Now disabled * [[Special:AbuseFilter/3]] (not needed since there are global filters that disallow this specific type of spam filter 3 would have catched) - Specific user page spam - 1,101 hits most recent 7 March 2026 - Still active - Kept enabled * [[Special:AbuseFilter/4]] (looking at the logs, there are too many false positives) - Questionable Language (profanity) - 6,055 hits including very recently - However it was logging hits without taking any actions - Edited to reduce likelihood of false positives by only applying filter to users with low (< 20) edit count and applied weak actions to tag and warn but not prevent publishing the content * [[Special:AbuseFilter/5]] (no hits since 2023) - Blocked Solicitation Links - 95 hits; none since 2023 - blocks specific historical spam sites - Non-active - Now disabled * Abuse filters 7, 8, 9, 10, 11, 12 (these filters are not needed anymore) - Not reviewed - They are currently disabled * [[Special:AbuseFilter/17]] (no hits since 2022) - Fundamental Physics Edits - 347 hits; none since 2022 - Non-active and very specific for a historical issue - Now disabled * [[Special:AbuseFilter/19]] (no hits since 2019) - Page Creation - 20 hits; none since 2019 - Retained for historical reference and possible future updates - Now disabled * [[Special:AbuseFilter/21]] (false positives, vandal currently inactive) - Globally Banned Editor (renamed to Low-edit Spam Monitor) - 2,829 hits including very recent - Only applies to users with less than 5 edits and takes no actions / monitoring only - Reviewing the details of the hits I don't see many false positives and have strengthened its actions to add a tag and warning -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:57, 9 April 2026 (UTC) == Block request == Please block ~2026-20985-80/~2026-21079-90/~2026-21223-88. Reason: Vandalism. [[User:Àncilu|Àncilu]] ([[User talk:Àncilu|discuss]] • [[Special:Contributions/Àncilu|contribs]]) 23:24, 5 April 2026 (UTC) :All edits should be deleted and the first is blocked by Atcovi. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 00:33, 6 April 2026 (UTC) == Antispam - Filter 12 == {{ping|Codename Noreste}} Thanks for contacting me with a suggested [[Special:AbuseFilter|abuse filter]] for the coupon spam we've been getting. A very much appreciated time saver. Per your suggestion, abuse filter 12 has been reactivated with your updated regex. It should tag and prevent page creation actions for coupon promo etc. Sincerely, James -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 10:33, 15 April 2026 (UTC) : @[[User:Jtneill|Jtneill]] Please do the following to filter 12: :# Enable the block action (temporary for unregistered users, indefinite for registered users) :# Disable both the disallow and tag actions, since that filter will be set to block : Thank you. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 13:42, 5 May 2026 (UTC) ::Excellent. Done: ::* Block duration for non-registered users = 1 week ::* Block duration for registered users = indefinite ::* Disallow and tagging disabled ::Thankyou. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 10:24, 9 May 2026 (UTC) == Urgent! error message This action has been automatically identified as potentially harmful, and therefore disallowed == While submitting the post this error was coming "This action has been automatically identified as potentially harmful, and therefore disallowed. If you believe your action was constructive, please inform an administrator of what you were trying to do. A brief description of the abuse rule which your action matched is: New User Created Page with External Link" How to resolve it? Here is the content: {{note|marketing material removed}} [[User:EasyshikshaMarketing|EasyshikshaMarketing]] ([[User talk:EasyshikshaMarketing|discuss]] • [[Special:Contributions/EasyshikshaMarketing|contribs]]) 05:14, 17 April 2026 (UTC) : That's because Wikiversity doesn't accept advertising. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 05:20, 17 April 2026 (UTC) ::So this is fine ::'''Online Internship and Digital Learning for Students''' ::Online learning has become an important part of modern education. With the help of the internet and digital tools, students can now study, practice, and gain experience without being physically present in a classroom. One key part of this system is the online internship, which helps students learn real-world skills along with their studies. ::An online internship allows students to work on tasks and projects through digital platforms. This makes learning more practical and useful, especially for those who want to understand how real work environments function. ::'''Background''' ::The concept of online learning developed from distance education, where students learned from remote locations. Over time, with the growth of digital technology, learning has become more interactive and flexible. ::The introduction of the online internship has added another important layer to digital education. It combines theoretical learning with practical experience, helping students prepare for future careers. ::During global events such as the COVID-19 pandemic, online education and online internship programs became essential. They helped students continue learning and gaining experience despite restrictions on physical movement. ::'''Importance of Online Internship''' ::An online internship plays an important role in student development. It helps bridge the gap between academic knowledge and practical skills. ::'''Some key points include:''' ::Students understand how real work is done ::They develop basic professional skills ::It supports career readiness ::It allows learning without location limits ::By participating in an online internship, students can improve their confidence and gain early exposure to different fields. ::'''Features of Online Internship-Based Learning''' ::Modern education platforms often include online internship opportunities as part of their learning system. These usually offer: ::Flexible schedules for students ::Access to learning from home ::Beginner-friendly tasks and projects ::A combination of theory and practice ::Such features make online internship programs suitable for a wide range of learners, including beginners. ::'''Learning Tasks''' ::Explore how online internship programs support student learning in digital environments. ::Identify how students can gain practical experience through an online internship ::Analyze the role of flexible learning in improving student engagement ::Understand how online education and internships work together ::'''References''' ::[https://ijpsl.in/wp-content/uploads/2020/09/E-Learning_Hanaaya-Navaneeth.pdf The Past, Present and Future of E-Learning: Hanaaya Varyani and Navaneeth M S] ::[https://ieeexplore.ieee.org/document/9788102 Current Trends and Future Perspectives of e-Learning in India] ::'''See Also''' ::[https://easyshiksha.com/online_courses/internship Online Internship] ::[https://easyshiksha.com/online_courses/ Online Courses] ::[https://easyshiksha.com/online_courses/kids-learning Kids Learning] ::[https://easyshiksha.com/career_helper/ Career Guidance] [[User:EasyshikshaMarketing|EasyshikshaMarketing]] ([[User talk:EasyshikshaMarketing|discuss]] • [[Special:Contributions/EasyshikshaMarketing|contribs]]) 05:27, 17 April 2026 (UTC) :::@[[User:EasyshikshaMarketing|EasyshikshaMarketing]] Wikiversity is a resource for education, or a space for education. However, your intention to link to another website is obvious, and such content does not belong here, as it contradicts the purpose of Wikiversity. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 11:38, 17 April 2026 (UTC) ==AI-generated images== Seeking your advice. Myself and students use some AI-generated images in the [[Motivation and emotion]] project. [[User:Dronebogus]] has been removing some of these images from Wikiversity pages and nominating them for deletion at Commons. As a result, some have been deleted and some have been kept. Dronebogus has made some useful edits and image suggestions for [[Motivation and emotion]] which I've appreciated and incorporated. However, there are other edits to remove an AI image by Dronebogus that I've reverted where I think the image is more educational than no image or an alternative image suggested by Dronebogus. There are a couple of pages where Dronebogus has reverted my reversion, so we are at risk of edit warring. We have briefly discussed and warned each other on our user talk pages, but it seems to come down to a difference in perception about the educational usefulness of the AI images. So, I'm asking here for others to please review the recent edit histories for these pages: * [[Motivation and emotion/Lectures/Brain and physiological needs]] * [[Motivation and emotion/Book/2025/Stockholm syndrome emotion]] and let us know what you think about the AI image suitability vs. using no image or alternative images suggested by Dronebogus. Sincerely, James -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:45, 26 April 2026 (UTC) :I think Jtneill needs to try harder to find non-AI alternatives both on Commons and the web. I’m not reiterating the well known problems with generative AI— you can read about those on Wikipedia and the broader Internet. Needless to say it’s kind of just inherently toxic. If you use it, it should be the last resort of last resorts. Just my stance, which I consider perfectly valid and reasonable. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 04:10, 27 April 2026 (UTC) == Request for Page Creation Approval - Research on Dr. Ian Stevenson == '''Hello Administrators,''' I am a digital archivist and researcher attempting to create an educational page regarding Dr. Ian Stevenson's scientific work, specifically titled "Where Reincarnation and Biology Intersect" and its official Vietnamese edition. My edits are purely constructive, aimed at integrating local research context with global structured data (including '''Wikidata Q139548587'''). The automated filter is currently preventing publication due to "new user external link limits." I have ensured that all citations, references, and images (sourced from Wikimedia Commons) are strictly academic and non-promotional. Could you please review my contributions and consider whitelisting my account or approving this specific page creation? Thank you for your assistance in preserving this research '''legacy''' within the open knowledge ecosystem. '''Best regards,''' [[User:Ian Stevenson777|Ian Stevenson777]] ([[User talk:Ian Stevenson777|discuss]] • [[Special:Contributions/Ian Stevenson777|contribs]]) 07:26, 28 April 2026 (UTC) : The page title should probably be in English, as this is English Wikiversity. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 14:19, 28 April 2026 (UTC) :@[[User:Ian Stevenson777|Ian Stevenson777]] I looked at the text and it seems to me that it is informative, but the user cannot learn anything from it because they would have to buy the book. In this regard, it seems to me that it is enough [[:w:Where Reincarnation and Biology Intersect|to have that article on Wikipedia]] and there is no need to duplicate it here on Wikiversity. If you are particularly interested with its Vietnamese translation, I would focus on Vietnamese Wikipedia, but it seems to me, that the article there [https://vi.wikipedia.org/w/index.php?title=N%C6%A1i_lu%C3%A2n_h%E1%BB%93i_v%C3%A0_sinh_h%E1%BB%8Dc_giao_nhau&oldid=75029095 have some problems at the moment]. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 12:51, 5 May 2026 (UTC) :This request is AI-generated and was made by a lock evading LTA. Please ignore it. [[User:NguoiDungKhongDinhDanh|NguoiDungKhongDinhDanh]] ([[User talk:NguoiDungKhongDinhDanh|discuss]] • [[Special:Contributions/NguoiDungKhongDinhDanh|contribs]]) 23:58, 13 May 2026 (UTC) == Call for custodians and bureaucrats == <div class="cd-moveMark">''Moved to [[Wikiversity:Notices for custodians#Call for custodians and bureaucrats]]. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 18:46, 12 May 2026 (UTC)''</div> == Interface administrator flag request == @[[User:Jtneill|Jtneill]], @[[User:Mu301|Mu301]] I am requesting IA flag to delete [[MediaWiki:Gadget-WikiSign.js]]. It was requested for deletion by [[User:Sophivorus|a trusted user]] and I think it can be deleted. Once I delete it, I will notify you and you can remove the flag. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:24, 10 May 2026 (UTC) :Can you enable two-factor authentication? :Policy: [[Wikiversity:Interface administrators]] -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 10:27, 10 May 2026 (UTC) ::Done @[[User:Jtneill|Jtneill]]. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:20, 11 May 2026 (UTC) :::You are an interface administrator for 24 hours. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 09:57, 11 May 2026 (UTC) ::::Thx @[[User:Jtneill|Jtneill]], I have just deleted the page. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 09:59, 11 May 2026 (UTC) == Block request == Hi. Please block @[[User:Skibidi Titan 5.0|Skibidi Titan 5.0]]. Reason : Vandalism. Thanks, [[User:Locpac|Locpac]] ([[User talk:Locpac|discuss]] • [[Special:Contributions/Locpac|contribs]]) 15:10, 10 May 2026 (UTC) :Thankyou. Done. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 15:25, 10 May 2026 (UTC) == ~2026-28496-53 == Please block [[Special:Contribs/~2026-28496-53]], vandalism. [[User:Tenshi Hinanawi|Tenshi Hinanawi]] ([[User talk:Tenshi Hinanawi|トーク]] • [[Special:Contributions/Tenshi Hinanawi|投稿記録]]) 17:02, 11 May 2026 (UTC) :{{done}} Thank you! —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 17:14, 11 May 2026 (UTC) == ~2026-28792-52 == Please block [[Special:Contribs/~2026-28792-52]], vandalism. Appears to be same user as above. [[User:Tenshi Hinanawi|Tenshi Hinanawi]] ([[User talk:Tenshi Hinanawi|トーク]] • [[Special:Contributions/Tenshi Hinanawi|投稿記録]]) 17:38, 12 May 2026 (UTC) : Blocked locally by Barras. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 17:46, 12 May 2026 (UTC) == New User Exceeded New Page Limit == Hello, my action of creating a Portal for Banjo learning (Portal:Banjo) was denied as I am a user whose had my account for a while but not written before today. My actions are constructive, but if you would rather me wait and let the system work as it intends to then that's okay too. --[[User:Kirby - Electrotechnics|Kirby - Electrotechnics]] ([[User talk:Kirby - Electrotechnics|discuss]] • [[Special:Contributions/Kirby - Electrotechnics|contribs]]) 01:13, 17 May 2026 (UTC) :That is standard and the rate limit will fall off as you stay and edit; it has some pretty easy barriers to cross. I can create a blank [[Portal:Banjo]] if you want, but to be clear, portal pages are usually much broader topics that can help orient you to specific pages. Do you think there will be that many pages about the topic of banjo playing? If not, then you can just continue editing [[Banjo]]. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 03:36, 17 May 2026 (UTC) == Disable [[Special:AbuseFilter/3]] == Please disable [[Special:AbuseFilter/3]] because it is redundant with [[m:Special:AbuseFilter/104|a global filter]] that disallows the same type of spam that filter 3 would have caught. Thank you. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 01:06, 19 May 2026 (UTC) : [[Special:AbuseFilter/history/3/diff/prev/454|Done]] -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 02:15, 19 May 2026 (UTC) == [[One man's look at concept]] == Please semi-protect that page indefinitely due to vandalism from an unregistered vandal. Thank you. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 18:52, 19 May 2026 (UTC) :I've protected the page, but I've decided to keep it to a year just because the vandalism seemed to have started in late 2025, and the longest protection imposed was just about 3 months. If vandalism persists after the protection, I'd be more than happy to protect it indefinitely. Thanks. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 21:24, 19 May 2026 (UTC) tr75kjihgzqp9catxnxipryspi11hci Social Victorians/Irish Aristocracy 0 329829 2812976 2812795 2026-06-05T22:11:22Z Scogdill 1331941 2812976 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. == 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> * Subsidiary Titles ** Marquess of Abercorn ** Viscount Hamilton ** Viscount Strabane, county Tyrone *Papers: PRONI for the Abercorn papers; 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. ==== [[Social Victorians/People/Londonderry|Marquess and Marchioness of Londonderry]] ==== * The Marquess and Marchioness attended the ball, 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 ==== [[Social Victorians/People/Lucan|Earl of Lucan]] ==== * Some members of the family attended the ball, and the family attended a number of social events at this time. * Papers ==== [[Social Victorians/People/Ormonde|Marquess and Marchioness of Ormonde]] ==== * 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 X was their manor, but they don't appear to have any papers. * Subsidiary Titles * Papers ==== [[Social Victorians/People/Antrim|Earl of Antrim]] ==== * Some members of this family attended the ball, though not the earl or countess. * Papers ==== [[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 ball, 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 ==== [[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 ball. 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 ball. Papers in PRONI. * 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 ==== [[Social Victorians/People/Gosford|Earl of Gosford]] ==== * The Earl and Countess of Gosford attended the ball, 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 ball. * 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 ball. * 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 ball 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 ==== Baron Carrington ==== * [[Social Victorians/People/Carrington|Charles Robert Wynn-Carington, 1st Marquess of Lincolnshire]] (1843–1928) attended the ball. * 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 ball 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}} tfrwfd4fdqgaelr4t00n9355hdc7gly Wikiversity:Inactivity policy 4 329965 2812850 2812784 2026-06-05T14:56:32Z Codename Noreste 2969951 Boldly reducing to one year per the colloquium. 2812850 wikitext text/x-wiki {{Draft}} Wikiversity [[Wikiversity:Support staff|support staff]] ([[Wikiversity:Curatorship|curators]], [[Wikiversity:Custodianship|custodians]] and [[Wikiversity:Bureaucratship|bureaucrats]]) are considered inactive if they have made no edits and logged actions within one year. == Process == # A notice is sent to the [[Wikiversity:Colloquium|Colloquium]], listing the inactive support staff. # An inactive support staff member is notified on their talk page, explaining that their rights may be removed. {{tlx|Inactive curator}} may be used for the notification. # If no response is received from the inactive support staff member within two weeks, whether through their talk page or the Colloquium, the rights will be removed. A custodian will remove the curator permission locally, while a steward will be asked to remove bureaucrat and/or custodian permissions (per [[m:Steward requests/Permissions#Removal of access]]). [[Category:Wikiversity administration|Inactivity policy]] p2o49ie1xxolt54put3hwcjaiqnsgcu Umuada 0 329971 2812833 2812828 2026-06-05T12:05:18Z Atcovi 276019 {{welcome and expand}} 2812833 wikitext text/x-wiki {{welcome and expand}} == Introduction == The married and unmarried daughters of a particular clan or village in [[w:ala Igbo|ala Igbo]]. While Igbo society is about tracing descent through the male, Ndi Umuada serve as a vital checks and balances in the society. They represent a collective authority that balances the political power held by the males. Their words and decisions are highly respected and are often final. ivl5xuqct0sjvz71icpageawi3tseby User talk:Wmbata 3 329972 2812841 2812827 2026-06-05T13:22:16Z Wmbata 3084293 /* Welcome */ Reply 2812841 wikitext text/x-wiki ==Welcome== {{Robelbox|theme=9|title='''[[Wikiversity:Welcome|Welcome]] to [[Wikiversity:What is Wikiversity|Wikiversity]], Wmbata!'''|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. 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See you around Wikiversity! ---- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:56, 5 June 2026 (UTC)</div> <!-- Template:Welcome --> {{Robelbox/close}} :Thanks @[[User:Jtneill|Jtneill]] [[User:Wmbata|Wmbata]] ([[User talk:Wmbata|discuss]] • [[Special:Contributions/Wmbata|contribs]]) 13:22, 5 June 2026 (UTC) f9hvka55fiod1nbv45olzztszhqqgrh Category:Motivation and emotion/Book/ADHD 14 329973 2812836 2026-06-05T12:07:50Z Jtneill 10242 Created page with "[[Category:Motivation and emotion/Book/Motivation]] [[Category:Motivation and emotion/Book/Neurodiversity]]" 2812836 wikitext text/x-wiki [[Category:Motivation and emotion/Book/Motivation]] [[Category:Motivation and emotion/Book/Neurodiversity]] 5ietbvzxq55i49fufago5nkbksi6d9u File:VLSI.Arith.2A.CLA.20260605.pdf 6 329974 2812843 2026-06-05T13:48:19Z Young1lim 21186 {{Information |Description=Carry Lookahead Adders 2A traditional (20260605 - 20260604) |Source={{own|Young1lim}} |Date=2026-06-05 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2812843 wikitext text/x-wiki == Summary == {{Information |Description=Carry Lookahead Adders 2A traditional (20260605 - 20260604) |Source={{own|Young1lim}} |Date=2026-06-05 |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}} emncdqmiolt2cu2lb0pt02dbs6qefob File:VLSI.Arith.2B.CLA.20260605.pdf 6 329975 2812844 2026-06-05T13:49:04Z Young1lim 21186 {{Information |Description=Carry Lookahead Adders 2B simplified (20260605 - 20260604) |Source={{own|Young1lim}} |Date=2026-06-05 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2812844 wikitext text/x-wiki == Summary == {{Information |Description=Carry Lookahead Adders 2B simplified (20260605 - 20260604) |Source={{own|Young1lim}} |Date=2026-06-05 |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}} ftmra9ipa8f6k5o9xs8fvvrf5x0usst File:C04.SA0.PtrOperator.1A.20260605.pdf 6 329976 2812846 2026-06-05T13:56:21Z Young1lim 21186 {{Information |Description=C04.SA0: Address and Dereference Operators (20260605 - 20260604) |Source={{own|Young1lim}} |Date=2026-06-05 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2812846 wikitext text/x-wiki == Summary == {{Information |Description=C04.SA0: Address and Dereference Operators (20260605 - 20260604) |Source={{own|Young1lim}} |Date=2026-06-05 |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}} a1gfax15mjyz8i5o5qgjca40ie4rrpp File:Laurent.5.Permutation.6C.20260605.pdf 6 329977 2812848 2026-06-05T14:01:36Z Young1lim 21186 {{Information |Description=Laurent.5: Permutation 6C (20260605 - 20260604) |Source={{own|Young1lim}} |Date=2026-06-05 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2812848 wikitext text/x-wiki == Summary == {{Information |Description=Laurent.5: Permutation 6C (20260605 - 20260604) |Source={{own|Young1lim}} |Date=2026-06-05 |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}} 08upakezbjx0fpcfhv90d4t2bed3mfs Category:Music technology 14 329978 2812852 2026-06-05T15:12:05Z Kirby - Electrotechnics 3074947 Created 'Category:Music technology' for pages to be put under 2812852 wikitext text/x-wiki [[Category:Music]] aa3fvhnseq7sszbitwo0qvms10njt7n 2813019 2812852 2026-06-06T00:37:19Z MathXplore 2888076 added [[Category:Technology]] using [[Help:Gadget-HotCat|HotCat]] 2813019 wikitext text/x-wiki [[Category:Music]] [[Category:Technology]] q8odbsbfajmh1xcvedk0cf5q22ur1lp Category:Music genres and styles 14 329979 2812856 2026-06-05T15:24:04Z Kirby - Electrotechnics 3074947 Creating this page to replace 'Category:Music styles', because people associate genres and styles together better, and easier categorizing for the average reader. 2812856 wikitext text/x-wiki [[Category:Music]] aa3fvhnseq7sszbitwo0qvms10njt7n Category:Euphonium 14 329980 2812886 2026-06-05T15:59:22Z Kirby - Electrotechnics 3074947 Created Category:Euphonium under 'Category:Music Instruments' 2812886 wikitext text/x-wiki [[Category:Music Instruments]] mjpqkpfkpdvf4kv5286h0i3yte86w98 2812887 2812886 2026-06-05T16:00:57Z Kirby - Electrotechnics 3074947 Incorrect capitalization 2812887 wikitext text/x-wiki [[Category:Music instruments]] q5wv1rb9alg0iw1wt59qc19wyow3eyn User talk:Chet cunningham 3 329982 2812892 2026-06-05T17:36:34Z Chet cunningham 3089504 /* wwII */ new section 2812892 wikitext text/x-wiki == wwII == world war 2 and the nazis [[User:Chet cunningham|Chet cunningham]] ([[User talk:Chet cunningham|discuss]] • [[Special:Contributions/Chet cunningham|contribs]]) 17:36, 5 June 2026 (UTC) k5bu6l2m623jczf0by6esi5lpp0v23s 2812893 2812892 2026-06-05T17:36:45Z Quinlan83 2913823 Requesting deletion 2812893 wikitext text/x-wiki {{delete|Test page}}== wwII == world war 2 and the nazis [[User:Chet cunningham|Chet cunningham]] ([[User talk:Chet cunningham|discuss]] • [[Special:Contributions/Chet cunningham|contribs]]) 17:36, 5 June 2026 (UTC) lg7zzl0ucytg3osin14o09mclu2e2wu 2812900 2812893 2026-06-05T17:40:24Z Quinlan83 2913823 fix 2812900 wikitext text/x-wiki {{Robelbox|theme=9|title=Welcome!|width=100%}} <div style="{{Robelbox/pad}}"> '''Hello and [[Wikiversity:Welcome|Welcome]] to [[Wikiversity:What is Wikiversity|Wikiversity]] Chet cunningham!''' You can [[Wikiversity:Contact|contact us]] with [[Wikiversity:Questions|questions]] at the [[Wikiversity:Colloquium|colloquium]] or [[User talk:Dave Braunschweig|me personally]] when you need [[Help:Contents|help]]. Please remember to [[Wikiversity:Signature|sign and date]] your finished comments when [[Wikiversity:Who are Wikiversity participants?|participating]] in [[Wikiversity:Talk page|discussions]]. The signature icon [[File:OOjs UI icon signature-ltr.svg]] above the edit window makes it simple. All users are expected to abide by our [[Wikiversity:Privacy policy|Privacy]], [[Wikiversity:Civility|Civility]], and the [[Foundation:Terms of Use|Terms of Use]] policies while at Wikiversity. 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You only need to [[Wikiversity:Be bold|be bold]] to contribute and to experiment with the [[wikiversity:sandbox|sandbox]] or [[special:mypage|your userpage]]. See you around Wikiversity! --[[User:Quinlan83|Quinlan83]] ([[User talk:Quinlan83|discussione]] • [[Special:Contributions/Quinlan83|contributi]]) 17:40, 5 June 2026 (UTC) <!-- Template:Welcome --> {{Robelbox/close}} g2itgprtwqfffyx0mf7odjohrd7huqt Category:Music education 14 329983 2812952 2026-06-05T21:37:19Z Kirby - Electrotechnics 3074947 Created 'Category:Music education' for pages to be put under 2812952 wikitext text/x-wiki [[Category:Music]] aa3fvhnseq7sszbitwo0qvms10njt7n 2813017 2812952 2026-06-06T00:36:03Z MathXplore 2888076 added [[Category:Education]] using [[Help:Gadget-HotCat|HotCat]] 2813017 wikitext text/x-wiki [[Category:Music]] [[Category:Education]] 5djlggsak9dfia4mbtixe7b7a24wwjy Category:Electronic Music 14 329984 2812958 2026-06-05T21:46:50Z Kirby - Electrotechnics 3074947 Created 'Category:Electronic Music' for pages to be put under 2812958 wikitext text/x-wiki [[Category:Music genres and styles]] qhimz008596xnk5444wagy28xmwcppl Category:Music resources 14 329985 2812962 2026-06-05T21:52:30Z Kirby - Electrotechnics 3074947 Created 'Category:Music resources' for pages to be put under 2812962 wikitext text/x-wiki [[Category:Music]] aa3fvhnseq7sszbitwo0qvms10njt7n Category:Musicians 14 329986 2812969 2026-06-05T22:00:27Z Kirby - Electrotechnics 3074947 Created 'Category:Musicians' for pages to be put under 2812969 wikitext text/x-wiki [[Category:Music]] aa3fvhnseq7sszbitwo0qvms10njt7n Category:Music appreciation 14 329987 2812979 2026-06-05T22:16:46Z Kirby - Electrotechnics 3074947 Created 'Category:Music appreciation' for pages to be put under 2812979 wikitext text/x-wiki [[Category:Music]] aa3fvhnseq7sszbitwo0qvms10njt7n Category:Songs 14 329988 2812990 2026-06-05T22:29:29Z Kirby - Electrotechnics 3074947 Created 'Category:Songs' for pages to be put under 2812990 wikitext text/x-wiki [[Category:Music]] aa3fvhnseq7sszbitwo0qvms10njt7n Category:Aural theory and ear training 14 329989 2812999 2026-06-06T00:22:03Z Kirby - Electrotechnics 3074947 Created 'Category:Aural theory and ear training' for pages to be put under 2812999 wikitext text/x-wiki [[Category:Music]] aa3fvhnseq7sszbitwo0qvms10njt7n Category:Music related pages 14 329990 2813002 2026-06-06T00:26:25Z Kirby - Electrotechnics 3074947 Created 'Category:Music related pages', because it sounds more permanent than "Music related projects" which will likely be absorbed into this page, or dispersed into another category. 2813002 wikitext text/x-wiki [[Category:Music]] aa3fvhnseq7sszbitwo0qvms10njt7n Talk:Theory of relativity/Kerr–Newman metric 1 329991 2813022 2026-06-06T03:21:00Z ~2026-33557-02 3089703 /* Mistake ? */ new section 2813022 wikitext text/x-wiki == Mistake ? == Hello, are you sure delta is squared in formula ? [[Special:Contributions/&#126;2026-33557-02|&#126;2026-33557-02]] ([[User talk:&#126;2026-33557-02|talk]]) 03:21, 6 June 2026 (UTC) k16jgfxvgd9hf6od6oauy8d67nwkmyq Template:Text diff 10 329993 2813031 2014-11-01T06:36:59Z Mxn 21595 Copied from [[Commons:Template:TextDiff]] (101455627) 2813031 wikitext text/x-wiki {{#invoke:Diff|main|{{{1|This is a test}}}|{{{2|This was a test!}}}}}<noinclude> {{documentation}} </noinclude> m51amq46hg9s9fvlgfxk5ywvc3b5m22 2813032 2813031 2023-08-30T18:00:05Z MusikBot II 2893438 Protected "[[Template:TextDiff]]": [[Wikipedia:High-risk templates|High-risk template or module]]: 252 transclusions ([[User:MusikBot II/TemplateProtector|more info]]) ([Edit=Require autoconfirmed or confirmed access] (indefinite)) 2813031 wikitext text/x-wiki {{#invoke:Diff|main|{{{1|This is a test}}}|{{{2|This was a test!}}}}}<noinclude> {{documentation}} </noinclude> m51amq46hg9s9fvlgfxk5ywvc3b5m22 2813033 2813032 2023-12-03T19:35:33Z Jonesey95 831143 try to make this template work better with block content, or with line breaks before the parameters 2813033 wikitext text/x-wiki {{#invoke:Diff|main|{{trim|1={{{1|This is a test}}}}}|{{trim|1={{{2|This was a test!}}}}}}}<noinclude> {{documentation}} </noinclude> eflktr2jy8z7rtue4veh6ob5it00nn7 2813034 2813033 2023-12-03T19:47:15Z Jonesey95 831143 does not resolve the problem in all cases 2813034 wikitext text/x-wiki {{#invoke:Diff|main|{{{1|This is a test}}}|{{{2|This was a test!}}}}}<noinclude> {{documentation}} </noinclude> m51amq46hg9s9fvlgfxk5ywvc3b5m22 2813035 2813034 2023-12-08T17:11:26Z HouseBlaster 2945622 HouseBlaster moved page [[Template:TextDiff]] to [[Template:Text diff]] over redirect: use plain english in title per [[WP:TPN]] 2813034 wikitext text/x-wiki {{#invoke:Diff|main|{{{1|This is a test}}}|{{{2|This was a test!}}}}}<noinclude> {{documentation}} </noinclude> m51amq46hg9s9fvlgfxk5ywvc3b5m22 2813036 2813035 2025-08-31T10:12:18Z Waddie96 2894833 [[Template:Request edit button/preload]] error check 2813036 wikitext text/x-wiki <!-- This error check is for [[Template:Request edit button/preload]]. -->{{#ifeq:{{{1}}}|ORIGINAL_WIKITEXT|{{error|Unnamed parameter {{para|1}} set to default value. Please change it.<br/>}}}}{{#ifeq:{{{2}}}|CHANGED_WIKITEXT|{{error|Unnamed parameter {{para|2}} set to default value. Please change it.}}}}{{#invoke:Diff|main|{{{1|This is a test}}}|{{{2|This was a test!}}}}}<noinclude> {{documentation}} </noinclude> o6liae7e1ghcx41cre8wttjkwdq53oc 2813037 2813036 2025-09-09T16:25:58Z Wbm1058 290388 do not transclude {{error}}s when users fail to properly use this template. just warn them. 2813037 wikitext text/x-wiki <!-- This user-input check is for [[Template:Request edit button/preload]]. -->{{#ifeq:{{{1}}}|ORIGINAL_WIKITEXT|{{warning sign}} Unnamed parameter {{para|1}} set to default value. Please change it. Failure to use {{tl|Text diff}} to specify your requested text changes, if not adequately described above, may lead to your request being denied.<br/>|{{#ifeq:{{{2}}}|CHANGED_WIKITEXT|{{warning sign}} Unnamed parameter {{para|2}} set to default value. Please change it. Failure to use {{tl|Text diff}} to specify your requested text changes, if not adequately described above, may lead to your request being denied.<br/>|{{#invoke:Diff|main|{{{1|This is a test}}}|{{{2|This was a test!}}}}}}}}}<noinclude> {{documentation}} </noinclude> reqb6byw8n5x095bbokxwb3qkqru8j4 2813038 2813037 2026-02-25T18:00:06Z MusikBot II 2893438 Changed protection settings for "[[Template:Text diff]]": [[Wikipedia:High-risk templates|High-risk template or module]]: 2500 transclusions ([[User:MusikBot II/TemplateProtector|more info]]) ([Edit=Require extended confirmed access] (indefinite) [Move=Require extended confirmed access] (indefinite)) 2813037 wikitext text/x-wiki <!-- This user-input check is for [[Template:Request edit button/preload]]. -->{{#ifeq:{{{1}}}|ORIGINAL_WIKITEXT|{{warning sign}} Unnamed parameter {{para|1}} set to default value. Please change it. Failure to use {{tl|Text diff}} to specify your requested text changes, if not adequately described above, may lead to your request being denied.<br/>|{{#ifeq:{{{2}}}|CHANGED_WIKITEXT|{{warning sign}} Unnamed parameter {{para|2}} set to default value. Please change it. Failure to use {{tl|Text diff}} to specify your requested text changes, if not adequately described above, may lead to your request being denied.<br/>|{{#invoke:Diff|main|{{{1|This is a test}}}|{{{2|This was a test!}}}}}}}}}<noinclude> {{documentation}} </noinclude> reqb6byw8n5x095bbokxwb3qkqru8j4 2813039 2813038 2026-06-06T08:30:11Z Jtneill 10242 8 revisions imported from [[:w:Template:Text_diff]] 2813037 wikitext text/x-wiki <!-- This user-input check is for [[Template:Request edit button/preload]]. -->{{#ifeq:{{{1}}}|ORIGINAL_WIKITEXT|{{warning sign}} Unnamed parameter {{para|1}} set to default value. Please change it. Failure to use {{tl|Text diff}} to specify your requested text changes, if not adequately described above, may lead to your request being denied.<br/>|{{#ifeq:{{{2}}}|CHANGED_WIKITEXT|{{warning sign}} Unnamed parameter {{para|2}} set to default value. Please change it. Failure to use {{tl|Text diff}} to specify your requested text changes, if not adequately described above, may lead to your request being denied.<br/>|{{#invoke:Diff|main|{{{1|This is a test}}}|{{{2|This was a test!}}}}}}}}}<noinclude> {{documentation}} </noinclude> reqb6byw8n5x095bbokxwb3qkqru8j4 2813221 2813032 2023-10-07T00:34:16Z Anomalocaris 1134554 Lint errors 2813221 wikitext text/x-wiki == Lint errors == This template is causing missing end tag, stripped tag, and misnested tag lint errors, and it seems that this is associated with line breaks or multiple blocks in either or both of the main parameters. Here are some pages that use this templates and have such errors: * [[Talk:Sri Sathya Sai International Organization]]: Missing end tag and Stripped tags for {{tag|ins}}. * [[Talk:IHH Healthcare]]: Missing end tag and Stripped tags for {{tag|ins}}. * [[Talk:Dog food]]: Missing end tag and Stripped tags for {{tag|del}}. * [[Talk:American Foundation for Suicide Prevention]]: HTML5-misnesting and Stripped tags for {{tag|ins}}. If this can fixed, it should be fixed. If it can't be fixed, it should be documented what markup causes these problems. —[[User:Anomalocaris|Anomalocaris]] ([[User talk:Anomalocaris|talk]]) 00:34, 7 October 2023 (UTC) htiamxrj1yoi8eomkja0kf2pis4tsi2 2813222 2813221 2023-10-07T00:36:11Z Anomalocaris 1134554 /* Lint errors */ also [[Template talk:Non-free 2D art]] 2813222 wikitext text/x-wiki == Lint errors == This template is causing missing end tag, stripped tag, and misnested tag lint errors, and it seems that this is associated with line breaks or multiple blocks in either or both of the main parameters. Here are some pages that use this templates and have such errors: * [[Talk:Sri Sathya Sai International Organization]]: Missing end tag and Stripped tags for {{tag|ins}}. * [[Talk:IHH Healthcare]]: Missing end tag and Stripped tags for {{tag|ins}}. * [[Talk:Dog food]]: Missing end tag and Stripped tags for {{tag|del}}. * [[Talk:American Foundation for Suicide Prevention]]: HTML5-misnesting and Stripped tags for {{tag|ins}}. * [[Template talk:Non-free 2D art]]: HTML5-misnesting and Stripped tags for {{tag|ins}}. If this can fixed, it should be fixed. If it can't be fixed, it should be documented what markup causes these problems. —[[User:Anomalocaris|Anomalocaris]] ([[User talk:Anomalocaris|talk]]) 00:34, 7 October 2023 (UTC) m2gjywv5tabz9db93v836amkhgwqvlp 2813223 2813034 2023-12-03T22:11:00Z Anomalocaris 1134554 /* Lint errors */ thanks [[User:Jonesey95|Jonesey95]] but need real fix 2813223 wikitext text/x-wiki == Lint errors == This template is causing missing end tag, stripped tag, and misnested tag lint errors, and it seems that this is associated with line breaks or multiple blocks in either or both of the main parameters. Here are some pages that use this templates and have such errors: * [[Talk:Sri Sathya Sai International Organization]]: Missing end tag and Stripped tags for {{tag|ins}}. * [[Talk:IHH Healthcare]]: Missing end tag and Stripped tags for {{tag|ins}}. * [[Talk:Dog food]]: Missing end tag and Stripped tags for {{tag|del}}. * [[Talk:American Foundation for Suicide Prevention]]: HTML5-misnesting and Stripped tags for {{tag|ins}}. * [[Template talk:Non-free 2D art]]: HTML5-misnesting and Stripped tags for {{tag|ins}}. If this can fixed, it should be fixed. If it can't be fixed, it should be documented what markup causes these problems. —[[User:Anomalocaris|Anomalocaris]] ([[User talk:Anomalocaris|talk]]) 00:34, 7 October 2023 (UTC) [[User:Jonesey95|Jonesey95]]: Thank you for fixing [[Template:TextDiff/doc]] to acknowledge that this template is good for inline use only. Unfortunately, it is already widely deployed with multi-paragraph arguments, so it would be good if somebody gets this thing to work in such contexts. —[[User:Anomalocaris|Anomalocaris]] ([[User talk:Anomalocaris|talk]]) 22:10, 3 December 2023 (UTC) 4qkl0eh7ef2rxwg4z2j33p5p2i0ufup 2813224 2813223 2023-12-03T23:30:03Z Jonesey95 831143 /* Lint errors */ some experimenting 2813224 wikitext text/x-wiki == Lint errors == This template is causing missing end tag, stripped tag, and misnested tag lint errors, and it seems that this is associated with line breaks or multiple blocks in either or both of the main parameters. Here are some pages that use this templates and have such errors: * [[Talk:Sri Sathya Sai International Organization]]: Missing end tag and Stripped tags for {{tag|ins}}. * [[Talk:IHH Healthcare]]: Missing end tag and Stripped tags for {{tag|ins}}. * [[Talk:Dog food]]: Missing end tag and Stripped tags for {{tag|del}}. * [[Talk:American Foundation for Suicide Prevention]]: HTML5-misnesting and Stripped tags for {{tag|ins}}. * [[Template talk:Non-free 2D art]]: HTML5-misnesting and Stripped tags for {{tag|ins}}. If this can fixed, it should be fixed. If it can't be fixed, it should be documented what markup causes these problems. —[[User:Anomalocaris|Anomalocaris]] ([[User talk:Anomalocaris|talk]]) 00:34, 7 October 2023 (UTC) [[User:Jonesey95|Jonesey95]]: Thank you for fixing [[Template:TextDiff/doc]] to acknowledge that this template is good for inline use only. Unfortunately, it is already widely deployed with multi-paragraph arguments, so it would be good if somebody gets this thing to work in such contexts. —[[User:Anomalocaris|Anomalocaris]] ([[User talk:Anomalocaris|talk]]) 22:10, 3 December 2023 (UTC) :As you might have seen, I did some experimenting in the template sandbox. It was not fruitful, for the most part. I don't know enough Lua or CSS, one of which might be able to help resolve the problem, to install a fix in the module. – [[User:Jonesey95|Jonesey95]] ([[User talk:Jonesey95|talk]]) 23:29, 3 December 2023 (UTC) lhx2lurywf2fttmvebnxg427dqu613k 2813225 2813224 2023-12-03T23:30:27Z Jonesey95 831143 /* Lint errors */ and 2813225 wikitext text/x-wiki == Lint errors == This template is causing missing end tag, stripped tag, and misnested tag lint errors, and it seems that this is associated with line breaks or multiple blocks in either or both of the main parameters. Here are some pages that use this templates and have such errors: * [[Talk:Sri Sathya Sai International Organization]]: Missing end tag and Stripped tags for {{tag|ins}}. * [[Talk:IHH Healthcare]]: Missing end tag and Stripped tags for {{tag|ins}}. * [[Talk:Dog food]]: Missing end tag and Stripped tags for {{tag|del}}. * [[Talk:American Foundation for Suicide Prevention]]: HTML5-misnesting and Stripped tags for {{tag|ins}}. * [[Template talk:Non-free 2D art]]: HTML5-misnesting and Stripped tags for {{tag|ins}}. If this can fixed, it should be fixed. If it can't be fixed, it should be documented what markup causes these problems. —[[User:Anomalocaris|Anomalocaris]] ([[User talk:Anomalocaris|talk]]) 00:34, 7 October 2023 (UTC) [[User:Jonesey95|Jonesey95]]: Thank you for fixing [[Template:TextDiff/doc]] to acknowledge that this template is good for inline use only. Unfortunately, it is already widely deployed with multi-paragraph arguments, so it would be good if somebody gets this thing to work in such contexts. —[[User:Anomalocaris|Anomalocaris]] ([[User talk:Anomalocaris|talk]]) 22:10, 3 December 2023 (UTC) :As you might have seen, I did some experimenting in the template sandbox and created a testcases page. It was not fruitful, for the most part. I don't know enough Lua or CSS, one of which might be able to help resolve the problem, to install a fix in the module. – [[User:Jonesey95|Jonesey95]] ([[User talk:Jonesey95|talk]]) 23:29, 3 December 2023 (UTC) 0agqqrhnj7g6ek0dj1a6sihcssvnjrk 2813226 2813035 2023-12-08T17:11:27Z HouseBlaster 2945622 HouseBlaster moved page [[Template talk:TextDiff]] to [[Template talk:Text diff]]: use plain english in title per [[WP:TPN]] 2813226 wikitext text/x-wiki == Lint errors == This template is causing missing end tag, stripped tag, and misnested tag lint errors, and it seems that this is associated with line breaks or multiple blocks in either or both of the main parameters. Here are some pages that use this templates and have such errors: * [[Talk:Sri Sathya Sai International Organization]]: Missing end tag and Stripped tags for {{tag|ins}}. * [[Talk:IHH Healthcare]]: Missing end tag and Stripped tags for {{tag|ins}}. * [[Talk:Dog food]]: Missing end tag and Stripped tags for {{tag|del}}. * [[Talk:American Foundation for Suicide Prevention]]: HTML5-misnesting and Stripped tags for {{tag|ins}}. * [[Template talk:Non-free 2D art]]: HTML5-misnesting and Stripped tags for {{tag|ins}}. If this can fixed, it should be fixed. If it can't be fixed, it should be documented what markup causes these problems. —[[User:Anomalocaris|Anomalocaris]] ([[User talk:Anomalocaris|talk]]) 00:34, 7 October 2023 (UTC) [[User:Jonesey95|Jonesey95]]: Thank you for fixing [[Template:TextDiff/doc]] to acknowledge that this template is good for inline use only. Unfortunately, it is already widely deployed with multi-paragraph arguments, so it would be good if somebody gets this thing to work in such contexts. —[[User:Anomalocaris|Anomalocaris]] ([[User talk:Anomalocaris|talk]]) 22:10, 3 December 2023 (UTC) :As you might have seen, I did some experimenting in the template sandbox and created a testcases page. It was not fruitful, for the most part. I don't know enough Lua or CSS, one of which might be able to help resolve the problem, to install a fix in the module. – [[User:Jonesey95|Jonesey95]] ([[User talk:Jonesey95|talk]]) 23:29, 3 December 2023 (UTC) 0agqqrhnj7g6ek0dj1a6sihcssvnjrk 2813227 2813226 2024-04-24T16:26:42Z W.andrea 461123 /* Why is my second test case failing? */ new section 2813227 wikitext text/x-wiki == Lint errors == This template is causing missing end tag, stripped tag, and misnested tag lint errors, and it seems that this is associated with line breaks or multiple blocks in either or both of the main parameters. Here are some pages that use this templates and have such errors: * [[Talk:Sri Sathya Sai International Organization]]: Missing end tag and Stripped tags for {{tag|ins}}. * [[Talk:IHH Healthcare]]: Missing end tag and Stripped tags for {{tag|ins}}. * [[Talk:Dog food]]: Missing end tag and Stripped tags for {{tag|del}}. * [[Talk:American Foundation for Suicide Prevention]]: HTML5-misnesting and Stripped tags for {{tag|ins}}. * [[Template talk:Non-free 2D art]]: HTML5-misnesting and Stripped tags for {{tag|ins}}. If this can fixed, it should be fixed. If it can't be fixed, it should be documented what markup causes these problems. —[[User:Anomalocaris|Anomalocaris]] ([[User talk:Anomalocaris|talk]]) 00:34, 7 October 2023 (UTC) [[User:Jonesey95|Jonesey95]]: Thank you for fixing [[Template:TextDiff/doc]] to acknowledge that this template is good for inline use only. Unfortunately, it is already widely deployed with multi-paragraph arguments, so it would be good if somebody gets this thing to work in such contexts. —[[User:Anomalocaris|Anomalocaris]] ([[User talk:Anomalocaris|talk]]) 22:10, 3 December 2023 (UTC) :As you might have seen, I did some experimenting in the template sandbox and created a testcases page. It was not fruitful, for the most part. I don't know enough Lua or CSS, one of which might be able to help resolve the problem, to install a fix in the module. – [[User:Jonesey95|Jonesey95]] ([[User talk:Jonesey95|talk]]) 23:29, 3 December 2023 (UTC) == Why is my second test case failing? == I just added two test cases and one is failing, but I can't figure out why. There's no difference in the rendered HTML as far as I can see. Where the first test case is working properly, that rules out a lot of possible problems, like [[Help:HTML in wikitext#thead, tfoot, tbody|tbody being unsupported]], which I just fixed. I'm guessing there's a quirk of the test system I'm not aware of. * [[Module talk:Diff/testcases|Results]] * [[Module:Diff/testcases|Test cases]] — [[User:W.andrea|W.andrea]] ([[User talk:W.andrea|talk]]) 16:26, 24 April 2024 (UTC) caci68wejunzio9i9n6cb81wv9sm1in 2813228 2813227 2024-04-24T16:30:27Z W.andrea 461123 /* Why is my second test case failing? */ add link to the test system 2813228 wikitext text/x-wiki == Lint errors == This template is causing missing end tag, stripped tag, and misnested tag lint errors, and it seems that this is associated with line breaks or multiple blocks in either or both of the main parameters. Here are some pages that use this templates and have such errors: * [[Talk:Sri Sathya Sai International Organization]]: Missing end tag and Stripped tags for {{tag|ins}}. * [[Talk:IHH Healthcare]]: Missing end tag and Stripped tags for {{tag|ins}}. * [[Talk:Dog food]]: Missing end tag and Stripped tags for {{tag|del}}. * [[Talk:American Foundation for Suicide Prevention]]: HTML5-misnesting and Stripped tags for {{tag|ins}}. * [[Template talk:Non-free 2D art]]: HTML5-misnesting and Stripped tags for {{tag|ins}}. If this can fixed, it should be fixed. If it can't be fixed, it should be documented what markup causes these problems. —[[User:Anomalocaris|Anomalocaris]] ([[User talk:Anomalocaris|talk]]) 00:34, 7 October 2023 (UTC) [[User:Jonesey95|Jonesey95]]: Thank you for fixing [[Template:TextDiff/doc]] to acknowledge that this template is good for inline use only. Unfortunately, it is already widely deployed with multi-paragraph arguments, so it would be good if somebody gets this thing to work in such contexts. —[[User:Anomalocaris|Anomalocaris]] ([[User talk:Anomalocaris|talk]]) 22:10, 3 December 2023 (UTC) :As you might have seen, I did some experimenting in the template sandbox and created a testcases page. It was not fruitful, for the most part. I don't know enough Lua or CSS, one of which might be able to help resolve the problem, to install a fix in the module. – [[User:Jonesey95|Jonesey95]] ([[User talk:Jonesey95|talk]]) 23:29, 3 December 2023 (UTC) == Why is my second test case failing? == I just added two test cases and one is failing, but I can't figure out why. There's no difference in the rendered HTML as far as I can see. Where the first test case is working properly, that rules out a lot of possible problems, like [[Help:HTML in wikitext#thead, tfoot, tbody|tbody being unsupported]], which I just fixed. I'm guessing there's a quirk of the [[Module:UnitTests|test system]] I'm not aware of. * [[Module talk:Diff/testcases|Results]] * [[Module:Diff/testcases|Test cases]] — [[User:W.andrea|W.andrea]] ([[User talk:W.andrea|talk]]) 16:26, 24 April 2024 (UTC) 090wgyq1aac0tt6v3duhoqv2ytu7ixm 2813229 2813228 2024-04-24T16:51:53Z W.andrea 461123 /* Why is my second test case failing? */ Add results from differs_at 2813229 wikitext text/x-wiki == Lint errors == This template is causing missing end tag, stripped tag, and misnested tag lint errors, and it seems that this is associated with line breaks or multiple blocks in either or both of the main parameters. Here are some pages that use this templates and have such errors: * [[Talk:Sri Sathya Sai International Organization]]: Missing end tag and Stripped tags for {{tag|ins}}. * [[Talk:IHH Healthcare]]: Missing end tag and Stripped tags for {{tag|ins}}. * [[Talk:Dog food]]: Missing end tag and Stripped tags for {{tag|del}}. * [[Talk:American Foundation for Suicide Prevention]]: HTML5-misnesting and Stripped tags for {{tag|ins}}. * [[Template talk:Non-free 2D art]]: HTML5-misnesting and Stripped tags for {{tag|ins}}. If this can fixed, it should be fixed. If it can't be fixed, it should be documented what markup causes these problems. —[[User:Anomalocaris|Anomalocaris]] ([[User talk:Anomalocaris|talk]]) 00:34, 7 October 2023 (UTC) [[User:Jonesey95|Jonesey95]]: Thank you for fixing [[Template:TextDiff/doc]] to acknowledge that this template is good for inline use only. Unfortunately, it is already widely deployed with multi-paragraph arguments, so it would be good if somebody gets this thing to work in such contexts. —[[User:Anomalocaris|Anomalocaris]] ([[User talk:Anomalocaris|talk]]) 22:10, 3 December 2023 (UTC) :As you might have seen, I did some experimenting in the template sandbox and created a testcases page. It was not fruitful, for the most part. I don't know enough Lua or CSS, one of which might be able to help resolve the problem, to install a fix in the module. – [[User:Jonesey95|Jonesey95]] ([[User talk:Jonesey95|talk]]) 23:29, 3 December 2023 (UTC) == Why is my second test case failing? == I just added two test cases and one is failing, but I can't figure out why. There's no difference in the rendered HTML as far as I can see. Where the first test case is working properly, that rules out a lot of possible problems, like [[Help:HTML in wikitext#thead, tfoot, tbody|tbody being unsupported]], which I just fixed. I'm guessing there's a quirk of the [[Module:UnitTests|test system]] I'm not aware of. * [[Module talk:Diff/testcases|Results]] * [[Module:Diff/testcases|Test cases]] I added <code>differs_at</code> to help pin down the problem, and it says character 717, but that points to the "c" in "one change", so I don't know what that means. Is it actually zero-indexed and it's pointing at the space? Maybe there's a non-breaking space at some level of the process? — [[User:W.andrea|W.andrea]] ([[User talk:W.andrea|talk]]) 16:26, 24 April 2024 (UTC) '''edited 16:51, 24 April 2024 (UTC)''' 2elcgb96tzuir4eu4w8rzxf3nfqengj 2813230 2813229 2024-04-24T16:52:27Z W.andrea 461123 /* Why is my second test case failing? */ No need to repeat date 2813230 wikitext text/x-wiki == Lint errors == This template is causing missing end tag, stripped tag, and misnested tag lint errors, and it seems that this is associated with line breaks or multiple blocks in either or both of the main parameters. Here are some pages that use this templates and have such errors: * [[Talk:Sri Sathya Sai International Organization]]: Missing end tag and Stripped tags for {{tag|ins}}. * [[Talk:IHH Healthcare]]: Missing end tag and Stripped tags for {{tag|ins}}. * [[Talk:Dog food]]: Missing end tag and Stripped tags for {{tag|del}}. * [[Talk:American Foundation for Suicide Prevention]]: HTML5-misnesting and Stripped tags for {{tag|ins}}. * [[Template talk:Non-free 2D art]]: HTML5-misnesting and Stripped tags for {{tag|ins}}. If this can fixed, it should be fixed. If it can't be fixed, it should be documented what markup causes these problems. —[[User:Anomalocaris|Anomalocaris]] ([[User talk:Anomalocaris|talk]]) 00:34, 7 October 2023 (UTC) [[User:Jonesey95|Jonesey95]]: Thank you for fixing [[Template:TextDiff/doc]] to acknowledge that this template is good for inline use only. Unfortunately, it is already widely deployed with multi-paragraph arguments, so it would be good if somebody gets this thing to work in such contexts. —[[User:Anomalocaris|Anomalocaris]] ([[User talk:Anomalocaris|talk]]) 22:10, 3 December 2023 (UTC) :As you might have seen, I did some experimenting in the template sandbox and created a testcases page. It was not fruitful, for the most part. I don't know enough Lua or CSS, one of which might be able to help resolve the problem, to install a fix in the module. – [[User:Jonesey95|Jonesey95]] ([[User talk:Jonesey95|talk]]) 23:29, 3 December 2023 (UTC) == Why is my second test case failing? == I just added two test cases and one is failing, but I can't figure out why. There's no difference in the rendered HTML as far as I can see. Where the first test case is working properly, that rules out a lot of possible problems, like [[Help:HTML in wikitext#thead, tfoot, tbody|tbody being unsupported]], which I just fixed. I'm guessing there's a quirk of the [[Module:UnitTests|test system]] I'm not aware of. * [[Module talk:Diff/testcases|Results]] * [[Module:Diff/testcases|Test cases]] I added <code>differs_at</code> to help pin down the problem, and it says character 717, but that points to the "c" in "one change", so I don't know what that means. Is it actually zero-indexed and it's pointing at the space? Maybe there's a non-breaking space at some level of the process? — [[User:W.andrea|W.andrea]] ([[User talk:W.andrea|talk]]) 16:26, 24 April 2024 (UTC) '''edited 16:51''' 20q5lblb12kuhbyp4rjx34fzqc4l1aq 2813231 2813230 2024-04-24T16:59:23Z W.andrea 461123 /* Why is my second test case failing? */ No longer needed. Found the problem. Space adjacent to a tag gets escaped. 2813231 wikitext text/x-wiki == Lint errors == This template is causing missing end tag, stripped tag, and misnested tag lint errors, and it seems that this is associated with line breaks or multiple blocks in either or both of the main parameters. Here are some pages that use this templates and have such errors: * [[Talk:Sri Sathya Sai International Organization]]: Missing end tag and Stripped tags for {{tag|ins}}. * [[Talk:IHH Healthcare]]: Missing end tag and Stripped tags for {{tag|ins}}. * [[Talk:Dog food]]: Missing end tag and Stripped tags for {{tag|del}}. * [[Talk:American Foundation for Suicide Prevention]]: HTML5-misnesting and Stripped tags for {{tag|ins}}. * [[Template talk:Non-free 2D art]]: HTML5-misnesting and Stripped tags for {{tag|ins}}. If this can fixed, it should be fixed. If it can't be fixed, it should be documented what markup causes these problems. —[[User:Anomalocaris|Anomalocaris]] ([[User talk:Anomalocaris|talk]]) 00:34, 7 October 2023 (UTC) [[User:Jonesey95|Jonesey95]]: Thank you for fixing [[Template:TextDiff/doc]] to acknowledge that this template is good for inline use only. Unfortunately, it is already widely deployed with multi-paragraph arguments, so it would be good if somebody gets this thing to work in such contexts. —[[User:Anomalocaris|Anomalocaris]] ([[User talk:Anomalocaris|talk]]) 22:10, 3 December 2023 (UTC) :As you might have seen, I did some experimenting in the template sandbox and created a testcases page. It was not fruitful, for the most part. I don't know enough Lua or CSS, one of which might be able to help resolve the problem, to install a fix in the module. – [[User:Jonesey95|Jonesey95]] ([[User talk:Jonesey95|talk]]) 23:29, 3 December 2023 (UTC) 0agqqrhnj7g6ek0dj1a6sihcssvnjrk 2813232 2813231 2024-07-08T07:09:28Z ToBeFree 494908 /* The highlighting doesn't always match MediaWiki's */ new section 2813232 wikitext text/x-wiki == Lint errors == This template is causing missing end tag, stripped tag, and misnested tag lint errors, and it seems that this is associated with line breaks or multiple blocks in either or both of the main parameters. Here are some pages that use this templates and have such errors: * [[Talk:Sri Sathya Sai International Organization]]: Missing end tag and Stripped tags for {{tag|ins}}. * [[Talk:IHH Healthcare]]: Missing end tag and Stripped tags for {{tag|ins}}. * [[Talk:Dog food]]: Missing end tag and Stripped tags for {{tag|del}}. * [[Talk:American Foundation for Suicide Prevention]]: HTML5-misnesting and Stripped tags for {{tag|ins}}. * [[Template talk:Non-free 2D art]]: HTML5-misnesting and Stripped tags for {{tag|ins}}. If this can fixed, it should be fixed. If it can't be fixed, it should be documented what markup causes these problems. —[[User:Anomalocaris|Anomalocaris]] ([[User talk:Anomalocaris|talk]]) 00:34, 7 October 2023 (UTC) [[User:Jonesey95|Jonesey95]]: Thank you for fixing [[Template:TextDiff/doc]] to acknowledge that this template is good for inline use only. Unfortunately, it is already widely deployed with multi-paragraph arguments, so it would be good if somebody gets this thing to work in such contexts. —[[User:Anomalocaris|Anomalocaris]] ([[User talk:Anomalocaris|talk]]) 22:10, 3 December 2023 (UTC) :As you might have seen, I did some experimenting in the template sandbox and created a testcases page. It was not fruitful, for the most part. I don't know enough Lua or CSS, one of which might be able to help resolve the problem, to install a fix in the module. – [[User:Jonesey95|Jonesey95]] ([[User talk:Jonesey95|talk]]) 23:29, 3 December 2023 (UTC) == The highlighting doesn't always match MediaWiki's == Example: [https://en.wikipedia.org/w/index.php?title=Wikipedia:Sandbox&diff=prev&oldid=1233276545&diffmode=source] In the example, the template currently highlights "correspondence)," on the right and the comma on the left. However, the comma was untouched and should thus probably not be highlighted. MediaWiki gets it right, the template currently doesn't. [[User:ToBeFree|&#126; ToBeFree]] ([[User talk:ToBeFree|talk]]) 07:09, 8 July 2024 (UTC) iv5e8bk1kygewvv4uearqq4yb5exik2 2813233 2813232 2024-07-08T07:11:16Z ToBeFree 494908 two 2813233 wikitext text/x-wiki == Lint errors == This template is causing missing end tag, stripped tag, and misnested tag lint errors, and it seems that this is associated with line breaks or multiple blocks in either or both of the main parameters. Here are some pages that use this templates and have such errors: * [[Talk:Sri Sathya Sai International Organization]]: Missing end tag and Stripped tags for {{tag|ins}}. * [[Talk:IHH Healthcare]]: Missing end tag and Stripped tags for {{tag|ins}}. * [[Talk:Dog food]]: Missing end tag and Stripped tags for {{tag|del}}. * [[Talk:American Foundation for Suicide Prevention]]: HTML5-misnesting and Stripped tags for {{tag|ins}}. * [[Template talk:Non-free 2D art]]: HTML5-misnesting and Stripped tags for {{tag|ins}}. If this can fixed, it should be fixed. If it can't be fixed, it should be documented what markup causes these problems. —[[User:Anomalocaris|Anomalocaris]] ([[User talk:Anomalocaris|talk]]) 00:34, 7 October 2023 (UTC) [[User:Jonesey95|Jonesey95]]: Thank you for fixing [[Template:TextDiff/doc]] to acknowledge that this template is good for inline use only. Unfortunately, it is already widely deployed with multi-paragraph arguments, so it would be good if somebody gets this thing to work in such contexts. —[[User:Anomalocaris|Anomalocaris]] ([[User talk:Anomalocaris|talk]]) 22:10, 3 December 2023 (UTC) :As you might have seen, I did some experimenting in the template sandbox and created a testcases page. It was not fruitful, for the most part. I don't know enough Lua or CSS, one of which might be able to help resolve the problem, to install a fix in the module. – [[User:Jonesey95|Jonesey95]] ([[User talk:Jonesey95|talk]]) 23:29, 3 December 2023 (UTC) == The highlighting doesn't always match MediaWiki's == Example: [https://en.wikipedia.org/w/index.php?title=Wikipedia:Sandbox&diff=prev&oldid=1233276545&diffmode=source] In the example, the template currently highlights "correspondence)" and the comma. However, both were untouched and should thus probably not be highlighted. MediaWiki gets it right, the template currently doesn't. [[User:ToBeFree|&#126; ToBeFree]] ([[User talk:ToBeFree|talk]]) 07:09, 8 July 2024 (UTC) phd3zjiskdqr70yhm4x5b60qwn2qjyt 2813234 2813233 2024-11-15T01:54:13Z Jonesey95 831143 new thoughts 2813234 wikitext text/x-wiki == Lint errors == This template is causing missing end tag, stripped tag, and misnested tag lint errors, and it seems that this is associated with line breaks or multiple blocks in either or both of the main parameters. Here are some pages that use this templates and have such errors: * [[Talk:Sri Sathya Sai International Organization]]: Missing end tag and Stripped tags for {{tag|ins}}. * [[Talk:IHH Healthcare]]: Missing end tag and Stripped tags for {{tag|ins}}. * [[Talk:Dog food]]: Missing end tag and Stripped tags for {{tag|del}}. * [[Talk:American Foundation for Suicide Prevention]]: HTML5-misnesting and Stripped tags for {{tag|ins}}. * [[Template talk:Non-free 2D art]]: HTML5-misnesting and Stripped tags for {{tag|ins}}. If this can fixed, it should be fixed. If it can't be fixed, it should be documented what markup causes these problems. —[[User:Anomalocaris|Anomalocaris]] ([[User talk:Anomalocaris|talk]]) 00:34, 7 October 2023 (UTC) [[User:Jonesey95|Jonesey95]]: Thank you for fixing [[Template:TextDiff/doc]] to acknowledge that this template is good for inline use only. Unfortunately, it is already widely deployed with multi-paragraph arguments, so it would be good if somebody gets this thing to work in such contexts. —[[User:Anomalocaris|Anomalocaris]] ([[User talk:Anomalocaris|talk]]) 22:10, 3 December 2023 (UTC) :As you might have seen, I did some experimenting in the template sandbox and created a testcases page. It was not fruitful, for the most part. I don't know enough Lua or CSS, one of which might be able to help resolve the problem, to install a fix in the module. – [[User:Jonesey95|Jonesey95]] ([[User talk:Jonesey95|talk]]) 23:29, 3 December 2023 (UTC) ::I am returning to this issue after some time. It still exists. I have replaced the [[Template:Text_diff/testcases#Minimal_test_with_block_content|minimal block-content test case with a smaller example]] that still shows an error. The relevant bit of the rendered wikitext appears to be an untranslated line break after "1947)": <br><syntaxhighlight lang="wikitext"> <div style="word-wrap: break-word; direction: ltr;">Jane Doe<ins class="diffchange diffchange-inline" style="background: var(--background-color-content-added,#a3d3ff); color: inherit; padding: 0.25em 0; font-weight: bold; text-decoration: initial;">&#32;</ins><ins class="diffchange diffchange-inline" style="background: var(--background-color-content-added,#a3d3ff); color: inherit; padding: 0.25em 0; font-weight: bold; text-decoration: initial;">(born</ins>&#32;<ins class="diffchange diffchange-inline" style="background: var(--background-color-content-added,#a3d3ff); color: inherit; padding: 0.25em 0; font-weight: bold; text-decoration: initial;">1947)</ins><ins class="diffchange diffchange-inline" style="background: var(--background-color-content-added,#a3d3ff); color: inherit; padding: 0.25em 0; font-weight: bold; text-decoration: initial;">&#32; &#10;&#10;</ins><ins class="diffchange diffchange-inline" style="background: var(--background-color-content-added,#a3d3ff); color: inherit; padding: 0.25em 0; font-weight: bold; text-decoration: initial;">&#61;&#61;Early</ins> </syntaxhighlight> ::Note the <code><nowiki>&amp;#32; </nowiki></code> (including a space) followed by a line break in the rendered code, then <code><nowiki>&amp;#10;&amp;#10;</ins></nowiki></code>, which represents two encoded line breaks and then a closing ins tag. ::The unencoded line break inside the {{tag|ins}} tag, which is an inline-only tag, as far as I know, may be the problem. I have a hypothesis that if the module code could encode the line break, the {{tag|ins}} tags would no longer contain line breaks and would not complain. If there are any Lua coders who can take a look at the sandbox module code and figure out a way to test this hypothesis, that may be helpful. ::The translation from plain text to encoded text may be performed by [[Extension:Scribunto/Lua_reference_manual#mw.text.nowiki|the Lua mw.text.nowiki]] function, if I am reading the code correctly. It may not be encoding enough of the line breaks. – [[User:Jonesey95|Jonesey95]] ([[User talk:Jonesey95|talk]]) 01:54, 15 November 2024 (UTC) == The highlighting doesn't always match MediaWiki's == Example: [https://en.wikipedia.org/w/index.php?title=Wikipedia:Sandbox&diff=prev&oldid=1233276545&diffmode=source] In the example, the template currently highlights "correspondence)" and the comma. However, both were untouched and should thus probably not be highlighted. MediaWiki gets it right, the template currently doesn't. [[User:ToBeFree|&#126; ToBeFree]] ([[User talk:ToBeFree|talk]]) 07:09, 8 July 2024 (UTC) 3ze8pwp27023ldkiwnhnibomw4ngvre 2813235 2813234 2024-11-15T01:54:34Z Jonesey95 831143 /* Lint errors */ link 2813235 wikitext text/x-wiki == Lint errors == This template is causing missing end tag, stripped tag, and misnested tag lint errors, and it seems that this is associated with line breaks or multiple blocks in either or both of the main parameters. Here are some pages that use this templates and have such errors: * [[Talk:Sri Sathya Sai International Organization]]: Missing end tag and Stripped tags for {{tag|ins}}. * [[Talk:IHH Healthcare]]: Missing end tag and Stripped tags for {{tag|ins}}. * [[Talk:Dog food]]: Missing end tag and Stripped tags for {{tag|del}}. * [[Talk:American Foundation for Suicide Prevention]]: HTML5-misnesting and Stripped tags for {{tag|ins}}. * [[Template talk:Non-free 2D art]]: HTML5-misnesting and Stripped tags for {{tag|ins}}. If this can fixed, it should be fixed. If it can't be fixed, it should be documented what markup causes these problems. —[[User:Anomalocaris|Anomalocaris]] ([[User talk:Anomalocaris|talk]]) 00:34, 7 October 2023 (UTC) [[User:Jonesey95|Jonesey95]]: Thank you for fixing [[Template:TextDiff/doc]] to acknowledge that this template is good for inline use only. Unfortunately, it is already widely deployed with multi-paragraph arguments, so it would be good if somebody gets this thing to work in such contexts. —[[User:Anomalocaris|Anomalocaris]] ([[User talk:Anomalocaris|talk]]) 22:10, 3 December 2023 (UTC) :As you might have seen, I did some experimenting in the template sandbox and created a testcases page. It was not fruitful, for the most part. I don't know enough Lua or CSS, one of which might be able to help resolve the problem, to install a fix in the module. – [[User:Jonesey95|Jonesey95]] ([[User talk:Jonesey95|talk]]) 23:29, 3 December 2023 (UTC) ::I am returning to this issue after some time. It still exists. I have replaced the [[Template:Text_diff/testcases#Minimal_test_with_block_content|minimal block-content test case with a smaller example]] that still shows an error. The relevant bit of the rendered wikitext appears to be an untranslated line break after "1947)": <br><syntaxhighlight lang="wikitext"> <div style="word-wrap: break-word; direction: ltr;">Jane Doe<ins class="diffchange diffchange-inline" style="background: var(--background-color-content-added,#a3d3ff); color: inherit; padding: 0.25em 0; font-weight: bold; text-decoration: initial;">&#32;</ins><ins class="diffchange diffchange-inline" style="background: var(--background-color-content-added,#a3d3ff); color: inherit; padding: 0.25em 0; font-weight: bold; text-decoration: initial;">(born</ins>&#32;<ins class="diffchange diffchange-inline" style="background: var(--background-color-content-added,#a3d3ff); color: inherit; padding: 0.25em 0; font-weight: bold; text-decoration: initial;">1947)</ins><ins class="diffchange diffchange-inline" style="background: var(--background-color-content-added,#a3d3ff); color: inherit; padding: 0.25em 0; font-weight: bold; text-decoration: initial;">&#32; &#10;&#10;</ins><ins class="diffchange diffchange-inline" style="background: var(--background-color-content-added,#a3d3ff); color: inherit; padding: 0.25em 0; font-weight: bold; text-decoration: initial;">&#61;&#61;Early</ins> </syntaxhighlight> ::Note the <code><nowiki>&amp;#32; </nowiki></code> (including a space) followed by a line break in the rendered code, then <code><nowiki>&amp;#10;&amp;#10;</ins></nowiki></code>, which represents two encoded line breaks and then a closing ins tag. ::The unencoded line break inside the {{tag|ins}} tag, which is an inline-only tag, as far as I know, may be the problem. I have a hypothesis that if the module code could encode the line break, the {{tag|ins}} tags would no longer contain line breaks and would not complain. If there are any Lua coders who can take a look at the sandbox module code and figure out a way to test this hypothesis, that may be helpful. ::The translation from plain text to encoded text may be performed by [[:mw:Extension:Scribunto/Lua_reference_manual#mw.text.nowiki|the Lua mw.text.nowiki]] function, if I am reading the code correctly. It may not be encoding enough of the line breaks. – [[User:Jonesey95|Jonesey95]] ([[User talk:Jonesey95|talk]]) 01:54, 15 November 2024 (UTC) == The highlighting doesn't always match MediaWiki's == Example: [https://en.wikipedia.org/w/index.php?title=Wikipedia:Sandbox&diff=prev&oldid=1233276545&diffmode=source] In the example, the template currently highlights "correspondence)" and the comma. However, both were untouched and should thus probably not be highlighted. MediaWiki gets it right, the template currently doesn't. [[User:ToBeFree|&#126; ToBeFree]] ([[User talk:ToBeFree|talk]]) 07:09, 8 July 2024 (UTC) 48nulghck7jutekt005r6nueazdqggc 2813236 2813235 2024-11-15T01:59:26Z Jonesey95 831143 /* Lint errors */ explain more 2813236 wikitext text/x-wiki == Lint errors == This template is causing missing end tag, stripped tag, and misnested tag lint errors, and it seems that this is associated with line breaks or multiple blocks in either or both of the main parameters. Here are some pages that use this templates and have such errors: * [[Talk:Sri Sathya Sai International Organization]]: Missing end tag and Stripped tags for {{tag|ins}}. * [[Talk:IHH Healthcare]]: Missing end tag and Stripped tags for {{tag|ins}}. * [[Talk:Dog food]]: Missing end tag and Stripped tags for {{tag|del}}. * [[Talk:American Foundation for Suicide Prevention]]: HTML5-misnesting and Stripped tags for {{tag|ins}}. * [[Template talk:Non-free 2D art]]: HTML5-misnesting and Stripped tags for {{tag|ins}}. If this can fixed, it should be fixed. If it can't be fixed, it should be documented what markup causes these problems. —[[User:Anomalocaris|Anomalocaris]] ([[User talk:Anomalocaris|talk]]) 00:34, 7 October 2023 (UTC) [[User:Jonesey95|Jonesey95]]: Thank you for fixing [[Template:TextDiff/doc]] to acknowledge that this template is good for inline use only. Unfortunately, it is already widely deployed with multi-paragraph arguments, so it would be good if somebody gets this thing to work in such contexts. —[[User:Anomalocaris|Anomalocaris]] ([[User talk:Anomalocaris|talk]]) 22:10, 3 December 2023 (UTC) :As you might have seen, I did some experimenting in the template sandbox and created a testcases page. It was not fruitful, for the most part. I don't know enough Lua or CSS, one of which might be able to help resolve the problem, to install a fix in the module. – [[User:Jonesey95|Jonesey95]] ([[User talk:Jonesey95|talk]]) 23:29, 3 December 2023 (UTC) ::I am returning to this issue after some time. It still exists. I have replaced the [[Template:Text_diff/testcases#Minimal_test_with_block_content|minimal block-content test case with a smaller example]] that still shows an error. The relevant bit of the rendered wikitext appears to be an untranslated line break after "1947)": <br><syntaxhighlight lang="wikitext"> <div style="word-wrap: break-word; direction: ltr;">Jane Doe<ins class="diffchange diffchange-inline" style="background: var(--background-color-content-added,#a3d3ff); color: inherit; padding: 0.25em 0; font-weight: bold; text-decoration: initial;">&#32;</ins><ins class="diffchange diffchange-inline" style="background: var(--background-color-content-added,#a3d3ff); color: inherit; padding: 0.25em 0; font-weight: bold; text-decoration: initial;">(born</ins>&#32;<ins class="diffchange diffchange-inline" style="background: var(--background-color-content-added,#a3d3ff); color: inherit; padding: 0.25em 0; font-weight: bold; text-decoration: initial;">1947)</ins><ins class="diffchange diffchange-inline" style="background: var(--background-color-content-added,#a3d3ff); color: inherit; padding: 0.25em 0; font-weight: bold; text-decoration: initial;">&#32; &#10;&#10;</ins><ins class="diffchange diffchange-inline" style="background: var(--background-color-content-added,#a3d3ff); color: inherit; padding: 0.25em 0; font-weight: bold; text-decoration: initial;">&#61;&#61;Early</ins> </syntaxhighlight> ::Note the <code><nowiki>&amp;#32; </nowiki></code> (an encoded space followed by an unencoded space) followed by a line break in the rendered code, then <code><nowiki>&amp;#10;&amp;#10;</ins></nowiki></code>, which represents two encoded line breaks and then a closing ins tag. ::The unencoded line break inside the {{tag|ins}} tag, which is an inline-only tag, as far as I know, may be the problem. I have a hypothesis that if the module code could encode the line break, the {{tag|ins}} tags would no longer contain line breaks and would not complain. If there are any Lua coders who can take a look at the sandbox module code and figure out a way to test this hypothesis, that may be helpful. ::The translation from plain text to encoded text may be performed by [[:mw:Extension:Scribunto/Lua_reference_manual#mw.text.nowiki|the Lua mw.text.nowiki]] function, if I am reading the code correctly. It may not be encoding enough of the line breaks. – [[User:Jonesey95|Jonesey95]] ([[User talk:Jonesey95|talk]]) 01:54, 15 November 2024 (UTC) == The highlighting doesn't always match MediaWiki's == Example: [https://en.wikipedia.org/w/index.php?title=Wikipedia:Sandbox&diff=prev&oldid=1233276545&diffmode=source] In the example, the template currently highlights "correspondence)" and the comma. However, both were untouched and should thus probably not be highlighted. MediaWiki gets it right, the template currently doesn't. [[User:ToBeFree|&#126; ToBeFree]] ([[User talk:ToBeFree|talk]]) 07:09, 8 July 2024 (UTC) hq0bynahlm7ot32nyxonjud0qoy6ihy 2813237 2813236 2024-11-15T04:47:54Z Johnuniq 98673 /* Lint errors */ will look 2813237 wikitext text/x-wiki == Lint errors == This template is causing missing end tag, stripped tag, and misnested tag lint errors, and it seems that this is associated with line breaks or multiple blocks in either or both of the main parameters. Here are some pages that use this templates and have such errors: * [[Talk:Sri Sathya Sai International Organization]]: Missing end tag and Stripped tags for {{tag|ins}}. * [[Talk:IHH Healthcare]]: Missing end tag and Stripped tags for {{tag|ins}}. * [[Talk:Dog food]]: Missing end tag and Stripped tags for {{tag|del}}. * [[Talk:American Foundation for Suicide Prevention]]: HTML5-misnesting and Stripped tags for {{tag|ins}}. * [[Template talk:Non-free 2D art]]: HTML5-misnesting and Stripped tags for {{tag|ins}}. If this can fixed, it should be fixed. If it can't be fixed, it should be documented what markup causes these problems. —[[User:Anomalocaris|Anomalocaris]] ([[User talk:Anomalocaris|talk]]) 00:34, 7 October 2023 (UTC) [[User:Jonesey95|Jonesey95]]: Thank you for fixing [[Template:TextDiff/doc]] to acknowledge that this template is good for inline use only. Unfortunately, it is already widely deployed with multi-paragraph arguments, so it would be good if somebody gets this thing to work in such contexts. —[[User:Anomalocaris|Anomalocaris]] ([[User talk:Anomalocaris|talk]]) 22:10, 3 December 2023 (UTC) :As you might have seen, I did some experimenting in the template sandbox and created a testcases page. It was not fruitful, for the most part. I don't know enough Lua or CSS, one of which might be able to help resolve the problem, to install a fix in the module. – [[User:Jonesey95|Jonesey95]] ([[User talk:Jonesey95|talk]]) 23:29, 3 December 2023 (UTC) ::I am returning to this issue after some time. It still exists. I have replaced the [[Template:Text_diff/testcases#Minimal_test_with_block_content|minimal block-content test case with a smaller example]] that still shows an error. The relevant bit of the rendered wikitext appears to be an untranslated line break after "1947)": <br><syntaxhighlight lang="wikitext"> <div style="word-wrap: break-word; direction: ltr;">Jane Doe<ins class="diffchange diffchange-inline" style="background: var(--background-color-content-added,#a3d3ff); color: inherit; padding: 0.25em 0; font-weight: bold; text-decoration: initial;">&#32;</ins><ins class="diffchange diffchange-inline" style="background: var(--background-color-content-added,#a3d3ff); color: inherit; padding: 0.25em 0; font-weight: bold; text-decoration: initial;">(born</ins>&#32;<ins class="diffchange diffchange-inline" style="background: var(--background-color-content-added,#a3d3ff); color: inherit; padding: 0.25em 0; font-weight: bold; text-decoration: initial;">1947)</ins><ins class="diffchange diffchange-inline" style="background: var(--background-color-content-added,#a3d3ff); color: inherit; padding: 0.25em 0; font-weight: bold; text-decoration: initial;">&#32; &#10;&#10;</ins><ins class="diffchange diffchange-inline" style="background: var(--background-color-content-added,#a3d3ff); color: inherit; padding: 0.25em 0; font-weight: bold; text-decoration: initial;">&#61;&#61;Early</ins> </syntaxhighlight> ::Note the <code><nowiki>&amp;#32; </nowiki></code> (an encoded space followed by an unencoded space) followed by a line break in the rendered code, then <code><nowiki>&amp;#10;&amp;#10;</ins></nowiki></code>, which represents two encoded line breaks and then a closing ins tag. ::The unencoded line break inside the {{tag|ins}} tag, which is an inline-only tag, as far as I know, may be the problem. I have a hypothesis that if the module code could encode the line break, the {{tag|ins}} tags would no longer contain line breaks and would not complain. If there are any Lua coders who can take a look at the sandbox module code and figure out a way to test this hypothesis, that may be helpful. ::The translation from plain text to encoded text may be performed by [[:mw:Extension:Scribunto/Lua_reference_manual#mw.text.nowiki|the Lua mw.text.nowiki]] function, if I am reading the code correctly. It may not be encoding enough of the line breaks. – [[User:Jonesey95|Jonesey95]] ([[User talk:Jonesey95|talk]]) 01:54, 15 November 2024 (UTC) :::I'll try to look at this over the weekend. Meanwhile, see [[User:Johnuniq/sandbox2]] where I posted just the minimal test using {{tl|Text diff/sandbox}}, then added something to show the wikitext from the template (it looks ok on a big screen). Page information shows stripped-tag and html5-misnesting errors for ins. [[User:Johnuniq|Johnuniq]] ([[User talk:Johnuniq|talk]]) 04:47, 15 November 2024 (UTC) == The highlighting doesn't always match MediaWiki's == Example: [https://en.wikipedia.org/w/index.php?title=Wikipedia:Sandbox&diff=prev&oldid=1233276545&diffmode=source] In the example, the template currently highlights "correspondence)" and the comma. However, both were untouched and should thus probably not be highlighted. MediaWiki gets it right, the template currently doesn't. [[User:ToBeFree|&#126; ToBeFree]] ([[User talk:ToBeFree|talk]]) 07:09, 8 July 2024 (UTC) jovf8k18zg1ibcat1ihumxhetm16of3 2813238 2813237 2024-11-15T04:55:17Z Jonesey95 831143 /* Lint errors */ accurate? 2813238 wikitext text/x-wiki == Lint errors == This template is causing missing end tag, stripped tag, and misnested tag lint errors, and it seems that this is associated with line breaks or multiple blocks in either or both of the main parameters. Here are some pages that use this templates and have such errors: * [[Talk:Sri Sathya Sai International Organization]]: Missing end tag and Stripped tags for {{tag|ins}}. * [[Talk:IHH Healthcare]]: Missing end tag and Stripped tags for {{tag|ins}}. * [[Talk:Dog food]]: Missing end tag and Stripped tags for {{tag|del}}. * [[Talk:American Foundation for Suicide Prevention]]: HTML5-misnesting and Stripped tags for {{tag|ins}}. * [[Template talk:Non-free 2D art]]: HTML5-misnesting and Stripped tags for {{tag|ins}}. If this can fixed, it should be fixed. If it can't be fixed, it should be documented what markup causes these problems. —[[User:Anomalocaris|Anomalocaris]] ([[User talk:Anomalocaris|talk]]) 00:34, 7 October 2023 (UTC) [[User:Jonesey95|Jonesey95]]: Thank you for fixing [[Template:TextDiff/doc]] to acknowledge that this template is good for inline use only. Unfortunately, it is already widely deployed with multi-paragraph arguments, so it would be good if somebody gets this thing to work in such contexts. —[[User:Anomalocaris|Anomalocaris]] ([[User talk:Anomalocaris|talk]]) 22:10, 3 December 2023 (UTC) :As you might have seen, I did some experimenting in the template sandbox and created a testcases page. It was not fruitful, for the most part. I don't know enough Lua or CSS, one of which might be able to help resolve the problem, to install a fix in the module. – [[User:Jonesey95|Jonesey95]] ([[User talk:Jonesey95|talk]]) 23:29, 3 December 2023 (UTC) ::I am returning to this issue after some time. It still exists. I have replaced the [[Template:Text_diff/testcases#Minimal_test_with_block_content|minimal block-content test case with a smaller example]] that still shows an error. The relevant bit of the rendered wikitext appears to be an untranslated line break after "1947)": <br><syntaxhighlight lang="wikitext"> <div style="word-wrap: break-word; direction: ltr;">Jane Doe<ins class="diffchange diffchange-inline" style="background: var(--background-color-content-added,#a3d3ff); color: inherit; padding: 0.25em 0; font-weight: bold; text-decoration: initial;">&#32;</ins><ins class="diffchange diffchange-inline" style="background: var(--background-color-content-added,#a3d3ff); color: inherit; padding: 0.25em 0; font-weight: bold; text-decoration: initial;">(born</ins>&#32;<ins class="diffchange diffchange-inline" style="background: var(--background-color-content-added,#a3d3ff); color: inherit; padding: 0.25em 0; font-weight: bold; text-decoration: initial;">1947)</ins><ins class="diffchange diffchange-inline" style="background: var(--background-color-content-added,#a3d3ff); color: inherit; padding: 0.25em 0; font-weight: bold; text-decoration: initial;">&#32; &#10;&#10;</ins><ins class="diffchange diffchange-inline" style="background: var(--background-color-content-added,#a3d3ff); color: inherit; padding: 0.25em 0; font-weight: bold; text-decoration: initial;">&#61;&#61;Early</ins> </syntaxhighlight> ::Note the <code><nowiki>&amp;#32; </nowiki></code> (an encoded space followed by an unencoded space) followed by a line break in the rendered code, then <code><nowiki>&amp;#10;&amp;#10;</ins></nowiki></code>, which represents two encoded line breaks and then a closing ins tag. ::The unencoded line break inside the {{tag|ins}} tag, which is an inline-only tag, as far as I know, may be the problem. I have a hypothesis that if the module code could encode the line break, the {{tag|ins}} tags would no longer contain line breaks and would not complain. If there are any Lua coders who can take a look at the sandbox module code and figure out a way to test this hypothesis, that may be helpful. ::The translation from plain text to encoded text may be performed by [[:mw:Extension:Scribunto/Lua_reference_manual#mw.text.nowiki|the Lua mw.text.nowiki]] function, if I am reading the code correctly. It may not be encoding enough of the line breaks. – [[User:Jonesey95|Jonesey95]] ([[User talk:Jonesey95|talk]]) 01:54, 15 November 2024 (UTC) :::I'll try to look at this over the weekend. Meanwhile, see [[User:Johnuniq/sandbox2]] where I posted just the minimal test using {{tl|Text diff/sandbox}}, then added something to show the wikitext from the template (it looks ok on a big screen). Page information shows stripped-tag and html5-misnesting errors for ins. [[User:Johnuniq|Johnuniq]] ([[User talk:Johnuniq|talk]]) 04:47, 15 November 2024 (UTC) ::::Thanks for looking at it. I wonder if that [[:Module:dump]] output is accurate. It does not have a line break in the <syntaxhighlight lang=wikitext>>&#32; &#10;&#10;</ins></syntaxhighlight> code after "1947)" but [[Special:ExpandTemplates]] does have a line break, which is where the stripped-tag and html5-misnesting errors for {{tag|ins}} are. – [[User:Jonesey95|Jonesey95]] ([[User talk:Jonesey95|talk]]) 04:55, 15 November 2024 (UTC) == The highlighting doesn't always match MediaWiki's == Example: [https://en.wikipedia.org/w/index.php?title=Wikipedia:Sandbox&diff=prev&oldid=1233276545&diffmode=source] In the example, the template currently highlights "correspondence)" and the comma. However, both were untouched and should thus probably not be highlighted. MediaWiki gets it right, the template currently doesn't. [[User:ToBeFree|&#126; ToBeFree]] ([[User talk:ToBeFree|talk]]) 07:09, 8 July 2024 (UTC) 6rk6commj52qoncz71ccxbp1jetxqjh 2813239 2813238 2024-11-15T05:53:10Z Johnuniq 98673 /* Lint errors */ dumphtml might not be a good idea 2813239 wikitext text/x-wiki == Lint errors == This template is causing missing end tag, stripped tag, and misnested tag lint errors, and it seems that this is associated with line breaks or multiple blocks in either or both of the main parameters. Here are some pages that use this templates and have such errors: * [[Talk:Sri Sathya Sai International Organization]]: Missing end tag and Stripped tags for {{tag|ins}}. * [[Talk:IHH Healthcare]]: Missing end tag and Stripped tags for {{tag|ins}}. * [[Talk:Dog food]]: Missing end tag and Stripped tags for {{tag|del}}. * [[Talk:American Foundation for Suicide Prevention]]: HTML5-misnesting and Stripped tags for {{tag|ins}}. * [[Template talk:Non-free 2D art]]: HTML5-misnesting and Stripped tags for {{tag|ins}}. If this can fixed, it should be fixed. If it can't be fixed, it should be documented what markup causes these problems. —[[User:Anomalocaris|Anomalocaris]] ([[User talk:Anomalocaris|talk]]) 00:34, 7 October 2023 (UTC) [[User:Jonesey95|Jonesey95]]: Thank you for fixing [[Template:TextDiff/doc]] to acknowledge that this template is good for inline use only. Unfortunately, it is already widely deployed with multi-paragraph arguments, so it would be good if somebody gets this thing to work in such contexts. —[[User:Anomalocaris|Anomalocaris]] ([[User talk:Anomalocaris|talk]]) 22:10, 3 December 2023 (UTC) :As you might have seen, I did some experimenting in the template sandbox and created a testcases page. It was not fruitful, for the most part. I don't know enough Lua or CSS, one of which might be able to help resolve the problem, to install a fix in the module. – [[User:Jonesey95|Jonesey95]] ([[User talk:Jonesey95|talk]]) 23:29, 3 December 2023 (UTC) ::I am returning to this issue after some time. It still exists. I have replaced the [[Template:Text_diff/testcases#Minimal_test_with_block_content|minimal block-content test case with a smaller example]] that still shows an error. The relevant bit of the rendered wikitext appears to be an untranslated line break after "1947)": <br><syntaxhighlight lang="wikitext"> <div style="word-wrap: break-word; direction: ltr;">Jane Doe<ins class="diffchange diffchange-inline" style="background: var(--background-color-content-added,#a3d3ff); color: inherit; padding: 0.25em 0; font-weight: bold; text-decoration: initial;">&#32;</ins><ins class="diffchange diffchange-inline" style="background: var(--background-color-content-added,#a3d3ff); color: inherit; padding: 0.25em 0; font-weight: bold; text-decoration: initial;">(born</ins>&#32;<ins class="diffchange diffchange-inline" style="background: var(--background-color-content-added,#a3d3ff); color: inherit; padding: 0.25em 0; font-weight: bold; text-decoration: initial;">1947)</ins><ins class="diffchange diffchange-inline" style="background: var(--background-color-content-added,#a3d3ff); color: inherit; padding: 0.25em 0; font-weight: bold; text-decoration: initial;">&#32; &#10;&#10;</ins><ins class="diffchange diffchange-inline" style="background: var(--background-color-content-added,#a3d3ff); color: inherit; padding: 0.25em 0; font-weight: bold; text-decoration: initial;">&#61;&#61;Early</ins> </syntaxhighlight> ::Note the <code><nowiki>&amp;#32; </nowiki></code> (an encoded space followed by an unencoded space) followed by a line break in the rendered code, then <code><nowiki>&amp;#10;&amp;#10;</ins></nowiki></code>, which represents two encoded line breaks and then a closing ins tag. ::The unencoded line break inside the {{tag|ins}} tag, which is an inline-only tag, as far as I know, may be the problem. I have a hypothesis that if the module code could encode the line break, the {{tag|ins}} tags would no longer contain line breaks and would not complain. If there are any Lua coders who can take a look at the sandbox module code and figure out a way to test this hypothesis, that may be helpful. ::The translation from plain text to encoded text may be performed by [[:mw:Extension:Scribunto/Lua_reference_manual#mw.text.nowiki|the Lua mw.text.nowiki]] function, if I am reading the code correctly. It may not be encoding enough of the line breaks. – [[User:Jonesey95|Jonesey95]] ([[User talk:Jonesey95|talk]]) 01:54, 15 November 2024 (UTC) :::I'll try to look at this over the weekend. Meanwhile, see [[User:Johnuniq/sandbox2]] where I posted just the minimal test using {{tl|Text diff/sandbox}}, then added something to show the wikitext from the template (it looks ok on a big screen). Page information shows stripped-tag and html5-misnesting errors for ins. [[User:Johnuniq|Johnuniq]] ([[User talk:Johnuniq|talk]]) 04:47, 15 November 2024 (UTC) ::::Thanks for looking at it. I wonder if that [[:Module:dump]] output is accurate. It does not have a line break in the <syntaxhighlight lang=wikitext>>&#32; &#10;&#10;</ins></syntaxhighlight> code after "1947)" but [[Special:ExpandTemplates]] does have a line break, which is where the stripped-tag and html5-misnesting errors for {{tag|ins}} are. – [[User:Jonesey95|Jonesey95]] ([[User talk:Jonesey95|talk]]) 04:55, 15 November 2024 (UTC) :::::Hmmm, it might not be accurate about that because dumphtml tries to show formatted html and it could be skipping a newline. I'll think about that. [[User:Johnuniq|Johnuniq]] ([[User talk:Johnuniq|talk]]) 05:53, 15 November 2024 (UTC) == The highlighting doesn't always match MediaWiki's == Example: [https://en.wikipedia.org/w/index.php?title=Wikipedia:Sandbox&diff=prev&oldid=1233276545&diffmode=source] In the example, the template currently highlights "correspondence)" and the comma. However, both were untouched and should thus probably not be highlighted. MediaWiki gets it right, the template currently doesn't. [[User:ToBeFree|&#126; ToBeFree]] ([[User talk:ToBeFree|talk]]) 07:09, 8 July 2024 (UTC) jz73uf9z8hj6ufgbgdf22crz2xqa6qa 2813240 2813239 2024-11-15T08:06:14Z Aidan9382 2958236 /* Lint errors */ Not the first time 2813240 wikitext text/x-wiki == Lint errors == This template is causing missing end tag, stripped tag, and misnested tag lint errors, and it seems that this is associated with line breaks or multiple blocks in either or both of the main parameters. Here are some pages that use this templates and have such errors: * [[Talk:Sri Sathya Sai International Organization]]: Missing end tag and Stripped tags for {{tag|ins}}. * [[Talk:IHH Healthcare]]: Missing end tag and Stripped tags for {{tag|ins}}. * [[Talk:Dog food]]: Missing end tag and Stripped tags for {{tag|del}}. * [[Talk:American Foundation for Suicide Prevention]]: HTML5-misnesting and Stripped tags for {{tag|ins}}. * [[Template talk:Non-free 2D art]]: HTML5-misnesting and Stripped tags for {{tag|ins}}. If this can fixed, it should be fixed. If it can't be fixed, it should be documented what markup causes these problems. —[[User:Anomalocaris|Anomalocaris]] ([[User talk:Anomalocaris|talk]]) 00:34, 7 October 2023 (UTC) [[User:Jonesey95|Jonesey95]]: Thank you for fixing [[Template:TextDiff/doc]] to acknowledge that this template is good for inline use only. Unfortunately, it is already widely deployed with multi-paragraph arguments, so it would be good if somebody gets this thing to work in such contexts. —[[User:Anomalocaris|Anomalocaris]] ([[User talk:Anomalocaris|talk]]) 22:10, 3 December 2023 (UTC) :As you might have seen, I did some experimenting in the template sandbox and created a testcases page. It was not fruitful, for the most part. I don't know enough Lua or CSS, one of which might be able to help resolve the problem, to install a fix in the module. – [[User:Jonesey95|Jonesey95]] ([[User talk:Jonesey95|talk]]) 23:29, 3 December 2023 (UTC) ::I am returning to this issue after some time. It still exists. I have replaced the [[Template:Text_diff/testcases#Minimal_test_with_block_content|minimal block-content test case with a smaller example]] that still shows an error. The relevant bit of the rendered wikitext appears to be an untranslated line break after "1947)": <br><syntaxhighlight lang="wikitext"> <div style="word-wrap: break-word; direction: ltr;">Jane Doe<ins class="diffchange diffchange-inline" style="background: var(--background-color-content-added,#a3d3ff); color: inherit; padding: 0.25em 0; font-weight: bold; text-decoration: initial;">&#32;</ins><ins class="diffchange diffchange-inline" style="background: var(--background-color-content-added,#a3d3ff); color: inherit; padding: 0.25em 0; font-weight: bold; text-decoration: initial;">(born</ins>&#32;<ins class="diffchange diffchange-inline" style="background: var(--background-color-content-added,#a3d3ff); color: inherit; padding: 0.25em 0; font-weight: bold; text-decoration: initial;">1947)</ins><ins class="diffchange diffchange-inline" style="background: var(--background-color-content-added,#a3d3ff); color: inherit; padding: 0.25em 0; font-weight: bold; text-decoration: initial;">&#32; &#10;&#10;</ins><ins class="diffchange diffchange-inline" style="background: var(--background-color-content-added,#a3d3ff); color: inherit; padding: 0.25em 0; font-weight: bold; text-decoration: initial;">&#61;&#61;Early</ins> </syntaxhighlight> ::Note the <code><nowiki>&amp;#32; </nowiki></code> (an encoded space followed by an unencoded space) followed by a line break in the rendered code, then <code><nowiki>&amp;#10;&amp;#10;</ins></nowiki></code>, which represents two encoded line breaks and then a closing ins tag. ::The unencoded line break inside the {{tag|ins}} tag, which is an inline-only tag, as far as I know, may be the problem. I have a hypothesis that if the module code could encode the line break, the {{tag|ins}} tags would no longer contain line breaks and would not complain. If there are any Lua coders who can take a look at the sandbox module code and figure out a way to test this hypothesis, that may be helpful. ::The translation from plain text to encoded text may be performed by [[:mw:Extension:Scribunto/Lua_reference_manual#mw.text.nowiki|the Lua mw.text.nowiki]] function, if I am reading the code correctly. It may not be encoding enough of the line breaks. – [[User:Jonesey95|Jonesey95]] ([[User talk:Jonesey95|talk]]) 01:54, 15 November 2024 (UTC) :::I'll try to look at this over the weekend. Meanwhile, see [[User:Johnuniq/sandbox2]] where I posted just the minimal test using {{tl|Text diff/sandbox}}, then added something to show the wikitext from the template (it looks ok on a big screen). Page information shows stripped-tag and html5-misnesting errors for ins. [[User:Johnuniq|Johnuniq]] ([[User talk:Johnuniq|talk]]) 04:47, 15 November 2024 (UTC) ::::Thanks for looking at it. I wonder if that [[:Module:dump]] output is accurate. It does not have a line break in the <syntaxhighlight lang=wikitext>>&#32; &#10;&#10;</ins></syntaxhighlight> code after "1947)" but [[Special:ExpandTemplates]] does have a line break, which is where the stripped-tag and html5-misnesting errors for {{tag|ins}} are. – [[User:Jonesey95|Jonesey95]] ([[User talk:Jonesey95|talk]]) 04:55, 15 November 2024 (UTC) :::::Hmmm, it might not be accurate about that because dumphtml tries to show formatted html and it could be skipping a newline. I'll think about that. [[User:Johnuniq|Johnuniq]] ([[User talk:Johnuniq|talk]]) 05:53, 15 November 2024 (UTC) This has also been previously brough up at [[Wikipedia talk:Linter/Archive 3#del and ins tag errors]]. All I really remember is something about newlines being part of the difference was causing issues, though I don't remember (or have) any ideas for a fix. [[User:Aidan9382|Aidan9382]] <sub>([[User talk:Aidan9382|talk]])</sub> 08:06, 15 November 2024 (UTC) == The highlighting doesn't always match MediaWiki's == Example: [https://en.wikipedia.org/w/index.php?title=Wikipedia:Sandbox&diff=prev&oldid=1233276545&diffmode=source] In the example, the template currently highlights "correspondence)" and the comma. However, both were untouched and should thus probably not be highlighted. MediaWiki gets it right, the template currently doesn't. [[User:ToBeFree|&#126; ToBeFree]] ([[User talk:ToBeFree|talk]]) 07:09, 8 July 2024 (UTC) kc601mxbp3y0thfxyh98bo97p01s355 2813241 2813240 2024-11-15T08:15:03Z Aidan9382 2958236 /* Lint errors */ Reply 2813241 wikitext text/x-wiki == Lint errors == This template is causing missing end tag, stripped tag, and misnested tag lint errors, and it seems that this is associated with line breaks or multiple blocks in either or both of the main parameters. Here are some pages that use this templates and have such errors: * [[Talk:Sri Sathya Sai International Organization]]: Missing end tag and Stripped tags for {{tag|ins}}. * [[Talk:IHH Healthcare]]: Missing end tag and Stripped tags for {{tag|ins}}. * [[Talk:Dog food]]: Missing end tag and Stripped tags for {{tag|del}}. * [[Talk:American Foundation for Suicide Prevention]]: HTML5-misnesting and Stripped tags for {{tag|ins}}. * [[Template talk:Non-free 2D art]]: HTML5-misnesting and Stripped tags for {{tag|ins}}. If this can fixed, it should be fixed. If it can't be fixed, it should be documented what markup causes these problems. —[[User:Anomalocaris|Anomalocaris]] ([[User talk:Anomalocaris|talk]]) 00:34, 7 October 2023 (UTC) [[User:Jonesey95|Jonesey95]]: Thank you for fixing [[Template:TextDiff/doc]] to acknowledge that this template is good for inline use only. Unfortunately, it is already widely deployed with multi-paragraph arguments, so it would be good if somebody gets this thing to work in such contexts. —[[User:Anomalocaris|Anomalocaris]] ([[User talk:Anomalocaris|talk]]) 22:10, 3 December 2023 (UTC) :As you might have seen, I did some experimenting in the template sandbox and created a testcases page. It was not fruitful, for the most part. I don't know enough Lua or CSS, one of which might be able to help resolve the problem, to install a fix in the module. – [[User:Jonesey95|Jonesey95]] ([[User talk:Jonesey95|talk]]) 23:29, 3 December 2023 (UTC) ::I am returning to this issue after some time. It still exists. I have replaced the [[Template:Text_diff/testcases#Minimal_test_with_block_content|minimal block-content test case with a smaller example]] that still shows an error. The relevant bit of the rendered wikitext appears to be an untranslated line break after "1947)": <br><syntaxhighlight lang="wikitext"> <div style="word-wrap: break-word; direction: ltr;">Jane Doe<ins class="diffchange diffchange-inline" style="background: var(--background-color-content-added,#a3d3ff); color: inherit; padding: 0.25em 0; font-weight: bold; text-decoration: initial;">&#32;</ins><ins class="diffchange diffchange-inline" style="background: var(--background-color-content-added,#a3d3ff); color: inherit; padding: 0.25em 0; font-weight: bold; text-decoration: initial;">(born</ins>&#32;<ins class="diffchange diffchange-inline" style="background: var(--background-color-content-added,#a3d3ff); color: inherit; padding: 0.25em 0; font-weight: bold; text-decoration: initial;">1947)</ins><ins class="diffchange diffchange-inline" style="background: var(--background-color-content-added,#a3d3ff); color: inherit; padding: 0.25em 0; font-weight: bold; text-decoration: initial;">&#32; &#10;&#10;</ins><ins class="diffchange diffchange-inline" style="background: var(--background-color-content-added,#a3d3ff); color: inherit; padding: 0.25em 0; font-weight: bold; text-decoration: initial;">&#61;&#61;Early</ins> </syntaxhighlight> ::Note the <code><nowiki>&amp;#32; </nowiki></code> (an encoded space followed by an unencoded space) followed by a line break in the rendered code, then <code><nowiki>&amp;#10;&amp;#10;</ins></nowiki></code>, which represents two encoded line breaks and then a closing ins tag. ::The unencoded line break inside the {{tag|ins}} tag, which is an inline-only tag, as far as I know, may be the problem. I have a hypothesis that if the module code could encode the line break, the {{tag|ins}} tags would no longer contain line breaks and would not complain. If there are any Lua coders who can take a look at the sandbox module code and figure out a way to test this hypothesis, that may be helpful. ::The translation from plain text to encoded text may be performed by [[:mw:Extension:Scribunto/Lua_reference_manual#mw.text.nowiki|the Lua mw.text.nowiki]] function, if I am reading the code correctly. It may not be encoding enough of the line breaks. – [[User:Jonesey95|Jonesey95]] ([[User talk:Jonesey95|talk]]) 01:54, 15 November 2024 (UTC) :::I'll try to look at this over the weekend. Meanwhile, see [[User:Johnuniq/sandbox2]] where I posted just the minimal test using {{tl|Text diff/sandbox}}, then added something to show the wikitext from the template (it looks ok on a big screen). Page information shows stripped-tag and html5-misnesting errors for ins. [[User:Johnuniq|Johnuniq]] ([[User talk:Johnuniq|talk]]) 04:47, 15 November 2024 (UTC) ::::Thanks for looking at it. I wonder if that [[:Module:dump]] output is accurate. It does not have a line break in the <syntaxhighlight lang=wikitext>>&#32; &#10;&#10;</ins></syntaxhighlight> code after "1947)" but [[Special:ExpandTemplates]] does have a line break, which is where the stripped-tag and html5-misnesting errors for {{tag|ins}} are. – [[User:Jonesey95|Jonesey95]] ([[User talk:Jonesey95|talk]]) 04:55, 15 November 2024 (UTC) :::::Hmmm, it might not be accurate about that because dumphtml tries to show formatted html and it could be skipping a newline. I'll think about that. [[User:Johnuniq|Johnuniq]] ([[User talk:Johnuniq|talk]]) 05:53, 15 November 2024 (UTC) This has also been previously brough up at [[Wikipedia talk:Linter/Archive 3#del and ins tag errors]]. All I really remember is something about newlines being part of the difference was causing issues, though I don't remember (or have) any ideas for a fix. [[User:Aidan9382|Aidan9382]] <sub>([[User talk:Aidan9382|talk]])</sub> 08:06, 15 November 2024 (UTC) :I've come up with a potential fix idea [[Special:Diff/1257509654|here]]. Not sure if it's the cleanest way of fixing it, but it seems to solve the linter issues and still seems to render fine. [[User:Aidan9382|Aidan9382]] <sub>([[User talk:Aidan9382|talk]])</sub> 08:15, 15 November 2024 (UTC) == The highlighting doesn't always match MediaWiki's == Example: [https://en.wikipedia.org/w/index.php?title=Wikipedia:Sandbox&diff=prev&oldid=1233276545&diffmode=source] In the example, the template currently highlights "correspondence)" and the comma. However, both were untouched and should thus probably not be highlighted. MediaWiki gets it right, the template currently doesn't. [[User:ToBeFree|&#126; ToBeFree]] ([[User talk:ToBeFree|talk]]) 07:09, 8 July 2024 (UTC) g0gvqy5efqv0l80n4pq9f83ebog0qlg 2813242 2813241 2024-11-15T08:22:41Z Aidan9382 2958236 /* Lint errors */ slight difference 2813242 wikitext text/x-wiki == Lint errors == This template is causing missing end tag, stripped tag, and misnested tag lint errors, and it seems that this is associated with line breaks or multiple blocks in either or both of the main parameters. Here are some pages that use this templates and have such errors: * [[Talk:Sri Sathya Sai International Organization]]: Missing end tag and Stripped tags for {{tag|ins}}. * [[Talk:IHH Healthcare]]: Missing end tag and Stripped tags for {{tag|ins}}. * [[Talk:Dog food]]: Missing end tag and Stripped tags for {{tag|del}}. * [[Talk:American Foundation for Suicide Prevention]]: HTML5-misnesting and Stripped tags for {{tag|ins}}. * [[Template talk:Non-free 2D art]]: HTML5-misnesting and Stripped tags for {{tag|ins}}. If this can fixed, it should be fixed. If it can't be fixed, it should be documented what markup causes these problems. —[[User:Anomalocaris|Anomalocaris]] ([[User talk:Anomalocaris|talk]]) 00:34, 7 October 2023 (UTC) [[User:Jonesey95|Jonesey95]]: Thank you for fixing [[Template:TextDiff/doc]] to acknowledge that this template is good for inline use only. Unfortunately, it is already widely deployed with multi-paragraph arguments, so it would be good if somebody gets this thing to work in such contexts. —[[User:Anomalocaris|Anomalocaris]] ([[User talk:Anomalocaris|talk]]) 22:10, 3 December 2023 (UTC) :As you might have seen, I did some experimenting in the template sandbox and created a testcases page. It was not fruitful, for the most part. I don't know enough Lua or CSS, one of which might be able to help resolve the problem, to install a fix in the module. – [[User:Jonesey95|Jonesey95]] ([[User talk:Jonesey95|talk]]) 23:29, 3 December 2023 (UTC) ::I am returning to this issue after some time. It still exists. I have replaced the [[Template:Text_diff/testcases#Minimal_test_with_block_content|minimal block-content test case with a smaller example]] that still shows an error. The relevant bit of the rendered wikitext appears to be an untranslated line break after "1947)": <br><syntaxhighlight lang="wikitext"> <div style="word-wrap: break-word; direction: ltr;">Jane Doe<ins class="diffchange diffchange-inline" style="background: var(--background-color-content-added,#a3d3ff); color: inherit; padding: 0.25em 0; font-weight: bold; text-decoration: initial;">&#32;</ins><ins class="diffchange diffchange-inline" style="background: var(--background-color-content-added,#a3d3ff); color: inherit; padding: 0.25em 0; font-weight: bold; text-decoration: initial;">(born</ins>&#32;<ins class="diffchange diffchange-inline" style="background: var(--background-color-content-added,#a3d3ff); color: inherit; padding: 0.25em 0; font-weight: bold; text-decoration: initial;">1947)</ins><ins class="diffchange diffchange-inline" style="background: var(--background-color-content-added,#a3d3ff); color: inherit; padding: 0.25em 0; font-weight: bold; text-decoration: initial;">&#32; &#10;&#10;</ins><ins class="diffchange diffchange-inline" style="background: var(--background-color-content-added,#a3d3ff); color: inherit; padding: 0.25em 0; font-weight: bold; text-decoration: initial;">&#61;&#61;Early</ins> </syntaxhighlight> ::Note the <code><nowiki>&amp;#32; </nowiki></code> (an encoded space followed by an unencoded space) followed by a line break in the rendered code, then <code><nowiki>&amp;#10;&amp;#10;</ins></nowiki></code>, which represents two encoded line breaks and then a closing ins tag. ::The unencoded line break inside the {{tag|ins}} tag, which is an inline-only tag, as far as I know, may be the problem. I have a hypothesis that if the module code could encode the line break, the {{tag|ins}} tags would no longer contain line breaks and would not complain. If there are any Lua coders who can take a look at the sandbox module code and figure out a way to test this hypothesis, that may be helpful. ::The translation from plain text to encoded text may be performed by [[:mw:Extension:Scribunto/Lua_reference_manual#mw.text.nowiki|the Lua mw.text.nowiki]] function, if I am reading the code correctly. It may not be encoding enough of the line breaks. – [[User:Jonesey95|Jonesey95]] ([[User talk:Jonesey95|talk]]) 01:54, 15 November 2024 (UTC) :::I'll try to look at this over the weekend. Meanwhile, see [[User:Johnuniq/sandbox2]] where I posted just the minimal test using {{tl|Text diff/sandbox}}, then added something to show the wikitext from the template (it looks ok on a big screen). Page information shows stripped-tag and html5-misnesting errors for ins. [[User:Johnuniq|Johnuniq]] ([[User talk:Johnuniq|talk]]) 04:47, 15 November 2024 (UTC) ::::Thanks for looking at it. I wonder if that [[:Module:dump]] output is accurate. It does not have a line break in the <syntaxhighlight lang=wikitext>>&#32; &#10;&#10;</ins></syntaxhighlight> code after "1947)" but [[Special:ExpandTemplates]] does have a line break, which is where the stripped-tag and html5-misnesting errors for {{tag|ins}} are. – [[User:Jonesey95|Jonesey95]] ([[User talk:Jonesey95|talk]]) 04:55, 15 November 2024 (UTC) :::::Hmmm, it might not be accurate about that because dumphtml tries to show formatted html and it could be skipping a newline. I'll think about that. [[User:Johnuniq|Johnuniq]] ([[User talk:Johnuniq|talk]]) 05:53, 15 November 2024 (UTC) This has also been previously brough up at [[Wikipedia talk:Linter/Archive 3#del and ins tag errors]]. All I really remember is something about newlines being part of the difference was causing issues, though I don't remember (or have) any ideas for a fix. [[User:Aidan9382|Aidan9382]] <sub>([[User talk:Aidan9382|talk]])</sub> 08:06, 15 November 2024 (UTC) :I've come up with a potential fix idea [[Special:Diff/1257509654|here]]. Not sure if it's the cleanest way of fixing it, but it seems to solve the linter issues and still seems to render fine, if a little bit different (stacks of newlines are a bit shorter). [[User:Aidan9382|Aidan9382]] <sub>([[User talk:Aidan9382|talk]])</sub> 08:15, 15 November 2024 (UTC) == The highlighting doesn't always match MediaWiki's == Example: [https://en.wikipedia.org/w/index.php?title=Wikipedia:Sandbox&diff=prev&oldid=1233276545&diffmode=source] In the example, the template currently highlights "correspondence)" and the comma. However, both were untouched and should thus probably not be highlighted. MediaWiki gets it right, the template currently doesn't. [[User:ToBeFree|&#126; ToBeFree]] ([[User talk:ToBeFree|talk]]) 07:09, 8 July 2024 (UTC) 4v7u2ikzubooi6pae9zel7tjsuas07a 2813243 2813242 2024-11-15T14:01:04Z Jonesey95 831143 /* Lint errors */ extra line break avoided 2813243 wikitext text/x-wiki == Lint errors == This template is causing missing end tag, stripped tag, and misnested tag lint errors, and it seems that this is associated with line breaks or multiple blocks in either or both of the main parameters. Here are some pages that use this templates and have such errors: * [[Talk:Sri Sathya Sai International Organization]]: Missing end tag and Stripped tags for {{tag|ins}}. * [[Talk:IHH Healthcare]]: Missing end tag and Stripped tags for {{tag|ins}}. * [[Talk:Dog food]]: Missing end tag and Stripped tags for {{tag|del}}. * [[Talk:American Foundation for Suicide Prevention]]: HTML5-misnesting and Stripped tags for {{tag|ins}}. * [[Template talk:Non-free 2D art]]: HTML5-misnesting and Stripped tags for {{tag|ins}}. If this can fixed, it should be fixed. If it can't be fixed, it should be documented what markup causes these problems. —[[User:Anomalocaris|Anomalocaris]] ([[User talk:Anomalocaris|talk]]) 00:34, 7 October 2023 (UTC) [[User:Jonesey95|Jonesey95]]: Thank you for fixing [[Template:TextDiff/doc]] to acknowledge that this template is good for inline use only. Unfortunately, it is already widely deployed with multi-paragraph arguments, so it would be good if somebody gets this thing to work in such contexts. —[[User:Anomalocaris|Anomalocaris]] ([[User talk:Anomalocaris|talk]]) 22:10, 3 December 2023 (UTC) :As you might have seen, I did some experimenting in the template sandbox and created a testcases page. It was not fruitful, for the most part. I don't know enough Lua or CSS, one of which might be able to help resolve the problem, to install a fix in the module. – [[User:Jonesey95|Jonesey95]] ([[User talk:Jonesey95|talk]]) 23:29, 3 December 2023 (UTC) ::I am returning to this issue after some time. It still exists. I have replaced the [[Template:Text_diff/testcases#Minimal_test_with_block_content|minimal block-content test case with a smaller example]] that still shows an error. The relevant bit of the rendered wikitext appears to be an untranslated line break after "1947)": <br><syntaxhighlight lang="wikitext"> <div style="word-wrap: break-word; direction: ltr;">Jane Doe<ins class="diffchange diffchange-inline" style="background: var(--background-color-content-added,#a3d3ff); color: inherit; padding: 0.25em 0; font-weight: bold; text-decoration: initial;">&#32;</ins><ins class="diffchange diffchange-inline" style="background: var(--background-color-content-added,#a3d3ff); color: inherit; padding: 0.25em 0; font-weight: bold; text-decoration: initial;">(born</ins>&#32;<ins class="diffchange diffchange-inline" style="background: var(--background-color-content-added,#a3d3ff); color: inherit; padding: 0.25em 0; font-weight: bold; text-decoration: initial;">1947)</ins><ins class="diffchange diffchange-inline" style="background: var(--background-color-content-added,#a3d3ff); color: inherit; padding: 0.25em 0; font-weight: bold; text-decoration: initial;">&#32; &#10;&#10;</ins><ins class="diffchange diffchange-inline" style="background: var(--background-color-content-added,#a3d3ff); color: inherit; padding: 0.25em 0; font-weight: bold; text-decoration: initial;">&#61;&#61;Early</ins> </syntaxhighlight> ::Note the <code><nowiki>&amp;#32; </nowiki></code> (an encoded space followed by an unencoded space) followed by a line break in the rendered code, then <code><nowiki>&amp;#10;&amp;#10;</ins></nowiki></code>, which represents two encoded line breaks and then a closing ins tag. ::The unencoded line break inside the {{tag|ins}} tag, which is an inline-only tag, as far as I know, may be the problem. I have a hypothesis that if the module code could encode the line break, the {{tag|ins}} tags would no longer contain line breaks and would not complain. If there are any Lua coders who can take a look at the sandbox module code and figure out a way to test this hypothesis, that may be helpful. ::The translation from plain text to encoded text may be performed by [[:mw:Extension:Scribunto/Lua_reference_manual#mw.text.nowiki|the Lua mw.text.nowiki]] function, if I am reading the code correctly. It may not be encoding enough of the line breaks. – [[User:Jonesey95|Jonesey95]] ([[User talk:Jonesey95|talk]]) 01:54, 15 November 2024 (UTC) :::I'll try to look at this over the weekend. Meanwhile, see [[User:Johnuniq/sandbox2]] where I posted just the minimal test using {{tl|Text diff/sandbox}}, then added something to show the wikitext from the template (it looks ok on a big screen). Page information shows stripped-tag and html5-misnesting errors for ins. [[User:Johnuniq|Johnuniq]] ([[User talk:Johnuniq|talk]]) 04:47, 15 November 2024 (UTC) ::::Thanks for looking at it. I wonder if that [[:Module:dump]] output is accurate. It does not have a line break in the <syntaxhighlight lang=wikitext>>&#32; &#10;&#10;</ins></syntaxhighlight> code after "1947)" but [[Special:ExpandTemplates]] does have a line break, which is where the stripped-tag and html5-misnesting errors for {{tag|ins}} are. – [[User:Jonesey95|Jonesey95]] ([[User talk:Jonesey95|talk]]) 04:55, 15 November 2024 (UTC) :::::Hmmm, it might not be accurate about that because dumphtml tries to show formatted html and it could be skipping a newline. I'll think about that. [[User:Johnuniq|Johnuniq]] ([[User talk:Johnuniq|talk]]) 05:53, 15 November 2024 (UTC) This has also been previously brough up at [[Wikipedia talk:Linter/Archive 3#del and ins tag errors]]. All I really remember is something about newlines being part of the difference was causing issues, though I don't remember (or have) any ideas for a fix. [[User:Aidan9382|Aidan9382]] <sub>([[User talk:Aidan9382|talk]])</sub> 08:06, 15 November 2024 (UTC) :I've come up with a potential fix idea [[Special:Diff/1257509654|here]]. Not sure if it's the cleanest way of fixing it, but it seems to solve the linter issues and still seems to render fine, if a little bit different (stacks of newlines are a bit shorter). [[User:Aidan9382|Aidan9382]] <sub>([[User talk:Aidan9382|talk]])</sub> 08:15, 15 November 2024 (UTC) ::That looks like a nice fix that targets the problem and does not affect anything else. It also appears to avoid displaying a spurious extra line break in the resulting diff. In the minimal case, the live template shows three line breaks after "Doe was born" in the diff, but the input contains only two line breaks, as does the sandbox diff. So the fix appears to be correcting a Linter error that was also resulting in a display error; two fixes! I'm happy with this change. I'll let {{U|Johnuniq}} take a look at it before making it live. – [[User:Jonesey95|Jonesey95]] ([[User talk:Jonesey95|talk]]) 14:01, 15 November 2024 (UTC) == The highlighting doesn't always match MediaWiki's == Example: [https://en.wikipedia.org/w/index.php?title=Wikipedia:Sandbox&diff=prev&oldid=1233276545&diffmode=source] In the example, the template currently highlights "correspondence)" and the comma. However, both were untouched and should thus probably not be highlighted. MediaWiki gets it right, the template currently doesn't. [[User:ToBeFree|&#126; ToBeFree]] ([[User talk:ToBeFree|talk]]) 07:09, 8 July 2024 (UTC) iexf1yosv8mpdv3b3frt74f16ipwu3z 2813244 2813243 2024-11-16T02:31:37Z Johnuniq 98673 /* Lint errors */ fix indents? 2813244 wikitext text/x-wiki == Lint errors == This template is causing missing end tag, stripped tag, and misnested tag lint errors, and it seems that this is associated with line breaks or multiple blocks in either or both of the main parameters. Here are some pages that use this templates and have such errors: * [[Talk:Sri Sathya Sai International Organization]]: Missing end tag and Stripped tags for {{tag|ins}}. * [[Talk:IHH Healthcare]]: Missing end tag and Stripped tags for {{tag|ins}}. * [[Talk:Dog food]]: Missing end tag and Stripped tags for {{tag|del}}. * [[Talk:American Foundation for Suicide Prevention]]: HTML5-misnesting and Stripped tags for {{tag|ins}}. * [[Template talk:Non-free 2D art]]: HTML5-misnesting and Stripped tags for {{tag|ins}}. If this can fixed, it should be fixed. If it can't be fixed, it should be documented what markup causes these problems. —[[User:Anomalocaris|Anomalocaris]] ([[User talk:Anomalocaris|talk]]) 00:34, 7 October 2023 (UTC) [[User:Jonesey95|Jonesey95]]: Thank you for fixing [[Template:TextDiff/doc]] to acknowledge that this template is good for inline use only. Unfortunately, it is already widely deployed with multi-paragraph arguments, so it would be good if somebody gets this thing to work in such contexts. —[[User:Anomalocaris|Anomalocaris]] ([[User talk:Anomalocaris|talk]]) 22:10, 3 December 2023 (UTC) :As you might have seen, I did some experimenting in the template sandbox and created a testcases page. It was not fruitful, for the most part. I don't know enough Lua or CSS, one of which might be able to help resolve the problem, to install a fix in the module. – [[User:Jonesey95|Jonesey95]] ([[User talk:Jonesey95|talk]]) 23:29, 3 December 2023 (UTC) ::I am returning to this issue after some time. It still exists. I have replaced the [[Template:Text_diff/testcases#Minimal_test_with_block_content|minimal block-content test case with a smaller example]] that still shows an error. The relevant bit of the rendered wikitext appears to be an untranslated line break after "1947)": <br><syntaxhighlight lang="wikitext"> <div style="word-wrap: break-word; direction: ltr;">Jane Doe<ins class="diffchange diffchange-inline" style="background: var(--background-color-content-added,#a3d3ff); color: inherit; padding: 0.25em 0; font-weight: bold; text-decoration: initial;">&#32;</ins><ins class="diffchange diffchange-inline" style="background: var(--background-color-content-added,#a3d3ff); color: inherit; padding: 0.25em 0; font-weight: bold; text-decoration: initial;">(born</ins>&#32;<ins class="diffchange diffchange-inline" style="background: var(--background-color-content-added,#a3d3ff); color: inherit; padding: 0.25em 0; font-weight: bold; text-decoration: initial;">1947)</ins><ins class="diffchange diffchange-inline" style="background: var(--background-color-content-added,#a3d3ff); color: inherit; padding: 0.25em 0; font-weight: bold; text-decoration: initial;">&#32; &#10;&#10;</ins><ins class="diffchange diffchange-inline" style="background: var(--background-color-content-added,#a3d3ff); color: inherit; padding: 0.25em 0; font-weight: bold; text-decoration: initial;">&#61;&#61;Early</ins> </syntaxhighlight> ::Note the <code><nowiki>&amp;#32; </nowiki></code> (an encoded space followed by an unencoded space) followed by a line break in the rendered code, then <code><nowiki>&amp;#10;&amp;#10;</ins></nowiki></code>, which represents two encoded line breaks and then a closing ins tag. ::The unencoded line break inside the {{tag|ins}} tag, which is an inline-only tag, as far as I know, may be the problem. I have a hypothesis that if the module code could encode the line break, the {{tag|ins}} tags would no longer contain line breaks and would not complain. If there are any Lua coders who can take a look at the sandbox module code and figure out a way to test this hypothesis, that may be helpful. ::The translation from plain text to encoded text may be performed by [[:mw:Extension:Scribunto/Lua_reference_manual#mw.text.nowiki|the Lua mw.text.nowiki]] function, if I am reading the code correctly. It may not be encoding enough of the line breaks. – [[User:Jonesey95|Jonesey95]] ([[User talk:Jonesey95|talk]]) 01:54, 15 November 2024 (UTC) :::I'll try to look at this over the weekend. Meanwhile, see [[User:Johnuniq/sandbox2]] where I posted just the minimal test using {{tl|Text diff/sandbox}}, then added something to show the wikitext from the template (it looks ok on a big screen). Page information shows stripped-tag and html5-misnesting errors for ins. [[User:Johnuniq|Johnuniq]] ([[User talk:Johnuniq|talk]]) 04:47, 15 November 2024 (UTC) ::::Thanks for looking at it. I wonder if that [[:Module:dump]] output is accurate. It does not have a line break in the <syntaxhighlight lang=wikitext>>&#32; &#10;&#10;</ins></syntaxhighlight> code after "1947)" but [[Special:ExpandTemplates]] does have a line break, which is where the stripped-tag and html5-misnesting errors for {{tag|ins}} are. – [[User:Jonesey95|Jonesey95]] ([[User talk:Jonesey95|talk]]) 04:55, 15 November 2024 (UTC) :::::Hmmm, it might not be accurate about that because dumphtml tries to show formatted html and it could be skipping a newline. I'll think about that. [[User:Johnuniq|Johnuniq]] ([[User talk:Johnuniq|talk]]) 05:53, 15 November 2024 (UTC) This has also been previously brough up at [[Wikipedia talk:Linter/Archive 3#del and ins tag errors]]. All I really remember is something about newlines being part of the difference was causing issues, though I don't remember (or have) any ideas for a fix. [[User:Aidan9382|Aidan9382]] <sub>([[User talk:Aidan9382|talk]])</sub> 08:06, 15 November 2024 (UTC) :I've come up with a potential fix idea [[Special:Diff/1257509654|here]]. Not sure if it's the cleanest way of fixing it, but it seems to solve the linter issues and still seems to render fine, if a little bit different (stacks of newlines are a bit shorter). [[User:Aidan9382|Aidan9382]] <sub>([[User talk:Aidan9382|talk]])</sub> 08:15, 15 November 2024 (UTC) ::That looks like a nice fix that targets the problem and does not affect anything else. It also appears to avoid displaying a spurious extra line break in the resulting diff. In the minimal case, the live template shows three line breaks after "Doe was born" in the diff, but the input contains only two line breaks, as does the sandbox diff. So the fix appears to be correcting a Linter error that was also resulting in a display error; two fixes! I'm happy with this change. I'll let {{U|Johnuniq}} take a look at it before making it live. – [[User:Jonesey95|Jonesey95]] ([[User talk:Jonesey95|talk]]) 14:01, 15 November 2024 (UTC) :::I did a trivial edit to update how the original MediaWiki author appears for compatibility with [[c:Module:Diff]], and to remove trailing whitespace. Should I fix the indents to use tabs? That would be more normal but would complicate comparing enwiki version with [[c:Module:Diff]]. I will need another 24 hours to look at the substantive issues. [[User:Johnuniq|Johnuniq]] ([[User talk:Johnuniq|talk]]) 02:31, 16 November 2024 (UTC) == The highlighting doesn't always match MediaWiki's == Example: [https://en.wikipedia.org/w/index.php?title=Wikipedia:Sandbox&diff=prev&oldid=1233276545&diffmode=source] In the example, the template currently highlights "correspondence)" and the comma. However, both were untouched and should thus probably not be highlighted. MediaWiki gets it right, the template currently doesn't. [[User:ToBeFree|&#126; ToBeFree]] ([[User talk:ToBeFree|talk]]) 07:09, 8 July 2024 (UTC) ggln2dei2xvo0wmuvyo9gdrafm6rrqn 2813245 2813244 2024-11-16T03:48:33Z Johnuniq 98673 /* Lint errors */ entab 2813245 wikitext text/x-wiki == Lint errors == This template is causing missing end tag, stripped tag, and misnested tag lint errors, and it seems that this is associated with line breaks or multiple blocks in either or both of the main parameters. Here are some pages that use this templates and have such errors: * [[Talk:Sri Sathya Sai International Organization]]: Missing end tag and Stripped tags for {{tag|ins}}. * [[Talk:IHH Healthcare]]: Missing end tag and Stripped tags for {{tag|ins}}. * [[Talk:Dog food]]: Missing end tag and Stripped tags for {{tag|del}}. * [[Talk:American Foundation for Suicide Prevention]]: HTML5-misnesting and Stripped tags for {{tag|ins}}. * [[Template talk:Non-free 2D art]]: HTML5-misnesting and Stripped tags for {{tag|ins}}. If this can fixed, it should be fixed. If it can't be fixed, it should be documented what markup causes these problems. —[[User:Anomalocaris|Anomalocaris]] ([[User talk:Anomalocaris|talk]]) 00:34, 7 October 2023 (UTC) [[User:Jonesey95|Jonesey95]]: Thank you for fixing [[Template:TextDiff/doc]] to acknowledge that this template is good for inline use only. Unfortunately, it is already widely deployed with multi-paragraph arguments, so it would be good if somebody gets this thing to work in such contexts. —[[User:Anomalocaris|Anomalocaris]] ([[User talk:Anomalocaris|talk]]) 22:10, 3 December 2023 (UTC) :As you might have seen, I did some experimenting in the template sandbox and created a testcases page. It was not fruitful, for the most part. I don't know enough Lua or CSS, one of which might be able to help resolve the problem, to install a fix in the module. – [[User:Jonesey95|Jonesey95]] ([[User talk:Jonesey95|talk]]) 23:29, 3 December 2023 (UTC) ::I am returning to this issue after some time. It still exists. I have replaced the [[Template:Text_diff/testcases#Minimal_test_with_block_content|minimal block-content test case with a smaller example]] that still shows an error. The relevant bit of the rendered wikitext appears to be an untranslated line break after "1947)": <br><syntaxhighlight lang="wikitext"> <div style="word-wrap: break-word; direction: ltr;">Jane Doe<ins class="diffchange diffchange-inline" style="background: var(--background-color-content-added,#a3d3ff); color: inherit; padding: 0.25em 0; font-weight: bold; text-decoration: initial;">&#32;</ins><ins class="diffchange diffchange-inline" style="background: var(--background-color-content-added,#a3d3ff); color: inherit; padding: 0.25em 0; font-weight: bold; text-decoration: initial;">(born</ins>&#32;<ins class="diffchange diffchange-inline" style="background: var(--background-color-content-added,#a3d3ff); color: inherit; padding: 0.25em 0; font-weight: bold; text-decoration: initial;">1947)</ins><ins class="diffchange diffchange-inline" style="background: var(--background-color-content-added,#a3d3ff); color: inherit; padding: 0.25em 0; font-weight: bold; text-decoration: initial;">&#32; &#10;&#10;</ins><ins class="diffchange diffchange-inline" style="background: var(--background-color-content-added,#a3d3ff); color: inherit; padding: 0.25em 0; font-weight: bold; text-decoration: initial;">&#61;&#61;Early</ins> </syntaxhighlight> ::Note the <code><nowiki>&amp;#32; </nowiki></code> (an encoded space followed by an unencoded space) followed by a line break in the rendered code, then <code><nowiki>&amp;#10;&amp;#10;</ins></nowiki></code>, which represents two encoded line breaks and then a closing ins tag. ::The unencoded line break inside the {{tag|ins}} tag, which is an inline-only tag, as far as I know, may be the problem. I have a hypothesis that if the module code could encode the line break, the {{tag|ins}} tags would no longer contain line breaks and would not complain. If there are any Lua coders who can take a look at the sandbox module code and figure out a way to test this hypothesis, that may be helpful. ::The translation from plain text to encoded text may be performed by [[:mw:Extension:Scribunto/Lua_reference_manual#mw.text.nowiki|the Lua mw.text.nowiki]] function, if I am reading the code correctly. It may not be encoding enough of the line breaks. – [[User:Jonesey95|Jonesey95]] ([[User talk:Jonesey95|talk]]) 01:54, 15 November 2024 (UTC) :::I'll try to look at this over the weekend. Meanwhile, see [[User:Johnuniq/sandbox2]] where I posted just the minimal test using {{tl|Text diff/sandbox}}, then added something to show the wikitext from the template (it looks ok on a big screen). Page information shows stripped-tag and html5-misnesting errors for ins. [[User:Johnuniq|Johnuniq]] ([[User talk:Johnuniq|talk]]) 04:47, 15 November 2024 (UTC) ::::Thanks for looking at it. I wonder if that [[:Module:dump]] output is accurate. It does not have a line break in the <syntaxhighlight lang=wikitext>>&#32; &#10;&#10;</ins></syntaxhighlight> code after "1947)" but [[Special:ExpandTemplates]] does have a line break, which is where the stripped-tag and html5-misnesting errors for {{tag|ins}} are. – [[User:Jonesey95|Jonesey95]] ([[User talk:Jonesey95|talk]]) 04:55, 15 November 2024 (UTC) :::::Hmmm, it might not be accurate about that because dumphtml tries to show formatted html and it could be skipping a newline. I'll think about that. [[User:Johnuniq|Johnuniq]] ([[User talk:Johnuniq|talk]]) 05:53, 15 November 2024 (UTC) This has also been previously brough up at [[Wikipedia talk:Linter/Archive 3#del and ins tag errors]]. All I really remember is something about newlines being part of the difference was causing issues, though I don't remember (or have) any ideas for a fix. [[User:Aidan9382|Aidan9382]] <sub>([[User talk:Aidan9382|talk]])</sub> 08:06, 15 November 2024 (UTC) :I've come up with a potential fix idea [[Special:Diff/1257509654|here]]. Not sure if it's the cleanest way of fixing it, but it seems to solve the linter issues and still seems to render fine, if a little bit different (stacks of newlines are a bit shorter). [[User:Aidan9382|Aidan9382]] <sub>([[User talk:Aidan9382|talk]])</sub> 08:15, 15 November 2024 (UTC) ::That looks like a nice fix that targets the problem and does not affect anything else. It also appears to avoid displaying a spurious extra line break in the resulting diff. In the minimal case, the live template shows three line breaks after "Doe was born" in the diff, but the input contains only two line breaks, as does the sandbox diff. So the fix appears to be correcting a Linter error that was also resulting in a display error; two fixes! I'm happy with this change. I'll let {{U|Johnuniq}} take a look at it before making it live. – [[User:Jonesey95|Jonesey95]] ([[User talk:Jonesey95|talk]]) 14:01, 15 November 2024 (UTC) :::I did a trivial edit to update how the original MediaWiki author appears for compatibility with [[c:Module:Diff]], and to remove trailing whitespace. Should I fix the indents to use tabs? That would be more normal but would complicate comparing enwiki version with [[c:Module:Diff]]. I will need another 24 hours to look at the substantive issues. [[User:Johnuniq|Johnuniq]] ([[User talk:Johnuniq|talk]]) 02:31, 16 November 2024 (UTC) ::::Rather than doing any work, I fixed the indents. Revert if unwanted. [[User:Johnuniq|Johnuniq]] ([[User talk:Johnuniq|talk]]) 03:48, 16 November 2024 (UTC) == The highlighting doesn't always match MediaWiki's == Example: [https://en.wikipedia.org/w/index.php?title=Wikipedia:Sandbox&diff=prev&oldid=1233276545&diffmode=source] In the example, the template currently highlights "correspondence)" and the comma. However, both were untouched and should thus probably not be highlighted. MediaWiki gets it right, the template currently doesn't. [[User:ToBeFree|&#126; ToBeFree]] ([[User talk:ToBeFree|talk]]) 07:09, 8 July 2024 (UTC) bnwg1a4juvjzaj3xjrgfaket7fjoazh 2813246 2813245 2024-11-16T14:08:02Z Jonesey95 831143 /* Lint errors */ while we are here 2813246 wikitext text/x-wiki == Lint errors == This template is causing missing end tag, stripped tag, and misnested tag lint errors, and it seems that this is associated with line breaks or multiple blocks in either or both of the main parameters. Here are some pages that use this templates and have such errors: * [[Talk:Sri Sathya Sai International Organization]]: Missing end tag and Stripped tags for {{tag|ins}}. * [[Talk:IHH Healthcare]]: Missing end tag and Stripped tags for {{tag|ins}}. * [[Talk:Dog food]]: Missing end tag and Stripped tags for {{tag|del}}. * [[Talk:American Foundation for Suicide Prevention]]: HTML5-misnesting and Stripped tags for {{tag|ins}}. * [[Template talk:Non-free 2D art]]: HTML5-misnesting and Stripped tags for {{tag|ins}}. If this can fixed, it should be fixed. If it can't be fixed, it should be documented what markup causes these problems. —[[User:Anomalocaris|Anomalocaris]] ([[User talk:Anomalocaris|talk]]) 00:34, 7 October 2023 (UTC) [[User:Jonesey95|Jonesey95]]: Thank you for fixing [[Template:TextDiff/doc]] to acknowledge that this template is good for inline use only. Unfortunately, it is already widely deployed with multi-paragraph arguments, so it would be good if somebody gets this thing to work in such contexts. —[[User:Anomalocaris|Anomalocaris]] ([[User talk:Anomalocaris|talk]]) 22:10, 3 December 2023 (UTC) :As you might have seen, I did some experimenting in the template sandbox and created a testcases page. It was not fruitful, for the most part. I don't know enough Lua or CSS, one of which might be able to help resolve the problem, to install a fix in the module. – [[User:Jonesey95|Jonesey95]] ([[User talk:Jonesey95|talk]]) 23:29, 3 December 2023 (UTC) ::I am returning to this issue after some time. It still exists. I have replaced the [[Template:Text_diff/testcases#Minimal_test_with_block_content|minimal block-content test case with a smaller example]] that still shows an error. The relevant bit of the rendered wikitext appears to be an untranslated line break after "1947)": <br><syntaxhighlight lang="wikitext"> <div style="word-wrap: break-word; direction: ltr;">Jane Doe<ins class="diffchange diffchange-inline" style="background: var(--background-color-content-added,#a3d3ff); color: inherit; padding: 0.25em 0; font-weight: bold; text-decoration: initial;">&#32;</ins><ins class="diffchange diffchange-inline" style="background: var(--background-color-content-added,#a3d3ff); color: inherit; padding: 0.25em 0; font-weight: bold; text-decoration: initial;">(born</ins>&#32;<ins class="diffchange diffchange-inline" style="background: var(--background-color-content-added,#a3d3ff); color: inherit; padding: 0.25em 0; font-weight: bold; text-decoration: initial;">1947)</ins><ins class="diffchange diffchange-inline" style="background: var(--background-color-content-added,#a3d3ff); color: inherit; padding: 0.25em 0; font-weight: bold; text-decoration: initial;">&#32; &#10;&#10;</ins><ins class="diffchange diffchange-inline" style="background: var(--background-color-content-added,#a3d3ff); color: inherit; padding: 0.25em 0; font-weight: bold; text-decoration: initial;">&#61;&#61;Early</ins> </syntaxhighlight> ::Note the <code><nowiki>&amp;#32; </nowiki></code> (an encoded space followed by an unencoded space) followed by a line break in the rendered code, then <code><nowiki>&amp;#10;&amp;#10;</ins></nowiki></code>, which represents two encoded line breaks and then a closing ins tag. ::The unencoded line break inside the {{tag|ins}} tag, which is an inline-only tag, as far as I know, may be the problem. I have a hypothesis that if the module code could encode the line break, the {{tag|ins}} tags would no longer contain line breaks and would not complain. If there are any Lua coders who can take a look at the sandbox module code and figure out a way to test this hypothesis, that may be helpful. ::The translation from plain text to encoded text may be performed by [[:mw:Extension:Scribunto/Lua_reference_manual#mw.text.nowiki|the Lua mw.text.nowiki]] function, if I am reading the code correctly. It may not be encoding enough of the line breaks. – [[User:Jonesey95|Jonesey95]] ([[User talk:Jonesey95|talk]]) 01:54, 15 November 2024 (UTC) :::I'll try to look at this over the weekend. Meanwhile, see [[User:Johnuniq/sandbox2]] where I posted just the minimal test using {{tl|Text diff/sandbox}}, then added something to show the wikitext from the template (it looks ok on a big screen). Page information shows stripped-tag and html5-misnesting errors for ins. [[User:Johnuniq|Johnuniq]] ([[User talk:Johnuniq|talk]]) 04:47, 15 November 2024 (UTC) ::::Thanks for looking at it. I wonder if that [[:Module:dump]] output is accurate. It does not have a line break in the <syntaxhighlight lang=wikitext>>&#32; &#10;&#10;</ins></syntaxhighlight> code after "1947)" but [[Special:ExpandTemplates]] does have a line break, which is where the stripped-tag and html5-misnesting errors for {{tag|ins}} are. – [[User:Jonesey95|Jonesey95]] ([[User talk:Jonesey95|talk]]) 04:55, 15 November 2024 (UTC) :::::Hmmm, it might not be accurate about that because dumphtml tries to show formatted html and it could be skipping a newline. I'll think about that. [[User:Johnuniq|Johnuniq]] ([[User talk:Johnuniq|talk]]) 05:53, 15 November 2024 (UTC) This has also been previously brough up at [[Wikipedia talk:Linter/Archive 3#del and ins tag errors]]. All I really remember is something about newlines being part of the difference was causing issues, though I don't remember (or have) any ideas for a fix. [[User:Aidan9382|Aidan9382]] <sub>([[User talk:Aidan9382|talk]])</sub> 08:06, 15 November 2024 (UTC) :I've come up with a potential fix idea [[Special:Diff/1257509654|here]]. Not sure if it's the cleanest way of fixing it, but it seems to solve the linter issues and still seems to render fine, if a little bit different (stacks of newlines are a bit shorter). [[User:Aidan9382|Aidan9382]] <sub>([[User talk:Aidan9382|talk]])</sub> 08:15, 15 November 2024 (UTC) ::That looks like a nice fix that targets the problem and does not affect anything else. It also appears to avoid displaying a spurious extra line break in the resulting diff. In the minimal case, the live template shows three line breaks after "Doe was born" in the diff, but the input contains only two line breaks, as does the sandbox diff. So the fix appears to be correcting a Linter error that was also resulting in a display error; two fixes! I'm happy with this change. I'll let {{U|Johnuniq}} take a look at it before making it live. – [[User:Jonesey95|Jonesey95]] ([[User talk:Jonesey95|talk]]) 14:01, 15 November 2024 (UTC) :::I did a trivial edit to update how the original MediaWiki author appears for compatibility with [[c:Module:Diff]], and to remove trailing whitespace. Should I fix the indents to use tabs? That would be more normal but would complicate comparing enwiki version with [[c:Module:Diff]]. I will need another 24 hours to look at the substantive issues. [[User:Johnuniq|Johnuniq]] ([[User talk:Johnuniq|talk]]) 02:31, 16 November 2024 (UTC) ::::Rather than doing any work, I fixed the indents. Revert if unwanted. [[User:Johnuniq|Johnuniq]] ([[User talk:Johnuniq|talk]]) 03:48, 16 November 2024 (UTC) ===Block content aligned vertically at the top=== Just for the record, if we are going to support block content, it seemed to me that both blocks should start their text at the top of the table cell, instead of being aligned vertically in the center. I have made a minor change to the td style in the sandbox. You can see the effect in the minimal block test case. This change should have no effect on in-line diffs or on the Linter error described above. – [[User:Jonesey95|Jonesey95]] ([[User talk:Jonesey95|talk]]) 14:08, 16 November 2024 (UTC) == The highlighting doesn't always match MediaWiki's == Example: [https://en.wikipedia.org/w/index.php?title=Wikipedia:Sandbox&diff=prev&oldid=1233276545&diffmode=source] In the example, the template currently highlights "correspondence)" and the comma. However, both were untouched and should thus probably not be highlighted. MediaWiki gets it right, the template currently doesn't. [[User:ToBeFree|&#126; ToBeFree]] ([[User talk:ToBeFree|talk]]) 07:09, 8 July 2024 (UTC) nya10k5logki8lda0yuab7fuybtyske 2813247 2813246 2024-11-17T02:50:36Z Johnuniq 98673 /* Lint errors */ ready 2813247 wikitext text/x-wiki == Lint errors == This template is causing missing end tag, stripped tag, and misnested tag lint errors, and it seems that this is associated with line breaks or multiple blocks in either or both of the main parameters. Here are some pages that use this templates and have such errors: * [[Talk:Sri Sathya Sai International Organization]]: Missing end tag and Stripped tags for {{tag|ins}}. * [[Talk:IHH Healthcare]]: Missing end tag and Stripped tags for {{tag|ins}}. * [[Talk:Dog food]]: Missing end tag and Stripped tags for {{tag|del}}. * [[Talk:American Foundation for Suicide Prevention]]: HTML5-misnesting and Stripped tags for {{tag|ins}}. * [[Template talk:Non-free 2D art]]: HTML5-misnesting and Stripped tags for {{tag|ins}}. If this can fixed, it should be fixed. If it can't be fixed, it should be documented what markup causes these problems. —[[User:Anomalocaris|Anomalocaris]] ([[User talk:Anomalocaris|talk]]) 00:34, 7 October 2023 (UTC) [[User:Jonesey95|Jonesey95]]: Thank you for fixing [[Template:TextDiff/doc]] to acknowledge that this template is good for inline use only. Unfortunately, it is already widely deployed with multi-paragraph arguments, so it would be good if somebody gets this thing to work in such contexts. —[[User:Anomalocaris|Anomalocaris]] ([[User talk:Anomalocaris|talk]]) 22:10, 3 December 2023 (UTC) :As you might have seen, I did some experimenting in the template sandbox and created a testcases page. It was not fruitful, for the most part. I don't know enough Lua or CSS, one of which might be able to help resolve the problem, to install a fix in the module. – [[User:Jonesey95|Jonesey95]] ([[User talk:Jonesey95|talk]]) 23:29, 3 December 2023 (UTC) ::I am returning to this issue after some time. It still exists. I have replaced the [[Template:Text_diff/testcases#Minimal_test_with_block_content|minimal block-content test case with a smaller example]] that still shows an error. The relevant bit of the rendered wikitext appears to be an untranslated line break after "1947)": <br><syntaxhighlight lang="wikitext"> <div style="word-wrap: break-word; direction: ltr;">Jane Doe<ins class="diffchange diffchange-inline" style="background: var(--background-color-content-added,#a3d3ff); color: inherit; padding: 0.25em 0; font-weight: bold; text-decoration: initial;">&#32;</ins><ins class="diffchange diffchange-inline" style="background: var(--background-color-content-added,#a3d3ff); color: inherit; padding: 0.25em 0; font-weight: bold; text-decoration: initial;">(born</ins>&#32;<ins class="diffchange diffchange-inline" style="background: var(--background-color-content-added,#a3d3ff); color: inherit; padding: 0.25em 0; font-weight: bold; text-decoration: initial;">1947)</ins><ins class="diffchange diffchange-inline" style="background: var(--background-color-content-added,#a3d3ff); color: inherit; padding: 0.25em 0; font-weight: bold; text-decoration: initial;">&#32; &#10;&#10;</ins><ins class="diffchange diffchange-inline" style="background: var(--background-color-content-added,#a3d3ff); color: inherit; padding: 0.25em 0; font-weight: bold; text-decoration: initial;">&#61;&#61;Early</ins> </syntaxhighlight> ::Note the <code><nowiki>&amp;#32; </nowiki></code> (an encoded space followed by an unencoded space) followed by a line break in the rendered code, then <code><nowiki>&amp;#10;&amp;#10;</ins></nowiki></code>, which represents two encoded line breaks and then a closing ins tag. ::The unencoded line break inside the {{tag|ins}} tag, which is an inline-only tag, as far as I know, may be the problem. I have a hypothesis that if the module code could encode the line break, the {{tag|ins}} tags would no longer contain line breaks and would not complain. If there are any Lua coders who can take a look at the sandbox module code and figure out a way to test this hypothesis, that may be helpful. ::The translation from plain text to encoded text may be performed by [[:mw:Extension:Scribunto/Lua_reference_manual#mw.text.nowiki|the Lua mw.text.nowiki]] function, if I am reading the code correctly. It may not be encoding enough of the line breaks. – [[User:Jonesey95|Jonesey95]] ([[User talk:Jonesey95|talk]]) 01:54, 15 November 2024 (UTC) :::I'll try to look at this over the weekend. Meanwhile, see [[User:Johnuniq/sandbox2]] where I posted just the minimal test using {{tl|Text diff/sandbox}}, then added something to show the wikitext from the template (it looks ok on a big screen). Page information shows stripped-tag and html5-misnesting errors for ins. [[User:Johnuniq|Johnuniq]] ([[User talk:Johnuniq|talk]]) 04:47, 15 November 2024 (UTC) ::::Thanks for looking at it. I wonder if that [[:Module:dump]] output is accurate. It does not have a line break in the <syntaxhighlight lang=wikitext>>&#32; &#10;&#10;</ins></syntaxhighlight> code after "1947)" but [[Special:ExpandTemplates]] does have a line break, which is where the stripped-tag and html5-misnesting errors for {{tag|ins}} are. – [[User:Jonesey95|Jonesey95]] ([[User talk:Jonesey95|talk]]) 04:55, 15 November 2024 (UTC) :::::Hmmm, it might not be accurate about that because dumphtml tries to show formatted html and it could be skipping a newline. I'll think about that. [[User:Johnuniq|Johnuniq]] ([[User talk:Johnuniq|talk]]) 05:53, 15 November 2024 (UTC) This has also been previously brough up at [[Wikipedia talk:Linter/Archive 3#del and ins tag errors]]. All I really remember is something about newlines being part of the difference was causing issues, though I don't remember (or have) any ideas for a fix. [[User:Aidan9382|Aidan9382]] <sub>([[User talk:Aidan9382|talk]])</sub> 08:06, 15 November 2024 (UTC) :I've come up with a potential fix idea [[Special:Diff/1257509654|here]]. Not sure if it's the cleanest way of fixing it, but it seems to solve the linter issues and still seems to render fine, if a little bit different (stacks of newlines are a bit shorter). [[User:Aidan9382|Aidan9382]] <sub>([[User talk:Aidan9382|talk]])</sub> 08:15, 15 November 2024 (UTC) ::That looks like a nice fix that targets the problem and does not affect anything else. It also appears to avoid displaying a spurious extra line break in the resulting diff. In the minimal case, the live template shows three line breaks after "Doe was born" in the diff, but the input contains only two line breaks, as does the sandbox diff. So the fix appears to be correcting a Linter error that was also resulting in a display error; two fixes! I'm happy with this change. I'll let {{U|Johnuniq}} take a look at it before making it live. – [[User:Jonesey95|Jonesey95]] ([[User talk:Jonesey95|talk]]) 14:01, 15 November 2024 (UTC) :::I did a trivial edit to update how the original MediaWiki author appears for compatibility with [[c:Module:Diff]], and to remove trailing whitespace. Should I fix the indents to use tabs? That would be more normal but would complicate comparing enwiki version with [[c:Module:Diff]]. I will need another 24 hours to look at the substantive issues. [[User:Johnuniq|Johnuniq]] ([[User talk:Johnuniq|talk]]) 02:31, 16 November 2024 (UTC) ::::Rather than doing any work, I fixed the indents. Revert if unwanted. [[User:Johnuniq|Johnuniq]] ([[User talk:Johnuniq|talk]]) 03:48, 16 November 2024 (UTC) ===Block content aligned vertically at the top=== Just for the record, if we are going to support block content, it seemed to me that both blocks should start their text at the top of the table cell, instead of being aligned vertically in the center. I have made a minor change to the td style in the sandbox. You can see the effect in the minimal block test case. This change should have no effect on in-line diffs or on the Linter error described above. – [[User:Jonesey95|Jonesey95]] ([[User talk:Jonesey95|talk]]) 14:08, 16 November 2024 (UTC) :I have finished examining the code and [[Module:Diff/sandbox]] is ready to be deployed. I haven't penetrated the module's innermost secrets but I got a fair understanding of what it is doing. I think Aidan9382's fix is all that's required for now. I wondered whether the algorithm could be improved to properly handle multiple lines but I think that would require quite a lot of pain. I gave up trying to fix the problem apparent in the simpler multiline demo at [[User:Johnuniq/sandbox2]] (it does not show "def hello" as being the same because some excess spaces confuse it). The module works well for its intended use. [[User:Johnuniq|Johnuniq]] ([[User talk:Johnuniq|talk]]) 02:50, 17 November 2024 (UTC) == The highlighting doesn't always match MediaWiki's == Example: [https://en.wikipedia.org/w/index.php?title=Wikipedia:Sandbox&diff=prev&oldid=1233276545&diffmode=source] In the example, the template currently highlights "correspondence)" and the comma. However, both were untouched and should thus probably not be highlighted. MediaWiki gets it right, the template currently doesn't. [[User:ToBeFree|&#126; ToBeFree]] ([[User talk:ToBeFree|talk]]) 07:09, 8 July 2024 (UTC) s9hzcoi0vcfikxt4ex6xrxg17is536o 2813248 2813247 2024-11-17T04:21:05Z Jonesey95 831143 /* Lint errors */ thanks 2813248 wikitext text/x-wiki == Lint errors == This template is causing missing end tag, stripped tag, and misnested tag lint errors, and it seems that this is associated with line breaks or multiple blocks in either or both of the main parameters. Here are some pages that use this templates and have such errors: * [[Talk:Sri Sathya Sai International Organization]]: Missing end tag and Stripped tags for {{tag|ins}}. * [[Talk:IHH Healthcare]]: Missing end tag and Stripped tags for {{tag|ins}}. * [[Talk:Dog food]]: Missing end tag and Stripped tags for {{tag|del}}. * [[Talk:American Foundation for Suicide Prevention]]: HTML5-misnesting and Stripped tags for {{tag|ins}}. * [[Template talk:Non-free 2D art]]: HTML5-misnesting and Stripped tags for {{tag|ins}}. If this can fixed, it should be fixed. If it can't be fixed, it should be documented what markup causes these problems. —[[User:Anomalocaris|Anomalocaris]] ([[User talk:Anomalocaris|talk]]) 00:34, 7 October 2023 (UTC) [[User:Jonesey95|Jonesey95]]: Thank you for fixing [[Template:TextDiff/doc]] to acknowledge that this template is good for inline use only. Unfortunately, it is already widely deployed with multi-paragraph arguments, so it would be good if somebody gets this thing to work in such contexts. —[[User:Anomalocaris|Anomalocaris]] ([[User talk:Anomalocaris|talk]]) 22:10, 3 December 2023 (UTC) :As you might have seen, I did some experimenting in the template sandbox and created a testcases page. It was not fruitful, for the most part. I don't know enough Lua or CSS, one of which might be able to help resolve the problem, to install a fix in the module. – [[User:Jonesey95|Jonesey95]] ([[User talk:Jonesey95|talk]]) 23:29, 3 December 2023 (UTC) ::I am returning to this issue after some time. It still exists. I have replaced the [[Template:Text_diff/testcases#Minimal_test_with_block_content|minimal block-content test case with a smaller example]] that still shows an error. The relevant bit of the rendered wikitext appears to be an untranslated line break after "1947)": <br><syntaxhighlight lang="wikitext"> <div style="word-wrap: break-word; direction: ltr;">Jane Doe<ins class="diffchange diffchange-inline" style="background: var(--background-color-content-added,#a3d3ff); color: inherit; padding: 0.25em 0; font-weight: bold; text-decoration: initial;">&#32;</ins><ins class="diffchange diffchange-inline" style="background: var(--background-color-content-added,#a3d3ff); color: inherit; padding: 0.25em 0; font-weight: bold; text-decoration: initial;">(born</ins>&#32;<ins class="diffchange diffchange-inline" style="background: var(--background-color-content-added,#a3d3ff); color: inherit; padding: 0.25em 0; font-weight: bold; text-decoration: initial;">1947)</ins><ins class="diffchange diffchange-inline" style="background: var(--background-color-content-added,#a3d3ff); color: inherit; padding: 0.25em 0; font-weight: bold; text-decoration: initial;">&#32; &#10;&#10;</ins><ins class="diffchange diffchange-inline" style="background: var(--background-color-content-added,#a3d3ff); color: inherit; padding: 0.25em 0; font-weight: bold; text-decoration: initial;">&#61;&#61;Early</ins> </syntaxhighlight> ::Note the <code><nowiki>&amp;#32; </nowiki></code> (an encoded space followed by an unencoded space) followed by a line break in the rendered code, then <code><nowiki>&amp;#10;&amp;#10;</ins></nowiki></code>, which represents two encoded line breaks and then a closing ins tag. ::The unencoded line break inside the {{tag|ins}} tag, which is an inline-only tag, as far as I know, may be the problem. I have a hypothesis that if the module code could encode the line break, the {{tag|ins}} tags would no longer contain line breaks and would not complain. If there are any Lua coders who can take a look at the sandbox module code and figure out a way to test this hypothesis, that may be helpful. ::The translation from plain text to encoded text may be performed by [[:mw:Extension:Scribunto/Lua_reference_manual#mw.text.nowiki|the Lua mw.text.nowiki]] function, if I am reading the code correctly. It may not be encoding enough of the line breaks. – [[User:Jonesey95|Jonesey95]] ([[User talk:Jonesey95|talk]]) 01:54, 15 November 2024 (UTC) :::I'll try to look at this over the weekend. Meanwhile, see [[User:Johnuniq/sandbox2]] where I posted just the minimal test using {{tl|Text diff/sandbox}}, then added something to show the wikitext from the template (it looks ok on a big screen). Page information shows stripped-tag and html5-misnesting errors for ins. [[User:Johnuniq|Johnuniq]] ([[User talk:Johnuniq|talk]]) 04:47, 15 November 2024 (UTC) ::::Thanks for looking at it. I wonder if that [[:Module:dump]] output is accurate. It does not have a line break in the <syntaxhighlight lang=wikitext>>&#32; &#10;&#10;</ins></syntaxhighlight> code after "1947)" but [[Special:ExpandTemplates]] does have a line break, which is where the stripped-tag and html5-misnesting errors for {{tag|ins}} are. – [[User:Jonesey95|Jonesey95]] ([[User talk:Jonesey95|talk]]) 04:55, 15 November 2024 (UTC) :::::Hmmm, it might not be accurate about that because dumphtml tries to show formatted html and it could be skipping a newline. I'll think about that. [[User:Johnuniq|Johnuniq]] ([[User talk:Johnuniq|talk]]) 05:53, 15 November 2024 (UTC) This has also been previously brough up at [[Wikipedia talk:Linter/Archive 3#del and ins tag errors]]. All I really remember is something about newlines being part of the difference was causing issues, though I don't remember (or have) any ideas for a fix. [[User:Aidan9382|Aidan9382]] <sub>([[User talk:Aidan9382|talk]])</sub> 08:06, 15 November 2024 (UTC) :I've come up with a potential fix idea [[Special:Diff/1257509654|here]]. Not sure if it's the cleanest way of fixing it, but it seems to solve the linter issues and still seems to render fine, if a little bit different (stacks of newlines are a bit shorter). [[User:Aidan9382|Aidan9382]] <sub>([[User talk:Aidan9382|talk]])</sub> 08:15, 15 November 2024 (UTC) ::That looks like a nice fix that targets the problem and does not affect anything else. It also appears to avoid displaying a spurious extra line break in the resulting diff. In the minimal case, the live template shows three line breaks after "Doe was born" in the diff, but the input contains only two line breaks, as does the sandbox diff. So the fix appears to be correcting a Linter error that was also resulting in a display error; two fixes! I'm happy with this change. I'll let {{U|Johnuniq}} take a look at it before making it live. – [[User:Jonesey95|Jonesey95]] ([[User talk:Jonesey95|talk]]) 14:01, 15 November 2024 (UTC) :::I did a trivial edit to update how the original MediaWiki author appears for compatibility with [[c:Module:Diff]], and to remove trailing whitespace. Should I fix the indents to use tabs? That would be more normal but would complicate comparing enwiki version with [[c:Module:Diff]]. I will need another 24 hours to look at the substantive issues. [[User:Johnuniq|Johnuniq]] ([[User talk:Johnuniq|talk]]) 02:31, 16 November 2024 (UTC) ::::Rather than doing any work, I fixed the indents. Revert if unwanted. [[User:Johnuniq|Johnuniq]] ([[User talk:Johnuniq|talk]]) 03:48, 16 November 2024 (UTC) ===Block content aligned vertically at the top=== Just for the record, if we are going to support block content, it seemed to me that both blocks should start their text at the top of the table cell, instead of being aligned vertically in the center. I have made a minor change to the td style in the sandbox. You can see the effect in the minimal block test case. This change should have no effect on in-line diffs or on the Linter error described above. – [[User:Jonesey95|Jonesey95]] ([[User talk:Jonesey95|talk]]) 14:08, 16 November 2024 (UTC) :I have finished examining the code and [[Module:Diff/sandbox]] is ready to be deployed. I haven't penetrated the module's innermost secrets but I got a fair understanding of what it is doing. I think Aidan9382's fix is all that's required for now. I wondered whether the algorithm could be improved to properly handle multiple lines but I think that would require quite a lot of pain. I gave up trying to fix the problem apparent in the simpler multiline demo at [[User:Johnuniq/sandbox2]] (it does not show "def hello" as being the same because some excess spaces confuse it). The module works well for its intended use. [[User:Johnuniq|Johnuniq]] ([[User talk:Johnuniq|talk]]) 02:50, 17 November 2024 (UTC) ::I have updated the live module. Please report any errors here. Thanks to both of you for your clever and quick work! – [[User:Jonesey95|Jonesey95]] ([[User talk:Jonesey95|talk]]) 04:21, 17 November 2024 (UTC) == The highlighting doesn't always match MediaWiki's == Example: [https://en.wikipedia.org/w/index.php?title=Wikipedia:Sandbox&diff=prev&oldid=1233276545&diffmode=source] In the example, the template currently highlights "correspondence)" and the comma. However, both were untouched and should thus probably not be highlighted. MediaWiki gets it right, the template currently doesn't. [[User:ToBeFree|&#126; ToBeFree]] ([[User talk:ToBeFree|talk]]) 07:09, 8 July 2024 (UTC) 0whx7no7jn7n29fppryzhsd7ddstivw 2813249 2813037 2025-10-26T18:08:57Z 2405:6E00:633:E7DF:24DD:ECDE:C536:C0E7 /* Semi-protected edit request on 26 October 2025 */ new section 2813249 wikitext text/x-wiki == Lint errors == This template is causing missing end tag, stripped tag, and misnested tag lint errors, and it seems that this is associated with line breaks or multiple blocks in either or both of the main parameters. Here are some pages that use this templates and have such errors: * [[Talk:Sri Sathya Sai International Organization]]: Missing end tag and Stripped tags for {{tag|ins}}. * [[Talk:IHH Healthcare]]: Missing end tag and Stripped tags for {{tag|ins}}. * [[Talk:Dog food]]: Missing end tag and Stripped tags for {{tag|del}}. * [[Talk:American Foundation for Suicide Prevention]]: HTML5-misnesting and Stripped tags for {{tag|ins}}. * [[Template talk:Non-free 2D art]]: HTML5-misnesting and Stripped tags for {{tag|ins}}. If this can fixed, it should be fixed. If it can't be fixed, it should be documented what markup causes these problems. —[[User:Anomalocaris|Anomalocaris]] ([[User talk:Anomalocaris|talk]]) 00:34, 7 October 2023 (UTC) [[User:Jonesey95|Jonesey95]]: Thank you for fixing [[Template:TextDiff/doc]] to acknowledge that this template is good for inline use only. Unfortunately, it is already widely deployed with multi-paragraph arguments, so it would be good if somebody gets this thing to work in such contexts. —[[User:Anomalocaris|Anomalocaris]] ([[User talk:Anomalocaris|talk]]) 22:10, 3 December 2023 (UTC) :As you might have seen, I did some experimenting in the template sandbox and created a testcases page. It was not fruitful, for the most part. I don't know enough Lua or CSS, one of which might be able to help resolve the problem, to install a fix in the module. – [[User:Jonesey95|Jonesey95]] ([[User talk:Jonesey95|talk]]) 23:29, 3 December 2023 (UTC) ::I am returning to this issue after some time. It still exists. I have replaced the [[Template:Text_diff/testcases#Minimal_test_with_block_content|minimal block-content test case with a smaller example]] that still shows an error. The relevant bit of the rendered wikitext appears to be an untranslated line break after "1947)": <br><syntaxhighlight lang="wikitext"> <div style="word-wrap: break-word; direction: ltr;">Jane Doe<ins class="diffchange diffchange-inline" style="background: var(--background-color-content-added,#a3d3ff); color: inherit; padding: 0.25em 0; font-weight: bold; text-decoration: initial;">&#32;</ins><ins class="diffchange diffchange-inline" style="background: var(--background-color-content-added,#a3d3ff); color: inherit; padding: 0.25em 0; font-weight: bold; text-decoration: initial;">(born</ins>&#32;<ins class="diffchange diffchange-inline" style="background: var(--background-color-content-added,#a3d3ff); color: inherit; padding: 0.25em 0; font-weight: bold; text-decoration: initial;">1947)</ins><ins class="diffchange diffchange-inline" style="background: var(--background-color-content-added,#a3d3ff); color: inherit; padding: 0.25em 0; font-weight: bold; text-decoration: initial;">&#32; &#10;&#10;</ins><ins class="diffchange diffchange-inline" style="background: var(--background-color-content-added,#a3d3ff); color: inherit; padding: 0.25em 0; font-weight: bold; text-decoration: initial;">&#61;&#61;Early</ins> </syntaxhighlight> ::Note the <code><nowiki>&amp;#32; </nowiki></code> (an encoded space followed by an unencoded space) followed by a line break in the rendered code, then <code><nowiki>&amp;#10;&amp;#10;</ins></nowiki></code>, which represents two encoded line breaks and then a closing ins tag. ::The unencoded line break inside the {{tag|ins}} tag, which is an inline-only tag, as far as I know, may be the problem. I have a hypothesis that if the module code could encode the line break, the {{tag|ins}} tags would no longer contain line breaks and would not complain. If there are any Lua coders who can take a look at the sandbox module code and figure out a way to test this hypothesis, that may be helpful. ::The translation from plain text to encoded text may be performed by [[:mw:Extension:Scribunto/Lua_reference_manual#mw.text.nowiki|the Lua mw.text.nowiki]] function, if I am reading the code correctly. It may not be encoding enough of the line breaks. – [[User:Jonesey95|Jonesey95]] ([[User talk:Jonesey95|talk]]) 01:54, 15 November 2024 (UTC) :::I'll try to look at this over the weekend. Meanwhile, see [[User:Johnuniq/sandbox2]] where I posted just the minimal test using {{tl|Text diff/sandbox}}, then added something to show the wikitext from the template (it looks ok on a big screen). Page information shows stripped-tag and html5-misnesting errors for ins. [[User:Johnuniq|Johnuniq]] ([[User talk:Johnuniq|talk]]) 04:47, 15 November 2024 (UTC) ::::Thanks for looking at it. I wonder if that [[:Module:dump]] output is accurate. It does not have a line break in the <syntaxhighlight lang=wikitext>>&#32; &#10;&#10;</ins></syntaxhighlight> code after "1947)" but [[Special:ExpandTemplates]] does have a line break, which is where the stripped-tag and html5-misnesting errors for {{tag|ins}} are. – [[User:Jonesey95|Jonesey95]] ([[User talk:Jonesey95|talk]]) 04:55, 15 November 2024 (UTC) :::::Hmmm, it might not be accurate about that because dumphtml tries to show formatted html and it could be skipping a newline. I'll think about that. [[User:Johnuniq|Johnuniq]] ([[User talk:Johnuniq|talk]]) 05:53, 15 November 2024 (UTC) This has also been previously brough up at [[Wikipedia talk:Linter/Archive 3#del and ins tag errors]]. All I really remember is something about newlines being part of the difference was causing issues, though I don't remember (or have) any ideas for a fix. [[User:Aidan9382|Aidan9382]] <sub>([[User talk:Aidan9382|talk]])</sub> 08:06, 15 November 2024 (UTC) :I've come up with a potential fix idea [[Special:Diff/1257509654|here]]. Not sure if it's the cleanest way of fixing it, but it seems to solve the linter issues and still seems to render fine, if a little bit different (stacks of newlines are a bit shorter). [[User:Aidan9382|Aidan9382]] <sub>([[User talk:Aidan9382|talk]])</sub> 08:15, 15 November 2024 (UTC) ::That looks like a nice fix that targets the problem and does not affect anything else. It also appears to avoid displaying a spurious extra line break in the resulting diff. In the minimal case, the live template shows three line breaks after "Doe was born" in the diff, but the input contains only two line breaks, as does the sandbox diff. So the fix appears to be correcting a Linter error that was also resulting in a display error; two fixes! I'm happy with this change. I'll let {{U|Johnuniq}} take a look at it before making it live. – [[User:Jonesey95|Jonesey95]] ([[User talk:Jonesey95|talk]]) 14:01, 15 November 2024 (UTC) :::I did a trivial edit to update how the original MediaWiki author appears for compatibility with [[c:Module:Diff]], and to remove trailing whitespace. Should I fix the indents to use tabs? That would be more normal but would complicate comparing enwiki version with [[c:Module:Diff]]. I will need another 24 hours to look at the substantive issues. [[User:Johnuniq|Johnuniq]] ([[User talk:Johnuniq|talk]]) 02:31, 16 November 2024 (UTC) ::::Rather than doing any work, I fixed the indents. Revert if unwanted. [[User:Johnuniq|Johnuniq]] ([[User talk:Johnuniq|talk]]) 03:48, 16 November 2024 (UTC) ===Block content aligned vertically at the top=== Just for the record, if we are going to support block content, it seemed to me that both blocks should start their text at the top of the table cell, instead of being aligned vertically in the center. I have made a minor change to the td style in the sandbox. You can see the effect in the minimal block test case. This change should have no effect on in-line diffs or on the Linter error described above. – [[User:Jonesey95|Jonesey95]] ([[User talk:Jonesey95|talk]]) 14:08, 16 November 2024 (UTC) :I have finished examining the code and [[Module:Diff/sandbox]] is ready to be deployed. I haven't penetrated the module's innermost secrets but I got a fair understanding of what it is doing. I think Aidan9382's fix is all that's required for now. I wondered whether the algorithm could be improved to properly handle multiple lines but I think that would require quite a lot of pain. I gave up trying to fix the problem apparent in the simpler multiline demo at [[User:Johnuniq/sandbox2]] (it does not show "def hello" as being the same because some excess spaces confuse it). The module works well for its intended use. [[User:Johnuniq|Johnuniq]] ([[User talk:Johnuniq|talk]]) 02:50, 17 November 2024 (UTC) ::I have updated the live module. Please report any errors here. Thanks to both of you for your clever and quick work! – [[User:Jonesey95|Jonesey95]] ([[User talk:Jonesey95|talk]]) 04:21, 17 November 2024 (UTC) == The highlighting doesn't always match MediaWiki's == Example: [https://en.wikipedia.org/w/index.php?title=Wikipedia:Sandbox&diff=prev&oldid=1233276545&diffmode=source] In the example, the template currently highlights "correspondence)" and the comma. However, both were untouched and should thus probably not be highlighted. MediaWiki gets it right, the template currently doesn't. [[User:ToBeFree|&#126; ToBeFree]] ([[User talk:ToBeFree|talk]]) 07:09, 8 July 2024 (UTC) == Semi-protected edit request on 26 October 2025 == {{edit semi-protected|Template:Text diff|answered=no}} '''Please fix it so that common tools like these don't look so messy.''' {{citation needed}} {{text diff | unsourced text. | unsourced text.{{citation needed}} }} {{additional citation needed}} {{text diff | poorly sourced text. | poorly sourced text.{{additional citation needed}} }} {{langx|de|Bayern}} {{text diff | Bayern | {{langx|de|Bayern}} }} {{Interlanguage link|Bavaria in German Wikipedia|de|Bayern}} {{text diff | Bavaria in German Wikipedia | {{Interlanguage link|Bavaria in German Wikipedia|de|Bayern}} }} {{see also|Bavaria #History}} {{text diff | [[Bavaria #History]] | {{see also|Bavaria #History}} }} [[Special:Contributions/2405:6E00:633:E7DF:24DD:ECDE:C536:C0E7|2405:6E00:633:E7DF:24DD:ECDE:C536:C0E7]] ([[User talk:2405:6E00:633:E7DF:24DD:ECDE:C536:C0E7|talk]]) 18:08, 26 October 2025 (UTC) 7do1r2vtdtsdqffr4iuti6umkz2uw9t 2813250 2813249 2025-10-27T02:14:43Z Jonesey95 831143 /* Semi-protected edit request on 26 October 2025 */ Responded to edit request 2813250 wikitext text/x-wiki == Lint errors == This template is causing missing end tag, stripped tag, and misnested tag lint errors, and it seems that this is associated with line breaks or multiple blocks in either or both of the main parameters. Here are some pages that use this templates and have such errors: * [[Talk:Sri Sathya Sai International Organization]]: Missing end tag and Stripped tags for {{tag|ins}}. * [[Talk:IHH Healthcare]]: Missing end tag and Stripped tags for {{tag|ins}}. * [[Talk:Dog food]]: Missing end tag and Stripped tags for {{tag|del}}. * [[Talk:American Foundation for Suicide Prevention]]: HTML5-misnesting and Stripped tags for {{tag|ins}}. * [[Template talk:Non-free 2D art]]: HTML5-misnesting and Stripped tags for {{tag|ins}}. If this can fixed, it should be fixed. If it can't be fixed, it should be documented what markup causes these problems. —[[User:Anomalocaris|Anomalocaris]] ([[User talk:Anomalocaris|talk]]) 00:34, 7 October 2023 (UTC) [[User:Jonesey95|Jonesey95]]: Thank you for fixing [[Template:TextDiff/doc]] to acknowledge that this template is good for inline use only. Unfortunately, it is already widely deployed with multi-paragraph arguments, so it would be good if somebody gets this thing to work in such contexts. —[[User:Anomalocaris|Anomalocaris]] ([[User talk:Anomalocaris|talk]]) 22:10, 3 December 2023 (UTC) :As you might have seen, I did some experimenting in the template sandbox and created a testcases page. It was not fruitful, for the most part. I don't know enough Lua or CSS, one of which might be able to help resolve the problem, to install a fix in the module. – [[User:Jonesey95|Jonesey95]] ([[User talk:Jonesey95|talk]]) 23:29, 3 December 2023 (UTC) ::I am returning to this issue after some time. It still exists. I have replaced the [[Template:Text_diff/testcases#Minimal_test_with_block_content|minimal block-content test case with a smaller example]] that still shows an error. The relevant bit of the rendered wikitext appears to be an untranslated line break after "1947)": <br><syntaxhighlight lang="wikitext"> <div style="word-wrap: break-word; direction: ltr;">Jane Doe<ins class="diffchange diffchange-inline" style="background: var(--background-color-content-added,#a3d3ff); color: inherit; padding: 0.25em 0; font-weight: bold; text-decoration: initial;">&#32;</ins><ins class="diffchange diffchange-inline" style="background: var(--background-color-content-added,#a3d3ff); color: inherit; padding: 0.25em 0; font-weight: bold; text-decoration: initial;">(born</ins>&#32;<ins class="diffchange diffchange-inline" style="background: var(--background-color-content-added,#a3d3ff); color: inherit; padding: 0.25em 0; font-weight: bold; text-decoration: initial;">1947)</ins><ins class="diffchange diffchange-inline" style="background: var(--background-color-content-added,#a3d3ff); color: inherit; padding: 0.25em 0; font-weight: bold; text-decoration: initial;">&#32; &#10;&#10;</ins><ins class="diffchange diffchange-inline" style="background: var(--background-color-content-added,#a3d3ff); color: inherit; padding: 0.25em 0; font-weight: bold; text-decoration: initial;">&#61;&#61;Early</ins> </syntaxhighlight> ::Note the <code><nowiki>&amp;#32; </nowiki></code> (an encoded space followed by an unencoded space) followed by a line break in the rendered code, then <code><nowiki>&amp;#10;&amp;#10;</ins></nowiki></code>, which represents two encoded line breaks and then a closing ins tag. ::The unencoded line break inside the {{tag|ins}} tag, which is an inline-only tag, as far as I know, may be the problem. I have a hypothesis that if the module code could encode the line break, the {{tag|ins}} tags would no longer contain line breaks and would not complain. If there are any Lua coders who can take a look at the sandbox module code and figure out a way to test this hypothesis, that may be helpful. ::The translation from plain text to encoded text may be performed by [[:mw:Extension:Scribunto/Lua_reference_manual#mw.text.nowiki|the Lua mw.text.nowiki]] function, if I am reading the code correctly. It may not be encoding enough of the line breaks. – [[User:Jonesey95|Jonesey95]] ([[User talk:Jonesey95|talk]]) 01:54, 15 November 2024 (UTC) :::I'll try to look at this over the weekend. Meanwhile, see [[User:Johnuniq/sandbox2]] where I posted just the minimal test using {{tl|Text diff/sandbox}}, then added something to show the wikitext from the template (it looks ok on a big screen). Page information shows stripped-tag and html5-misnesting errors for ins. [[User:Johnuniq|Johnuniq]] ([[User talk:Johnuniq|talk]]) 04:47, 15 November 2024 (UTC) ::::Thanks for looking at it. I wonder if that [[:Module:dump]] output is accurate. It does not have a line break in the <syntaxhighlight lang=wikitext>>&#32; &#10;&#10;</ins></syntaxhighlight> code after "1947)" but [[Special:ExpandTemplates]] does have a line break, which is where the stripped-tag and html5-misnesting errors for {{tag|ins}} are. – [[User:Jonesey95|Jonesey95]] ([[User talk:Jonesey95|talk]]) 04:55, 15 November 2024 (UTC) :::::Hmmm, it might not be accurate about that because dumphtml tries to show formatted html and it could be skipping a newline. I'll think about that. [[User:Johnuniq|Johnuniq]] ([[User talk:Johnuniq|talk]]) 05:53, 15 November 2024 (UTC) This has also been previously brough up at [[Wikipedia talk:Linter/Archive 3#del and ins tag errors]]. All I really remember is something about newlines being part of the difference was causing issues, though I don't remember (or have) any ideas for a fix. [[User:Aidan9382|Aidan9382]] <sub>([[User talk:Aidan9382|talk]])</sub> 08:06, 15 November 2024 (UTC) :I've come up with a potential fix idea [[Special:Diff/1257509654|here]]. Not sure if it's the cleanest way of fixing it, but it seems to solve the linter issues and still seems to render fine, if a little bit different (stacks of newlines are a bit shorter). [[User:Aidan9382|Aidan9382]] <sub>([[User talk:Aidan9382|talk]])</sub> 08:15, 15 November 2024 (UTC) ::That looks like a nice fix that targets the problem and does not affect anything else. It also appears to avoid displaying a spurious extra line break in the resulting diff. In the minimal case, the live template shows three line breaks after "Doe was born" in the diff, but the input contains only two line breaks, as does the sandbox diff. So the fix appears to be correcting a Linter error that was also resulting in a display error; two fixes! I'm happy with this change. I'll let {{U|Johnuniq}} take a look at it before making it live. – [[User:Jonesey95|Jonesey95]] ([[User talk:Jonesey95|talk]]) 14:01, 15 November 2024 (UTC) :::I did a trivial edit to update how the original MediaWiki author appears for compatibility with [[c:Module:Diff]], and to remove trailing whitespace. Should I fix the indents to use tabs? That would be more normal but would complicate comparing enwiki version with [[c:Module:Diff]]. I will need another 24 hours to look at the substantive issues. [[User:Johnuniq|Johnuniq]] ([[User talk:Johnuniq|talk]]) 02:31, 16 November 2024 (UTC) ::::Rather than doing any work, I fixed the indents. Revert if unwanted. [[User:Johnuniq|Johnuniq]] ([[User talk:Johnuniq|talk]]) 03:48, 16 November 2024 (UTC) ===Block content aligned vertically at the top=== Just for the record, if we are going to support block content, it seemed to me that both blocks should start their text at the top of the table cell, instead of being aligned vertically in the center. I have made a minor change to the td style in the sandbox. You can see the effect in the minimal block test case. This change should have no effect on in-line diffs or on the Linter error described above. – [[User:Jonesey95|Jonesey95]] ([[User talk:Jonesey95|talk]]) 14:08, 16 November 2024 (UTC) :I have finished examining the code and [[Module:Diff/sandbox]] is ready to be deployed. I haven't penetrated the module's innermost secrets but I got a fair understanding of what it is doing. I think Aidan9382's fix is all that's required for now. I wondered whether the algorithm could be improved to properly handle multiple lines but I think that would require quite a lot of pain. I gave up trying to fix the problem apparent in the simpler multiline demo at [[User:Johnuniq/sandbox2]] (it does not show "def hello" as being the same because some excess spaces confuse it). The module works well for its intended use. [[User:Johnuniq|Johnuniq]] ([[User talk:Johnuniq|talk]]) 02:50, 17 November 2024 (UTC) ::I have updated the live module. Please report any errors here. Thanks to both of you for your clever and quick work! – [[User:Jonesey95|Jonesey95]] ([[User talk:Jonesey95|talk]]) 04:21, 17 November 2024 (UTC) == The highlighting doesn't always match MediaWiki's == Example: [https://en.wikipedia.org/w/index.php?title=Wikipedia:Sandbox&diff=prev&oldid=1233276545&diffmode=source] In the example, the template currently highlights "correspondence)" and the comma. However, both were untouched and should thus probably not be highlighted. MediaWiki gets it right, the template currently doesn't. [[User:ToBeFree|&#126; ToBeFree]] ([[User talk:ToBeFree|talk]]) 07:09, 8 July 2024 (UTC) == Semi-protected edit request on 26 October 2025 == {{edit semi-protected|Template:Text diff|answered=yes}} '''Please fix it so that common tools like these don't look so messy.''' {{citation needed}} {{text diff | unsourced text. | unsourced text.{{citation needed}} }} {{additional citation needed}} {{text diff | poorly sourced text. | poorly sourced text.{{additional citation needed}} }} {{langx|de|Bayern}} {{text diff | Bayern | {{langx|de|Bayern}} }} {{Interlanguage link|Bavaria in German Wikipedia|de|Bayern}} {{text diff | Bavaria in German Wikipedia | {{Interlanguage link|Bavaria in German Wikipedia|de|Bayern}} }} {{see also|Bavaria #History}} {{text diff | [[Bavaria #History]] | {{see also|Bavaria #History}} }} [[Special:Contributions/2405:6E00:633:E7DF:24DD:ECDE:C536:C0E7|2405:6E00:633:E7DF:24DD:ECDE:C536:C0E7]] ([[User talk:2405:6E00:633:E7DF:24DD:ECDE:C536:C0E7|talk]]) 18:08, 26 October 2025 (UTC) :{{Not done}}''':''' this is the [[Help:Talk pages|talk page]] for discussing improvements to the [[Help:Template|template]] {{tlx|Text diff}}. Please make your request at the talk page for the article concerned.<!-- Template:ESp --> – [[User:Jonesey95|Jonesey95]] ([[User talk:Jonesey95|talk]]) 02:14, 27 October 2025 (UTC) 2bd5ckbxh9e2acsgbqgdsoecih8w8cn 2813251 2813039 2026-06-06T08:30:24Z Jtneill 10242 30 revisions imported from [[:w:Template_talk:Text_diff]] 2813037 wikitext text/x-wiki <!-- This user-input check is for [[Template:Request edit button/preload]]. -->{{#ifeq:{{{1}}}|ORIGINAL_WIKITEXT|{{warning sign}} Unnamed parameter {{para|1}} set to default value. Please change it. Failure to use {{tl|Text diff}} to specify your requested text changes, if not adequately described above, may lead to your request being denied.<br/>|{{#ifeq:{{{2}}}|CHANGED_WIKITEXT|{{warning sign}} Unnamed parameter {{para|2}} set to default value. Please change it. Failure to use {{tl|Text diff}} to specify your requested text changes, if not adequately described above, may lead to your request being denied.<br/>|{{#invoke:Diff|main|{{{1|This is a test}}}|{{{2|This was a test!}}}}}}}}}<noinclude> {{documentation}} </noinclude> reqb6byw8n5x095bbokxwb3qkqru8j4 Template:Documentation/styles.css 10 329994 2813091 2019-08-04T17:04:12Z Izno 21264 start a page 2813091 sanitized-css text/css .template-documentation { clear: both; margin: 1em 0 0 0; border: 1px solid #a2a9b1; background-color: #ecfcf4; padding: 1em; } 41wnji59p5r8ytvr1n39t1dz3y04e0x 2813092 2813091 2019-08-04T17:25:25Z Izno 21264 add some more 2813092 sanitized-css text/css .template-documentation { clear: both; margin: 1em 0 0 0; border: 1px solid #a2a9b1; background-color: #ecfcf4; padding: 1em; } .template-documentation-heading { } #documentation-meta-data { background-color: #ecfcf4; } #documentation-meta-data .mbox-text { font-style: italic; } 9ljry4odfzvve50d7nk9kyz7xlm2mi7 2813093 2813092 2019-08-04T17:34:21Z Izno 21264 add end here 2813093 sanitized-css text/css .template-documentation { clear: both; margin: 1em 0 0 0; border: 1px solid #a2a9b1; background-color: #ecfcf4; padding: 1em; } .template-documentation-heading { } /* So right or left floating items don't stick out of the doc box. */ .template-documentation-end { clear: both; } #documentation-meta-data { background-color: #ecfcf4; } #documentation-meta-data .mbox-text { font-style: italic; } 3x4zh5wqa6ksu9vecobtgxcrdovqaj8 2813094 2813093 2019-08-04T17:41:20Z Izno 21264 add 2813094 sanitized-css text/css .template-documentation { clear: both; margin: 1em 0 0 0; border: 1px solid #a2a9b1; background-color: #ecfcf4; padding: 1em; } .template-documentation-heading { } /* So right or left floating items don't stick out of the doc box. */ .template-documentation-sandbox-start, .template-documentation-end { clear: both; } #documentation-meta-data { background-color: #ecfcf4; } #documentation-meta-data .mbox-text { font-style: italic; } 3g6xz5pxkqs6qlp5ji4ux05itl64mof 2813095 2813094 2019-08-04T17:48:30Z Izno 21264 add small 2813095 sanitized-css text/css .template-documentation { clear: both; margin: 1em 0 0 0; border: 1px solid #a2a9b1; background-color: #ecfcf4; padding: 1em; } .template-documentation-heading { } /* So right or left floating items don't stick out of the doc box. */ .template-documentation-sandbox-start, .template-documentation-end { clear: both; } #documentation-meta-data { background-color: #ecfcf4; } #documentation-meta-data .mbox-text { font-style: italic; } #documentation-meta-data .mbox-text .template-documentation-toolbar-link { font-style: normal; font-size: 85%; } 95erhlsepx5auczp07bx6vasbhyx6go 2813096 2813095 2019-08-04T18:10:10Z Izno 21264 add these styles 2813096 sanitized-css text/css .template-documentation { clear: both; margin: 1em 0 0 0; border: 1px solid #a2a9b1; background-color: #ecfcf4; padding: 1em; } .template-documentation-startbox { padding-bottom: 3px; border-bottom: 1px solid #aaa; margin-bottom: 1ex; } .template-documentation-startbox .template-heading { font-weight: bold; font-size: 125%; } .template-documentation-startbox .other-heading { font-size: 150%; } /* So right or left floating items don't stick out of the doc box. */ .template-documentation-sandbox-start, .template-documentation-end { clear: both; } #documentation-meta-data { background-color: #ecfcf4; } #documentation-meta-data .mbox-text { font-style: italic; } #documentation-meta-data .mbox-text .template-documentation-toolbar-link { font-style: normal; font-size: 85%; } 2uk921kosolrp6smt0k1ia1vwzws7ts 2813097 2813096 2019-08-04T18:10:43Z Izno 21264 move that doc 2813097 sanitized-css text/css .template-documentation { clear: both; margin: 1em 0 0 0; border: 1px solid #a2a9b1; background-color: #ecfcf4; padding: 1em; } .template-documentation-startbox { padding-bottom: 3px; border-bottom: 1px solid #aaa; margin-bottom: 1ex; } .template-documentation-startbox .template-heading { font-weight: bold; font-size: 125%; } .template-documentation-startbox .other-heading { font-size: 150%; } .template-documentation-sandbox-start, .template-documentation-end /* So right or left floating items don't stick out of the doc box.*/ { clear: both; } #documentation-meta-data { background-color: #ecfcf4; } #documentation-meta-data .mbox-text { font-style: italic; } #documentation-meta-data .mbox-text .template-documentation-toolbar-link { font-style: normal; font-size: 85%; } psfpjqsoahj384d97roqggub6zirhti 2813098 2813097 2020-11-11T21:05:28Z Izno 21264 rework this just to convince myself 2813098 sanitized-css text/css .template-documentation { clear: both; margin: 1em 0 0 0; border: 1px solid #a2a9b1; background-color: #ecfcf4; padding: 1em; } /* .template-documentation-startbox { padding-bottom: 3px; border-bottom: 1px solid #aaa; margin-bottom: 1ex; } .template-documentation-startbox .template-heading { font-weight: bold; font-size: 125%; } .template-documentation-startbox .other-heading { font-size: 150%; } .template-documentation-sandbox-start, .template-documentation-end /* So right or left floating items don't stick out of the doc box.*/ /* { clear: both; } */ #documentation-meta-data { background-color: #ecfcf4; } #documentation-meta-data .mbox-text { font-style: italic; } #documentation-meta-data .mbox-text .template-documentation-toolbar-link { font-style: normal; font-size: 85%; } e2qcmz43fmmvawa38bjj7o81kcpdrq3 2813099 2813098 2020-11-11T21:22:43Z Izno 21264 and resave as this 2813099 sanitized-css text/css .template-documentation { clear: both; margin: 1em 0 0 0; border: 1px solid #a2a9b1; background-color: #ecfcf4; padding: 1em; } .template-documentation-startbox { padding-bottom: 3px; border-bottom: 1px solid #aaa; margin-bottom: 1ex; } .template-documentation-startbox .template-heading { font-weight: bold; font-size: 125%; } .template-documentation-startbox .other-heading { font-size: 150%; } .template-documentation-sandbox-start, /* So right or left floating items don't stick out of the doc box.*/ .template-documentation-end { clear: both; } #documentation-meta-data { background-color: #ecfcf4; } #documentation-meta-data .mbox-text { font-style: italic; } #documentation-meta-data .mbox-text .template-documentation-toolbar-link { font-style: normal; font-size: 85%; } ltc8s18ojauyraj1y3dbtyoua1mgdux 2813100 2813099 2020-11-12T04:32:27Z Izno 21264 consistent styles for main and metadata boxes 2813100 sanitized-css text/css .template-documentation, .template-documentation-metadata { border: 1px solid #a2a9b1; background-color: #ecfcf4; padding: 1em; clear: both; } .template-documentation { margin: 1em 0 0 0; } .template-documentation-startbox { padding-bottom: 3px; border-bottom: 1px solid #aaa; margin-bottom: 1ex; } .template-documentation-heading { font-weight: bold; font-size: 125%; } .template-documentation-other-heading { font-size: 150%; } .template-documentation-sandbox-start, /* So right or left floating items don't stick out of the doc box.*/ .template-documentation-end { clear: both; } .template-documentation-metadata { margin: 0.2em 0; font-style: italic; } .template-documentation-toolbar { font-style: normal; font-size: 85%; } 26bud8eoddt0pvia0zp1pyu6bhkuhkf 2813101 2813100 2020-11-12T05:14:10Z Izno 21264 like this padding better, indeed; sync right-left tho 2813101 sanitized-css text/css .template-documentation, .template-documentation-metadata { border: 1px solid #a2a9b1; background-color: #ecfcf4; clear: both; } .template-documentation { margin: 1em 0 0 0; padding: 1em; } .template-documentation-startbox { padding-bottom: 3px; border-bottom: 1px solid #aaa; margin-bottom: 1ex; } .template-documentation-heading { font-weight: bold; font-size: 125%; } .template-documentation-other-heading { font-size: 150%; } .template-documentation-sandbox-start, /* So right or left floating items don't stick out of the doc box.*/ .template-documentation-end { clear: both; } .template-documentation-metadata { margin: 0.2em 0; font-style: italic; padding: 0.25em 1em; } .template-documentation-toolbar { font-style: normal; font-size: 85%; } 95m6lv0e4hzbwrrya7tujxzk4cmpg95 2813102 2813101 2020-11-12T05:16:18Z Izno 21264 bump to an even 0.3 2813102 sanitized-css text/css .template-documentation, .template-documentation-metadata { border: 1px solid #a2a9b1; background-color: #ecfcf4; clear: both; } .template-documentation { margin: 1em 0 0 0; padding: 1em; } .template-documentation-startbox { padding-bottom: 3px; border-bottom: 1px solid #aaa; margin-bottom: 1ex; } .template-documentation-heading { font-weight: bold; font-size: 125%; } .template-documentation-other-heading { font-size: 150%; } .template-documentation-sandbox-start, /* So right or left floating items don't stick out of the doc box.*/ .template-documentation-end { clear: both; } .template-documentation-metadata { margin: 0.2em 0; font-style: italic; padding: 0.3em 1em; } .template-documentation-toolbar { font-style: normal; font-size: 85%; } j06bjsmnc1fgvbfazhpqnlu8rb1iskq 2813103 2813102 2020-11-12T06:34:28Z Izno 21264 make this just for the spans 2813103 sanitized-css text/css .template-documentation, .template-documentation-metadata { border: 1px solid #a2a9b1; background-color: #ecfcf4; clear: both; } .template-documentation { margin: 1em 0 0 0; padding: 1em; } .template-documentation-startbox { padding-bottom: 3px; border-bottom: 1px solid #aaa; margin-bottom: 1ex; } .template-documentation-heading { font-weight: bold; font-size: 125%; } .template-documentation-other-heading { font-size: 150%; } .template-documentation-sandbox-start, /* So right or left floating items don't stick out of the doc box.*/ .template-documentation-end { clear: both; } .template-documentation-metadata { margin: 0.2em 0; font-style: italic; padding: 0.3em 1em; } .template-documentation-metadata > span { font-style: normal; font-size: 85%; } nvqt1si94e1tdmgpzrr9wltqabfx17h 2813104 2813103 2020-11-12T06:35:53Z Izno 21264 Undid revision 988284245 by [[Special:Contributions/Izno|Izno]] ([[User talk:Izno|talk]]) nope 2813104 sanitized-css text/css .template-documentation, .template-documentation-metadata { border: 1px solid #a2a9b1; background-color: #ecfcf4; clear: both; } .template-documentation { margin: 1em 0 0 0; padding: 1em; } .template-documentation-startbox { padding-bottom: 3px; border-bottom: 1px solid #aaa; margin-bottom: 1ex; } .template-documentation-heading { font-weight: bold; font-size: 125%; } .template-documentation-other-heading { font-size: 150%; } .template-documentation-sandbox-start, /* So right or left floating items don't stick out of the doc box.*/ .template-documentation-end { clear: both; } .template-documentation-metadata { margin: 0.2em 0; font-style: italic; padding: 0.3em 1em; } .template-documentation-toolbar { font-style: normal; font-size: 85%; } j06bjsmnc1fgvbfazhpqnlu8rb1iskq 2813105 2813104 2020-11-12T06:47:18Z Izno 21264 and bump padding 2813105 sanitized-css text/css .template-documentation, .template-documentation-metadata { border: 1px solid #a2a9b1; background-color: #ecfcf4; clear: both; } .template-documentation { margin: 1em 0 0 0; padding: 1em; } .template-documentation-startbox { padding-bottom: 3px; border-bottom: 1px solid #aaa; margin-bottom: 1ex; } .template-documentation-heading { font-weight: bold; font-size: 125%; } .template-documentation-other-heading { font-size: 150%; } .template-documentation-clear { /* Don't want things to stick out where they shouldn't. */ clear: both; } .template-documentation-metadata { margin: 0.2em 0; font-style: italic; padding: 0.4em 1em; } .template-documentation-toolbar { font-style: normal; font-size: 85%; } fygw83ot86d2f5rcbmg6wo3yfqhanue 2813106 2813105 2020-11-12T06:51:08Z Izno 21264 Izno moved page [[Module:Documentation/sandbox/styles.css]] to [[Module:Documentation/styles.css]] without leaving a redirect 2813105 sanitized-css text/css .template-documentation, .template-documentation-metadata { border: 1px solid #a2a9b1; background-color: #ecfcf4; clear: both; } .template-documentation { margin: 1em 0 0 0; padding: 1em; } .template-documentation-startbox { padding-bottom: 3px; border-bottom: 1px solid #aaa; margin-bottom: 1ex; } .template-documentation-heading { font-weight: bold; font-size: 125%; } .template-documentation-other-heading { font-size: 150%; } .template-documentation-clear { /* Don't want things to stick out where they shouldn't. */ clear: both; } .template-documentation-metadata { margin: 0.2em 0; font-style: italic; padding: 0.4em 1em; } .template-documentation-toolbar { font-style: normal; font-size: 85%; } fygw83ot86d2f5rcbmg6wo3yfqhanue 2813107 2813106 2020-11-12T07:59:44Z Izno 21264 not quite live yet 2813107 sanitized-css text/css /* Not quite live so don't protect yet. */ .template-documentation, .template-documentation-metadata { border: 1px solid #a2a9b1; background-color: #ecfcf4; clear: both; } .template-documentation { margin: 1em 0 0 0; padding: 1em; } .template-documentation-startbox { padding-bottom: 3px; border-bottom: 1px solid #aaa; margin-bottom: 1ex; } .template-documentation-heading { font-weight: bold; font-size: 125%; } .template-documentation-other-heading { font-size: 150%; } .template-documentation-clear { /* Don't want things to stick out where they shouldn't. */ clear: both; } .template-documentation-metadata { margin: 0.2em 0; font-style: italic; padding: 0.4em 1em; } .template-documentation-toolbar { font-style: normal; font-size: 85%; } 09u7fu24nrrth8tqpqla9ccu0jm3r7z 2813108 2813107 2020-11-14T21:35:39Z Izno 21264 remove other-heading per talk page 2813108 sanitized-css text/css /* Not quite live so don't protect yet. */ .template-documentation, .template-documentation-metadata { border: 1px solid #a2a9b1; background-color: #ecfcf4; clear: both; } .template-documentation { margin: 1em 0 0 0; padding: 1em; } .template-documentation-startbox { padding-bottom: 3px; border-bottom: 1px solid #aaa; margin-bottom: 1ex; } .template-documentation-heading { font-weight: bold; font-size: 125%; } .template-documentation-clear { /* Don't want things to stick out where they shouldn't. */ clear: both; } .template-documentation-metadata { margin: 0.2em 0; font-style: italic; padding: 0.4em 1em; } .template-documentation-toolbar { font-style: normal; font-size: 85%; } 6huio2iei2rfximkz38kzgvj0knddce 2813109 2813108 2020-11-16T18:29:05Z Izno 21264 shorten to documentation 2813109 sanitized-css text/css /* Not quite live so don't protect yet. */ .documentation, .documentation-metadata { border: 1px solid #a2a9b1; background-color: #ecfcf4; clear: both; } .documentation { margin: 1em 0 0 0; padding: 1em; } .documentation-metadata { margin: 0.2em 0; /* same margin left-right as .documentation */ font-style: italic; padding: 0.4em 1em; /* same padding left-right as .documentation */ } .documentation-startbox { padding-bottom: 3px; border-bottom: 1px solid #aaa; margin-bottom: 1ex; } .documentation-heading { font-weight: bold; font-size: 125%; } .documentation-clear { /* Don't want things to stick out where they shouldn't. */ clear: both; } .documentation-link-toolbar { font-style: normal; font-size: 85%; } 9mdu33bqqp0z3g8i6s68mp4ylo61omb 2813110 2813109 2020-11-17T01:14:45Z Izno 21264 -toolbar 2813110 sanitized-css text/css /* Not quite live so don't protect yet. */ .documentation, .documentation-metadata { border: 1px solid #a2a9b1; background-color: #ecfcf4; clear: both; } .documentation { margin: 1em 0 0 0; padding: 1em; } .documentation-metadata { margin: 0.2em 0; /* same margin left-right as .documentation */ font-style: italic; padding: 0.4em 1em; /* same padding left-right as .documentation */ } .documentation-startbox { padding-bottom: 3px; border-bottom: 1px solid #aaa; margin-bottom: 1ex; } .documentation-heading { font-weight: bold; font-size: 125%; } .documentation-clear { /* Don't want things to stick out where they shouldn't. */ clear: both; } .documentation-toolbar { font-style: normal; font-size: 85%; } 9ky1fmc65yoioohag6fi0753klq6nk6 2813111 2813110 2020-11-19T20:16:57Z Izno 21264 Protected "[[Module:Documentation/styles.css]]": match module ([Edit=Require administrator access] (indefinite) [Move=Require administrator access] (indefinite)) 2813110 sanitized-css text/css /* Not quite live so don't protect yet. */ .documentation, .documentation-metadata { border: 1px solid #a2a9b1; background-color: #ecfcf4; clear: both; } .documentation { margin: 1em 0 0 0; padding: 1em; } .documentation-metadata { margin: 0.2em 0; /* same margin left-right as .documentation */ font-style: italic; padding: 0.4em 1em; /* same padding left-right as .documentation */ } .documentation-startbox { padding-bottom: 3px; border-bottom: 1px solid #aaa; margin-bottom: 1ex; } .documentation-heading { font-weight: bold; font-size: 125%; } .documentation-clear { /* Don't want things to stick out where they shouldn't. */ clear: both; } .documentation-toolbar { font-style: normal; font-size: 85%; } 9ky1fmc65yoioohag6fi0753klq6nk6 2813112 2813111 2020-11-19T20:17:59Z Izno 21264 pp small 2813112 sanitized-css text/css /* {{pp|small=yes}} */ .documentation, .documentation-metadata { border: 1px solid #a2a9b1; background-color: #ecfcf4; clear: both; } .documentation { margin: 1em 0 0 0; padding: 1em; } .documentation-metadata { margin: 0.2em 0; /* same margin left-right as .documentation */ font-style: italic; padding: 0.4em 1em; /* same padding left-right as .documentation */ } .documentation-startbox { padding-bottom: 3px; border-bottom: 1px solid #aaa; margin-bottom: 1ex; } .documentation-heading { font-weight: bold; font-size: 125%; } .documentation-clear { /* Don't want things to stick out where they shouldn't. */ clear: both; } .documentation-toolbar { font-style: normal; font-size: 85%; } o2ia525en7wbq32mq4fpdevtl5d6pcb 2813113 2813112 2020-11-19T20:21:58Z Izno 21264 Changed protection level for "[[Module:Documentation/styles.css]]": actually match module ([Edit=Require template editor access] (indefinite) [Move=Require template editor access] (indefinite)) 2813112 sanitized-css text/css /* {{pp|small=yes}} */ .documentation, .documentation-metadata { border: 1px solid #a2a9b1; background-color: #ecfcf4; clear: both; } .documentation { margin: 1em 0 0 0; padding: 1em; } .documentation-metadata { margin: 0.2em 0; /* same margin left-right as .documentation */ font-style: italic; padding: 0.4em 1em; /* same padding left-right as .documentation */ } .documentation-startbox { padding-bottom: 3px; border-bottom: 1px solid #aaa; margin-bottom: 1ex; } .documentation-heading { font-weight: bold; font-size: 125%; } .documentation-clear { /* Don't want things to stick out where they shouldn't. */ clear: both; } .documentation-toolbar { font-style: normal; font-size: 85%; } o2ia525en7wbq32mq4fpdevtl5d6pcb 2813114 2813113 2024-03-10T21:00:17Z Jdlrobson 400633 Night mode styles 2813114 sanitized-css text/css /* {{pp|small=yes}} */ .documentation, .documentation-metadata { border: 1px solid #a2a9b1; background-color: #ecfcf4; clear: both; } .documentation { margin: 1em 0 0 0; padding: 1em; } .documentation-metadata { margin: 0.2em 0; /* same margin left-right as .documentation */ font-style: italic; padding: 0.4em 1em; /* same padding left-right as .documentation */ } .documentation-startbox { padding-bottom: 3px; border-bottom: 1px solid #aaa; margin-bottom: 1ex; } .documentation-heading { font-weight: bold; font-size: 125%; } .documentation-clear { /* Don't want things to stick out where they shouldn't. */ clear: both; } .documentation-toolbar { font-style: normal; font-size: 85%; } html.skin-night-mode-clientpref-1 .documentation, html.skin-night-mode-clientpref-1 .documentation-metadata { background: none; color: inherit; } @media (prefers-color-scheme: dark) { html.skin-night-mode-clientpref-2 .documentation, html.skin-night-mode-clientpref-2 .documentation-metadata { background: none; color: inherit; } } 3tzft3jsj5wagechm2ngn6sz9a24go7 2813115 2813114 2024-03-11T03:40:55Z Izno 21264 clean 2813115 sanitized-css text/css /* {{pp|small=yes}} */ .documentation, .documentation-metadata { border: 1px solid #a2a9b1; background-color: #ecfcf4; color: inherit; clear: both; } .documentation { margin: 1em 0 0 0; padding: 1em; } .documentation-metadata { margin: 0.2em 0; /* same margin left-right as .documentation */ font-style: italic; padding: 0.4em 1em; /* same padding left-right as .documentation */ } .documentation-startbox { padding-bottom: 3px; border-bottom: 1px solid #aaa; margin-bottom: 1ex; } .documentation-heading { font-weight: bold; font-size: 125%; } .documentation-clear { /* Don't want things to stick out where they shouldn't. */ clear: both; } .documentation-toolbar { font-style: normal; font-size: 85%; } html.skin-night-mode-clientpref-1 .documentation, html.skin-night-mode-clientpref-1 .documentation-metadata { background: none; } @media (prefers-color-scheme: dark) { html.skin-night-mode-clientpref-2 .documentation, html.skin-night-mode-clientpref-2 .documentation-metadata { background: none; } } a9rekskqhtelchuzw1p82cazjb65uod 2813116 2813115 2024-03-11T03:46:25Z Izno 21264 don't actually see a reason for this color either 2813116 sanitized-css text/css /* {{pp|small=yes}} */ .documentation, .documentation-metadata { border: 1px solid #a2a9b1; background-color: #ecfcf4; clear: both; } .documentation { margin: 1em 0 0 0; padding: 1em; } .documentation-metadata { margin: 0.2em 0; /* same margin left-right as .documentation */ font-style: italic; padding: 0.4em 1em; /* same padding left-right as .documentation */ } .documentation-startbox { padding-bottom: 3px; border-bottom: 1px solid #aaa; margin-bottom: 1ex; } .documentation-heading { font-weight: bold; font-size: 125%; } .documentation-clear { /* Don't want things to stick out where they shouldn't. */ clear: both; } .documentation-toolbar { font-style: normal; font-size: 85%; } html.skin-night-mode-clientpref-1 .documentation, html.skin-night-mode-clientpref-1 .documentation-metadata { background: none; } @media (prefers-color-scheme: dark) { html.skin-night-mode-clientpref-2 .documentation, html.skin-night-mode-clientpref-2 .documentation-metadata { background: none; } } 4cwd2l9ns12sis7tt9q7marpv9l2zdh 2813117 2813116 2024-03-11T03:51:33Z Izno 21264 inherit these instead 2813117 sanitized-css text/css /* {{pp|small=yes}} */ .documentation, .documentation-metadata { border: 1px solid #a2a9b1; background-color: #ecfcf4; clear: both; } .documentation { margin: 1em 0 0 0; padding: 1em; } .documentation-metadata { margin: 0.2em 0; /* same margin left-right as .documentation */ font-style: italic; padding: 0.4em 1em; /* same padding left-right as .documentation */ } .documentation-startbox { padding-bottom: 3px; border-bottom: 1px solid #aaa; margin-bottom: 1ex; } .documentation-heading { font-weight: bold; font-size: 125%; } .documentation-clear { /* Don't want things to stick out where they shouldn't. */ clear: both; } .documentation-toolbar { font-style: normal; font-size: 85%; } html.skin-night-mode-clientpref-1 .documentation, html.skin-night-mode-clientpref-1 .documentation-metadata { background: inherit; } @media (prefers-color-scheme: dark) { html.skin-night-mode-clientpref-2 .documentation, html.skin-night-mode-clientpref-2 .documentation-metadata { background: inherit; } } 3upma61vv8m9hsnys1kij9z7jmk7lp8 2813118 2813117 2024-03-11T03:56:36Z Izno 21264 adj 2813118 sanitized-css text/css /* {{pp|small=yes}} */ .documentation, .documentation-metadata { border: 1px solid #a2a9b1; background-color: #ecfcf4; clear: both; } .documentation { margin: 1em 0 0 0; padding: 1em; } .documentation-metadata { margin: 0.2em 0; /* same margin left-right as .documentation */ font-style: italic; padding: 0.4em 1em; /* same padding left-right as .documentation */ } .documentation-startbox { padding-bottom: 3px; border-bottom: 1px solid #aaa; margin-bottom: 1ex; } .documentation-heading { font-weight: bold; font-size: 125%; } .documentation-clear { /* Don't want things to stick out where they shouldn't. */ clear: both; } .documentation-toolbar { font-style: normal; font-size: 85%; } html.skin-night-mode-clientpref-1 .documentation, html.skin-night-mode-clientpref-1 .documentation-metadata { background-color: inherit; } @media (prefers-color-scheme: dark) { html.skin-night-mode-clientpref-2 .documentation, html.skin-night-mode-clientpref-2 .documentation-metadata { background-color: inherit; } } h2isik8zny820rm2eu1x1kefrphpcjh 2813119 2813118 2024-03-20T03:28:32Z Jdlrobson 400633 Classes have been updated. More information on [[mw:Recommendations_for_night_mode_compatibility_on_Wikimedia_wikis]] 2813119 sanitized-css text/css /* {{pp|small=yes}} */ .documentation, .documentation-metadata { border: 1px solid #a2a9b1; background-color: #ecfcf4; clear: both; } .documentation { margin: 1em 0 0 0; padding: 1em; } .documentation-metadata { margin: 0.2em 0; /* same margin left-right as .documentation */ font-style: italic; padding: 0.4em 1em; /* same padding left-right as .documentation */ } .documentation-startbox { padding-bottom: 3px; border-bottom: 1px solid #aaa; margin-bottom: 1ex; } .documentation-heading { font-weight: bold; font-size: 125%; } .documentation-clear { /* Don't want things to stick out where they shouldn't. */ clear: both; } .documentation-toolbar { font-style: normal; font-size: 85%; } html.skin-theme-clientpref-night .documentation, html.skin-theme-clientpref-night .documentation-metadata { background-color: inherit; } @media (prefers-color-scheme: dark) { html.skin-theme-clientpref-os .documentation, html.skin-theme-clientpref-os .documentation-metadata { background-color: inherit; } } ir1q0ztql6sgzaf48j10wd9yzxqm39i 2813120 2813119 2024-05-25T05:13:51Z Izno 21264 use the same green as the main page 2813120 sanitized-css text/css /* {{pp|small=yes}} */ .documentation, .documentation-metadata { border: 1px solid #a2a9b1; background-color: #ecfcf4; clear: both; } .documentation { margin: 1em 0 0 0; padding: 1em; } .documentation-metadata { margin: 0.2em 0; /* same margin left-right as .documentation */ font-style: italic; padding: 0.4em 1em; /* same padding left-right as .documentation */ } .documentation-startbox { padding-bottom: 3px; border-bottom: 1px solid #aaa; margin-bottom: 1ex; } .documentation-heading { font-weight: bold; font-size: 125%; } .documentation-clear { /* Don't want things to stick out where they shouldn't. */ clear: both; } .documentation-toolbar { font-style: normal; font-size: 85%; } html.skin-theme-clientpref-night .documentation, html.skin-theme-clientpref-night .documentation-metadata { background-color: #0b1e1c; } @media (prefers-color-scheme: dark) { html.skin-theme-clientpref-os .documentation, html.skin-theme-clientpref-os .documentation-metadata { background-color: #0b1e1c; } } 3x03uk4lmse5lx5vr01rbusm0jf6e26 2813121 2813120 2024-05-25T05:16:23Z Izno 21264 Reverted edit by [[Special:Contribs/Izno|Izno]] ([[User talk:Izno|talk]]) to last version by Jdlrobson 2813121 sanitized-css text/css /* {{pp|small=yes}} */ .documentation, .documentation-metadata { border: 1px solid #a2a9b1; background-color: #ecfcf4; clear: both; } .documentation { margin: 1em 0 0 0; padding: 1em; } .documentation-metadata { margin: 0.2em 0; /* same margin left-right as .documentation */ font-style: italic; padding: 0.4em 1em; /* same padding left-right as .documentation */ } .documentation-startbox { padding-bottom: 3px; border-bottom: 1px solid #aaa; margin-bottom: 1ex; } .documentation-heading { font-weight: bold; font-size: 125%; } .documentation-clear { /* Don't want things to stick out where they shouldn't. */ clear: both; } .documentation-toolbar { font-style: normal; font-size: 85%; } html.skin-theme-clientpref-night .documentation, html.skin-theme-clientpref-night .documentation-metadata { background-color: inherit; } @media (prefers-color-scheme: dark) { html.skin-theme-clientpref-os .documentation, html.skin-theme-clientpref-os .documentation-metadata { background-color: inherit; } } ir1q0ztql6sgzaf48j10wd9yzxqm39i 2813122 2813121 2024-05-25T05:16:40Z Izno 21264 Undid revision [[Special:Diff/1225551552|1225551552]] by [[Special:Contributions/Izno|Izno]] ([[User talk:Izno|talk]]) sigh 2813122 sanitized-css text/css /* {{pp|small=yes}} */ .documentation, .documentation-metadata { border: 1px solid #a2a9b1; background-color: #ecfcf4; clear: both; } .documentation { margin: 1em 0 0 0; padding: 1em; } .documentation-metadata { margin: 0.2em 0; /* same margin left-right as .documentation */ font-style: italic; padding: 0.4em 1em; /* same padding left-right as .documentation */ } .documentation-startbox { padding-bottom: 3px; border-bottom: 1px solid #aaa; margin-bottom: 1ex; } .documentation-heading { font-weight: bold; font-size: 125%; } .documentation-clear { /* Don't want things to stick out where they shouldn't. */ clear: both; } .documentation-toolbar { font-style: normal; font-size: 85%; } html.skin-theme-clientpref-night .documentation, html.skin-theme-clientpref-night .documentation-metadata { background-color: #0b1e1c; } @media (prefers-color-scheme: dark) { html.skin-theme-clientpref-os .documentation, html.skin-theme-clientpref-os .documentation-metadata { background-color: #0b1e1c; } } 3x03uk4lmse5lx5vr01rbusm0jf6e26 2813123 2813122 2024-07-31T22:51:48Z Jon (WMF) 398586 Fixes [[phab:T369874]] via automated script 2813123 sanitized-css text/css /* {{pp|small=yes}} */ .documentation, .documentation-metadata { border: 1px solid #a2a9b1; background-color: #ecfcf4; clear: both; } .documentation { margin: 1em 0 0 0; padding: 1em; } .documentation-metadata { margin: 0.2em 0; /* same margin left-right as .documentation */ font-style: italic; padding: 0.4em 1em; /* same padding left-right as .documentation */ } .documentation-startbox { padding-bottom: 3px; border-bottom: 1px solid #aaa; margin-bottom: 1ex; } .documentation-heading { font-weight: bold; font-size: 125%; } .documentation-clear { /* Don't want things to stick out where they shouldn't. */ clear: both; } .documentation-toolbar { font-style: normal; font-size: 85%; } @media screen { html.skin-theme-clientpref-night .documentation, html.skin-theme-clientpref-night .documentation-metadata { background-color: #0b1e1c; } } @media screen and ( prefers-color-scheme: dark) { html.skin-theme-clientpref-os .documentation, html.skin-theme-clientpref-os .documentation-metadata { background-color: #0b1e1c; } } 3igw4iwzymtlarkxz89qi8dgj0mdzjy 2813124 2813123 2025-02-22T07:52:19Z Taavi 2867356 use codex tokens for borders to improve dark mode support 2813124 sanitized-css text/css /* {{pp|small=yes}} */ .documentation, .documentation-metadata { border: 1px solid var( --border-color-base, #a2a9b1 ); background-color: #ecfcf4; clear: both; } .documentation { margin: 1em 0 0 0; padding: 1em; } .documentation-metadata { margin: 0.2em 0; /* same margin left-right as .documentation */ font-style: italic; padding: 0.4em 1em; /* same padding left-right as .documentation */ } .documentation-startbox { padding-bottom: 3px; border-bottom: 1px solid var( --border-color-base, #a2a9b1 ); margin-bottom: 1ex; } .documentation-heading { font-weight: bold; font-size: 125%; } .documentation-clear { /* Don't want things to stick out where they shouldn't. */ clear: both; } .documentation-toolbar { font-style: normal; font-size: 85%; } @media screen { html.skin-theme-clientpref-night .documentation, html.skin-theme-clientpref-night .documentation-metadata { background-color: #0b1e1c; } } @media screen and ( prefers-color-scheme: dark ) { html.skin-theme-clientpref-os .documentation, html.skin-theme-clientpref-os .documentation-metadata { background-color: #0b1e1c; } } i8lops10ml7ifgplup23473ri87jccv 2813125 2813124 2025-10-02T12:44:28Z Jonesey95 831143 Fix [[Special:LintErrors|Linter]] errors related to text color in dark mode; this generally works but may cause problems, in which case revert or try specifying a different CSS value. 2813125 sanitized-css text/css /* {{pp|small=yes}} */ .documentation, .documentation-metadata { border: 1px solid var( --border-color-base, #a2a9b1 ); background-color: #ecfcf4; color:inherit; clear: both; } .documentation { margin: 1em 0 0 0; padding: 1em; } .documentation-metadata { margin: 0.2em 0; /* same margin left-right as .documentation */ font-style: italic; padding: 0.4em 1em; /* same padding left-right as .documentation */ } .documentation-startbox { padding-bottom: 3px; border-bottom: 1px solid var( --border-color-base, #a2a9b1 ); margin-bottom: 1ex; } .documentation-heading { font-weight: bold; font-size: 125%; } .documentation-clear { /* Don't want things to stick out where they shouldn't. */ clear: both; } .documentation-toolbar { font-style: normal; font-size: 85%; } @media screen { html.skin-theme-clientpref-night .documentation, html.skin-theme-clientpref-night .documentation-metadata { background-color: #0b1e1c; color:inherit; } } @media screen and ( prefers-color-scheme: dark ) { html.skin-theme-clientpref-os .documentation, html.skin-theme-clientpref-os .documentation-metadata { background-color: #0b1e1c; color:inherit; } } c21gvdpc307rvalbk05ah9j5kvouvl3 2813126 2813125 2026-06-06T08:30:16Z Jtneill 10242 35 revisions imported from [[:w:Module:Documentation/styles.css]] 2813125 sanitized-css text/css /* {{pp|small=yes}} */ .documentation, .documentation-metadata { border: 1px solid var( --border-color-base, #a2a9b1 ); background-color: #ecfcf4; color:inherit; clear: both; } .documentation { margin: 1em 0 0 0; padding: 1em; } .documentation-metadata { margin: 0.2em 0; /* same margin left-right as .documentation */ font-style: italic; padding: 0.4em 1em; /* same padding left-right as .documentation */ } .documentation-startbox { padding-bottom: 3px; border-bottom: 1px solid var( --border-color-base, #a2a9b1 ); margin-bottom: 1ex; } .documentation-heading { font-weight: bold; font-size: 125%; } .documentation-clear { /* Don't want things to stick out where they shouldn't. */ clear: both; } .documentation-toolbar { font-style: normal; font-size: 85%; } @media screen { html.skin-theme-clientpref-night .documentation, html.skin-theme-clientpref-night .documentation-metadata { background-color: #0b1e1c; color:inherit; } } @media screen and ( prefers-color-scheme: dark ) { html.skin-theme-clientpref-os .documentation, html.skin-theme-clientpref-os .documentation-metadata { background-color: #0b1e1c; color:inherit; } } c21gvdpc307rvalbk05ah9j5kvouvl3 Template:Templatedata header 10 329995 2813134 2020-08-18T14:12:38Z Jonesey95 831143 create redirect for what my hands want to type. Silly InterCaps. 2813134 wikitext text/x-wiki #REDIRECT [[Template:TemplateData header]] 4f5j00d4fbzl99id7id41c4xn7tyk03 2813135 2813134 2024-09-20T21:04:15Z 41.223.117.67 [[WP:AES|←]]Changed redirect target from [[Template:TemplateData header]] to [[KCN]] 2813135 wikitext text/x-wiki #REDIRECT [[KCN]] o39uelmw8va0srrn1rbcr3k8kxz0wdp 2813136 2813135 2024-09-20T21:06:58Z 41.223.119.35 [[WP:AES|←]]Changed redirect target from [[KCN]] to [[K.C Nathan]] 2813136 wikitext text/x-wiki #REDIRECT [[K.C Nathan]] ggn6ahghd14nfkmflh5nmu53438uciy 2813137 2813136 2024-09-20T21:12:39Z Bkonrad 45117 Rvv 2813137 wikitext text/x-wiki #REDIRECT [[Template:TemplateData header]] 4f5j00d4fbzl99id7id41c4xn7tyk03 2813138 2813137 2024-09-20T21:25:57Z Nachaka 3089800 [[WP:AES|←]]Changed redirect target from [[Template:TemplateData header]] to [[K.C Nathan]] 2813138 wikitext text/x-wiki #REDIRECT [[K.C Nathan]] ggn6ahghd14nfkmflh5nmu53438uciy 2813139 2813138 2024-09-20T21:49:24Z Knitsey 2948541 Reverted 1 edit by [[Special:Contributions/Nachaka|Nachaka]] ([[User talk:Nachaka|talk]]) 2813139 wikitext text/x-wiki #REDIRECT [[Template:TemplateData header]] 4f5j00d4fbzl99id7id41c4xn7tyk03 2813140 2813139 2026-06-06T08:30:19Z Jtneill 10242 6 revisions imported from [[:w:Template:Templatedata_header]] 2813139 wikitext text/x-wiki #REDIRECT [[Template:TemplateData header]] 4f5j00d4fbzl99id7id41c4xn7tyk03 Template:Message box/ombox.css 10 329996 2813152 2022-05-08T01:06:21Z Izno 21264 ombox 2813152 sanitized-css text/css .ombox { margin: 4px 10%; border-collapse: collapse; border: 1px solid #a2a9b1; /* Default "notice" gray */ background-color: #f8f9fa; box-sizing: border-box; } /* For the "small=yes" option. */ .ombox.mbox-small { /* @noflip */ clear: right; /* @noflip */ float: right; /* @noflip */ margin: 4px 0 4px 1em; box-sizing: border-box; width: 238px; font-size: 88%; line-height: 1.25em; } .ombox-speedy { border: 2px solid #b32424; /* Red */ background-color: #fee7e6; /* Pink */ } .ombox-delete { border: 2px solid #b32424; /* Red */ } .ombox-content { border: 1px solid #f28500; /* Orange */ } .ombox-style { border: 1px solid #fc3; /* Yellow */ } .ombox-move { border: 1px solid #9932cc; /* Purple */ } .ombox-protection { border: 2px solid #a2a9b1; /* Gray-gold */ } .ombox .mbox-text { border: none; /* @noflip */ padding: 0.25em 0.9em; width: 100%; } .ombox .mbox-image { border: none; /* @noflip */ padding: 2px 0 2px 0.9em; text-align: center; } .ombox .mbox-imageright { border: none; /* @noflip */ padding: 2px 0.9em 2px 0; text-align: center; } /* An empty narrow cell */ .ombox .mbox-empty-cell { border: none; padding: 0; width: 1px; } ci24bo898ytloivs9lgalv6z4hip8du 2813153 2813152 2022-05-08T01:07:31Z Izno 21264 -dupe 2813153 sanitized-css text/css .ombox { margin: 4px 10%; border-collapse: collapse; border: 1px solid #a2a9b1; /* Default "notice" gray */ background-color: #f8f9fa; box-sizing: border-box; } /* For the "small=yes" option. */ .ombox.mbox-small { /* @noflip */ clear: right; /* @noflip */ float: right; /* @noflip */ margin: 4px 0 4px 1em; width: 238px; font-size: 88%; line-height: 1.25em; } .ombox-speedy { border: 2px solid #b32424; /* Red */ background-color: #fee7e6; /* Pink */ } .ombox-delete { border: 2px solid #b32424; /* Red */ } .ombox-content { border: 1px solid #f28500; /* Orange */ } .ombox-style { border: 1px solid #fc3; /* Yellow */ } .ombox-move { border: 1px solid #9932cc; /* Purple */ } .ombox-protection { border: 2px solid #a2a9b1; /* Gray-gold */ } .ombox .mbox-text { border: none; /* @noflip */ padding: 0.25em 0.9em; width: 100%; } .ombox .mbox-image { border: none; /* @noflip */ padding: 2px 0 2px 0.9em; text-align: center; } .ombox .mbox-imageright { border: none; /* @noflip */ padding: 2px 0.9em 2px 0; text-align: center; } /* An empty narrow cell */ .ombox .mbox-empty-cell { border: none; padding: 0; width: 1px; } 6nvzht2pyxwys5byyx3iz1p5ighi212 2813154 2813153 2022-05-27T17:33:58Z Izno 21264 add pp 2813154 sanitized-css text/css /* {{pp|small=y}} */ .ombox { margin: 4px 10%; border-collapse: collapse; border: 1px solid #a2a9b1; /* Default "notice" gray */ background-color: #f8f9fa; box-sizing: border-box; } /* For the "small=yes" option. */ .ombox.mbox-small { /* @noflip */ clear: right; /* @noflip */ float: right; /* @noflip */ margin: 4px 0 4px 1em; width: 238px; font-size: 88%; line-height: 1.25em; } .ombox-speedy { border: 2px solid #b32424; /* Red */ background-color: #fee7e6; /* Pink */ } .ombox-delete { border: 2px solid #b32424; /* Red */ } .ombox-content { border: 1px solid #f28500; /* Orange */ } .ombox-style { border: 1px solid #fc3; /* Yellow */ } .ombox-move { border: 1px solid #9932cc; /* Purple */ } .ombox-protection { border: 2px solid #a2a9b1; /* Gray-gold */ } .ombox .mbox-text { border: none; /* @noflip */ padding: 0.25em 0.9em; width: 100%; } .ombox .mbox-image { border: none; /* @noflip */ padding: 2px 0 2px 0.9em; text-align: center; } .ombox .mbox-imageright { border: none; /* @noflip */ padding: 2px 0.9em 2px 0; text-align: center; } /* An empty narrow cell */ .ombox .mbox-empty-cell { border: none; padding: 0; width: 1px; } 7gqx2puzdhuyxquvc8imufso41jejl3 2813155 2813154 2022-05-27T17:34:51Z Izno 21264 Protected "[[Module:Message box/ombox.css]]": match parent ([Edit=Require administrator access] (indefinite)) 2813154 sanitized-css text/css /* {{pp|small=y}} */ .ombox { margin: 4px 10%; border-collapse: collapse; border: 1px solid #a2a9b1; /* Default "notice" gray */ background-color: #f8f9fa; box-sizing: border-box; } /* For the "small=yes" option. */ .ombox.mbox-small { /* @noflip */ clear: right; /* @noflip */ float: right; /* @noflip */ margin: 4px 0 4px 1em; width: 238px; font-size: 88%; line-height: 1.25em; } .ombox-speedy { border: 2px solid #b32424; /* Red */ background-color: #fee7e6; /* Pink */ } .ombox-delete { border: 2px solid #b32424; /* Red */ } .ombox-content { border: 1px solid #f28500; /* Orange */ } .ombox-style { border: 1px solid #fc3; /* Yellow */ } .ombox-move { border: 1px solid #9932cc; /* Purple */ } .ombox-protection { border: 2px solid #a2a9b1; /* Gray-gold */ } .ombox .mbox-text { border: none; /* @noflip */ padding: 0.25em 0.9em; width: 100%; } .ombox .mbox-image { border: none; /* @noflip */ padding: 2px 0 2px 0.9em; text-align: center; } .ombox .mbox-imageright { border: none; /* @noflip */ padding: 2px 0.9em 2px 0; text-align: center; } /* An empty narrow cell */ .ombox .mbox-empty-cell { border: none; padding: 0; width: 1px; } 7gqx2puzdhuyxquvc8imufso41jejl3 2813156 2813155 2022-05-27T17:38:38Z Izno 21264 Changed protection settings for "[[Module:Message box/ombox.css]]": and also move ([Edit=Require administrator access] (indefinite) [Move=Require administrator access] (indefinite)) 2813154 sanitized-css text/css /* {{pp|small=y}} */ .ombox { margin: 4px 10%; border-collapse: collapse; border: 1px solid #a2a9b1; /* Default "notice" gray */ background-color: #f8f9fa; box-sizing: border-box; } /* For the "small=yes" option. */ .ombox.mbox-small { /* @noflip */ clear: right; /* @noflip */ float: right; /* @noflip */ margin: 4px 0 4px 1em; width: 238px; font-size: 88%; line-height: 1.25em; } .ombox-speedy { border: 2px solid #b32424; /* Red */ background-color: #fee7e6; /* Pink */ } .ombox-delete { border: 2px solid #b32424; /* Red */ } .ombox-content { border: 1px solid #f28500; /* Orange */ } .ombox-style { border: 1px solid #fc3; /* Yellow */ } .ombox-move { border: 1px solid #9932cc; /* Purple */ } .ombox-protection { border: 2px solid #a2a9b1; /* Gray-gold */ } .ombox .mbox-text { border: none; /* @noflip */ padding: 0.25em 0.9em; width: 100%; } .ombox .mbox-image { border: none; /* @noflip */ padding: 2px 0 2px 0.9em; text-align: center; } .ombox .mbox-imageright { border: none; /* @noflip */ padding: 2px 0.9em 2px 0; text-align: center; } /* An empty narrow cell */ .ombox .mbox-empty-cell { border: none; padding: 0; width: 1px; } 7gqx2puzdhuyxquvc8imufso41jejl3 2813157 2813156 2022-07-09T18:34:08Z Izno 21264 invalid type css 2813157 sanitized-css text/css /* {{pp|small=y}} */ .ombox { margin: 4px 10%; border-collapse: collapse; border: 1px solid #a2a9b1; /* Default "notice" gray */ background-color: #f8f9fa; box-sizing: border-box; } /* For the "small=yes" option. */ .ombox.mbox-small { /* @noflip */ clear: right; /* @noflip */ float: right; /* @noflip */ margin: 4px 0 4px 1em; width: 238px; font-size: 88%; line-height: 1.25em; } .ombox-speedy { border: 2px solid #b32424; /* Red */ background-color: #fee7e6; /* Pink */ } .ombox-delete { border: 2px solid #b32424; /* Red */ } .ombox-content { border: 1px solid #f28500; /* Orange */ } .ombox-style { border: 1px solid #fc3; /* Yellow */ } .ombox-move { border: 1px solid #9932cc; /* Purple */ } .ombox-protection { border: 2px solid #a2a9b1; /* Gray-gold */ } .ombox .mbox-text { border: none; /* @noflip */ padding: 0.25em 0.9em; width: 100%; } .ombox .mbox-image { border: none; /* @noflip */ padding: 2px 0 2px 0.9em; text-align: center; } .ombox .mbox-imageright { border: none; /* @noflip */ padding: 2px 0.9em 2px 0; text-align: center; } /* An empty narrow cell */ .ombox .mbox-empty-cell { border: none; padding: 0; width: 1px; } .ombox .mbox-invalid-type { text-align: center; } 3ux5g5k02v43o5ahyuseaigu0p3juig 2813158 2813157 2022-07-10T15:25:06Z Izno 21264 margin to media query 2813158 sanitized-css text/css /* {{pp|small=y}} */ .ombox { margin: 4px 0; border-collapse: collapse; border: 1px solid #a2a9b1; /* Default "notice" gray */ background-color: #f8f9fa; box-sizing: border-box; } /* For the "small=yes" option. */ .ombox.mbox-small { /* @noflip */ clear: right; /* @noflip */ float: right; /* @noflip */ margin: 4px 0 4px 1em; width: 238px; font-size: 88%; line-height: 1.25em; } .ombox-speedy { border: 2px solid #b32424; /* Red */ background-color: #fee7e6; /* Pink */ } .ombox-delete { border: 2px solid #b32424; /* Red */ } .ombox-content { border: 1px solid #f28500; /* Orange */ } .ombox-style { border: 1px solid #fc3; /* Yellow */ } .ombox-move { border: 1px solid #9932cc; /* Purple */ } .ombox-protection { border: 2px solid #a2a9b1; /* Gray-gold */ } .ombox .mbox-text { border: none; /* @noflip */ padding: 0.25em 0.9em; width: 100%; } .ombox .mbox-image { border: none; /* @noflip */ padding: 2px 0 2px 0.9em; text-align: center; } .ombox .mbox-imageright { border: none; /* @noflip */ padding: 2px 0.9em 2px 0; text-align: center; } /* An empty narrow cell */ .ombox .mbox-empty-cell { border: none; padding: 0; width: 1px; } .ombox .mbox-invalid-type { text-align: center; } @media (min-width: 720px) { .ombox { margin: 4px 10%; } } 0rc06h7hmk60fb8yqkyizm46vs03igq 2813159 2813158 2022-07-11T18:40:17Z Izno 21264 and move mbox-small to 720px here as well 2813159 sanitized-css text/css /* {{pp|small=y}} */ .ombox { margin: 4px 0; border-collapse: collapse; border: 1px solid #a2a9b1; /* Default "notice" gray */ background-color: #f8f9fa; box-sizing: border-box; } /* For the "small=yes" option. */ .ombox.mbox-small { font-size: 88%; line-height: 1.25em; } .ombox-speedy { border: 2px solid #b32424; /* Red */ background-color: #fee7e6; /* Pink */ } .ombox-delete { border: 2px solid #b32424; /* Red */ } .ombox-content { border: 1px solid #f28500; /* Orange */ } .ombox-style { border: 1px solid #fc3; /* Yellow */ } .ombox-move { border: 1px solid #9932cc; /* Purple */ } .ombox-protection { border: 2px solid #a2a9b1; /* Gray-gold */ } .ombox .mbox-text { border: none; /* @noflip */ padding: 0.25em 0.9em; width: 100%; } .ombox .mbox-image { border: none; /* @noflip */ padding: 2px 0 2px 0.9em; text-align: center; } .ombox .mbox-imageright { border: none; /* @noflip */ padding: 2px 0.9em 2px 0; text-align: center; } /* An empty narrow cell */ .ombox .mbox-empty-cell { border: none; padding: 0; width: 1px; } .ombox .mbox-invalid-type { text-align: center; } @media (min-width: 720px) { .ombox { margin: 4px 10%; } .ombox.mbox-small { /* @noflip */ clear: right; /* @noflip */ float: right; /* @noflip */ margin: 4px 0 4px 1em; width: 238px; } } gt34qcz2etl1lglsfax1xmoaasgmdxe 2813160 2813159 2024-05-06T20:53:30Z MSGJ 121057 set color: #202122 per suggestion 2813160 sanitized-css text/css /* {{pp|small=y}} */ .ombox { margin: 4px 0; border-collapse: collapse; border: 1px solid #a2a9b1; /* Default "notice" gray */ background-color: #f8f9fa; box-sizing: border-box; color: #202122; } /* For the "small=yes" option. */ .ombox.mbox-small { font-size: 88%; line-height: 1.25em; } .ombox-speedy { border: 2px solid #b32424; /* Red */ background-color: #fee7e6; /* Pink */ } .ombox-delete { border: 2px solid #b32424; /* Red */ } .ombox-content { border: 1px solid #f28500; /* Orange */ } .ombox-style { border: 1px solid #fc3; /* Yellow */ } .ombox-move { border: 1px solid #9932cc; /* Purple */ } .ombox-protection { border: 2px solid #a2a9b1; /* Gray-gold */ } .ombox .mbox-text { border: none; /* @noflip */ padding: 0.25em 0.9em; width: 100%; } .ombox .mbox-image { border: none; /* @noflip */ padding: 2px 0 2px 0.9em; text-align: center; } .ombox .mbox-imageright { border: none; /* @noflip */ padding: 2px 0.9em 2px 0; text-align: center; } /* An empty narrow cell */ .ombox .mbox-empty-cell { border: none; padding: 0; width: 1px; } .ombox .mbox-invalid-type { text-align: center; } @media (min-width: 720px) { .ombox { margin: 4px 10%; } .ombox.mbox-small { /* @noflip */ clear: right; /* @noflip */ float: right; /* @noflip */ margin: 4px 0 4px 1em; width: 238px; } } 9k7m17zmq4hm67vbtjezw6q9ixgoo4w 2813161 2813160 2024-05-21T16:57:58Z Izno 21264 live 2813161 sanitized-css text/css /* {{pp|small=y}} */ .ombox { margin: 4px 0; border-collapse: collapse; border: 1px solid #a2a9b1; /* Default "notice" gray */ background-color: var(--background-color-neutral-subtle, #f8f9fa); box-sizing: border-box; } /* For the "small=yes" option. */ .ombox.mbox-small { font-size: 88%; line-height: 1.25em; } .ombox-speedy { border: 2px solid #b32424; /* Red */ background-color: #fee7e6; /* Pink */ } .ombox-delete { border: 2px solid #b32424; /* Red */ } .ombox-content { border: 1px solid #f28500; /* Orange */ } .ombox-style { border: 1px solid #fc3; /* Yellow */ } .ombox-move { border: 1px solid #9932cc; /* Purple */ } .ombox-protection { border: 2px solid #a2a9b1; /* Gray-gold */ } .ombox .mbox-text { border: none; /* @noflip */ padding: 0.25em 0.9em; width: 100%; } .ombox .mbox-image { border: none; /* @noflip */ padding: 2px 0 2px 0.9em; text-align: center; } .ombox .mbox-imageright { border: none; /* @noflip */ padding: 2px 0.9em 2px 0; text-align: center; } /* An empty narrow cell */ .ombox .mbox-empty-cell { border: none; padding: 0; width: 1px; } .ombox .mbox-invalid-type { text-align: center; } @media (min-width: 720px) { .ombox { margin: 4px 10%; } .ombox.mbox-small { /* @noflip */ clear: right; /* @noflip */ float: right; /* @noflip */ margin: 4px 0 4px 1em; width: 238px; } } html.skin-theme-clientpref-night .ombox-speedy { background-color: #310402; /* Dark red, same hue/saturation as light */ } @media (prefers-color-scheme: dark) { html.skin-theme-clientpref-os .ombox-speedy { background-color: #310402; /* Dark red, same hue/saturation as light */ } } c0ah0g75ql8rxakz2f1gr23beydrjpq 2813162 2813161 2024-06-03T16:13:03Z Jon (WMF) 398586 [[mw:Recommendations_for_night_mode_compatibility_on_Wikimedia_wikis]] Always define color alongside background color (fixes issues in Village pump ombox)) 2813162 sanitized-css text/css /* {{pp|small=y}} */ .ombox { margin: 4px 0; border-collapse: collapse; border: 1px solid #a2a9b1; /* Default "notice" gray */ background-color: var(--background-color-neutral-subtle, #f8f9fa); box-sizing: border-box; color: var(--color-base); } /* For the "small=yes" option. */ .ombox.mbox-small { font-size: 88%; line-height: 1.25em; } .ombox-speedy { border: 2px solid #b32424; /* Red */ background-color: #fee7e6; /* Pink */ } .ombox-delete { border: 2px solid #b32424; /* Red */ } .ombox-content { border: 1px solid #f28500; /* Orange */ } .ombox-style { border: 1px solid #fc3; /* Yellow */ } .ombox-move { border: 1px solid #9932cc; /* Purple */ } .ombox-protection { border: 2px solid #a2a9b1; /* Gray-gold */ } .ombox .mbox-text { border: none; /* @noflip */ padding: 0.25em 0.9em; width: 100%; } .ombox .mbox-image { border: none; /* @noflip */ padding: 2px 0 2px 0.9em; text-align: center; } .ombox .mbox-imageright { border: none; /* @noflip */ padding: 2px 0.9em 2px 0; text-align: center; } /* An empty narrow cell */ .ombox .mbox-empty-cell { border: none; padding: 0; width: 1px; } .ombox .mbox-invalid-type { text-align: center; } @media (min-width: 720px) { .ombox { margin: 4px 10%; } .ombox.mbox-small { /* @noflip */ clear: right; /* @noflip */ float: right; /* @noflip */ margin: 4px 0 4px 1em; width: 238px; } } html.skin-theme-clientpref-night .ombox-speedy { background-color: #310402; /* Dark red, same hue/saturation as light */ } @media (prefers-color-scheme: dark) { html.skin-theme-clientpref-os .ombox-speedy { background-color: #310402; /* Dark red, same hue/saturation as light */ } } avkdukr75imj0nrtp1l98dsbldb7fte 2813163 2813162 2024-06-14T04:52:02Z Izno 21264 not sure that was needed but let's set the fallback 2813163 sanitized-css text/css /* {{pp|small=y}} */ .ombox { margin: 4px 0; border-collapse: collapse; border: 1px solid #a2a9b1; /* Default "notice" gray */ background-color: var(--background-color-neutral-subtle, #f8f9fa); box-sizing: border-box; color: var(--color-base, #202122); } /* For the "small=yes" option. */ .ombox.mbox-small { font-size: 88%; line-height: 1.25em; } .ombox-speedy { border: 2px solid #b32424; /* Red */ background-color: #fee7e6; /* Pink */ } .ombox-delete { border: 2px solid #b32424; /* Red */ } .ombox-content { border: 1px solid #f28500; /* Orange */ } .ombox-style { border: 1px solid #fc3; /* Yellow */ } .ombox-move { border: 1px solid #9932cc; /* Purple */ } .ombox-protection { border: 2px solid #a2a9b1; /* Gray-gold */ } .ombox .mbox-text { border: none; /* @noflip */ padding: 0.25em 0.9em; width: 100%; } .ombox .mbox-image { border: none; /* @noflip */ padding: 2px 0 2px 0.9em; text-align: center; } .ombox .mbox-imageright { border: none; /* @noflip */ padding: 2px 0.9em 2px 0; text-align: center; } /* An empty narrow cell */ .ombox .mbox-empty-cell { border: none; padding: 0; width: 1px; } .ombox .mbox-invalid-type { text-align: center; } @media (min-width: 720px) { .ombox { margin: 4px 10%; } .ombox.mbox-small { /* @noflip */ clear: right; /* @noflip */ float: right; /* @noflip */ margin: 4px 0 4px 1em; width: 238px; } } html.skin-theme-clientpref-night .ombox-speedy { background-color: #310402; /* Dark red, same hue/saturation as light */ } @media (prefers-color-scheme: dark) { html.skin-theme-clientpref-os .ombox-speedy { background-color: #310402; /* Dark red, same hue/saturation as light */ } } mtaroy813aac5dag9txyirqdzrf7yem 2813164 2813163 2024-06-14T15:58:47Z Jon (WMF) 398586 Fixes [[phab:T367463]] 2813164 sanitized-css text/css /* {{pp|small=y}} */ .ombox { margin: 4px 0; border-collapse: collapse; border: 1px solid #a2a9b1; /* Default "notice" gray */ background-color: var(--background-color-neutral-subtle, #f8f9fa); box-sizing: border-box; color: var(--color-base, #202122); } /* For the "small=yes" option. */ .ombox.mbox-small { font-size: 88%; line-height: 1.25em; } .ombox-speedy { border: 2px solid #b32424; /* Red */ background-color: #fee7e6; /* Pink */ } .ombox-delete { border: 2px solid #b32424; /* Red */ } .ombox-content { border: 1px solid #f28500; /* Orange */ } .ombox-style { border: 1px solid #fc3; /* Yellow */ } .ombox-move { border: 1px solid #9932cc; /* Purple */ } .ombox-protection { border: 2px solid #a2a9b1; /* Gray-gold */ } .ombox .mbox-text { border: none; /* @noflip */ padding: 0.25em 0.9em; width: 100%; } .ombox .mbox-image { border: none; /* @noflip */ padding: 2px 0 2px 0.9em; text-align: center; } .ombox .mbox-imageright { border: none; /* @noflip */ padding: 2px 0.9em 2px 0; text-align: center; } /* An empty narrow cell */ .ombox .mbox-empty-cell { border: none; padding: 0; width: 1px; } .ombox .mbox-invalid-type { text-align: center; } @media (min-width: 720px) { .ombox { margin: 4px 10%; } .ombox.mbox-small { /* @noflip */ clear: right; /* @noflip */ float: right; /* @noflip */ margin: 4px 0 4px 1em; width: 238px; } } /** T367463 */ body.skin--responsive table.ombox img { max-width: none !important; } html.skin-theme-clientpref-night .ombox-speedy { background-color: #310402; /* Dark red, same hue/saturation as light */ } @media (prefers-color-scheme: dark) { html.skin-theme-clientpref-os .ombox-speedy { background-color: #310402; /* Dark red, same hue/saturation as light */ } } msy4j1qs05qfauryrbm7z8erd6xm3xq 2813165 2813164 2024-08-03T21:24:20Z Izno 21264 only on screen 2813165 sanitized-css text/css /* {{pp|small=y}} */ .ombox { margin: 4px 0; border-collapse: collapse; border: 1px solid #a2a9b1; /* Default "notice" gray */ background-color: var(--background-color-neutral-subtle, #f8f9fa); box-sizing: border-box; color: var(--color-base, #202122); } /* For the "small=yes" option. */ .ombox.mbox-small { font-size: 88%; line-height: 1.25em; } .ombox-speedy { border: 2px solid #b32424; /* Red */ background-color: #fee7e6; /* Pink */ } .ombox-delete { border: 2px solid #b32424; /* Red */ } .ombox-content { border: 1px solid #f28500; /* Orange */ } .ombox-style { border: 1px solid #fc3; /* Yellow */ } .ombox-move { border: 1px solid #9932cc; /* Purple */ } .ombox-protection { border: 2px solid #a2a9b1; /* Gray-gold */ } .ombox .mbox-text { border: none; /* @noflip */ padding: 0.25em 0.9em; width: 100%; } .ombox .mbox-image { border: none; /* @noflip */ padding: 2px 0 2px 0.9em; text-align: center; } .ombox .mbox-imageright { border: none; /* @noflip */ padding: 2px 0.9em 2px 0; text-align: center; } /* An empty narrow cell */ .ombox .mbox-empty-cell { border: none; padding: 0; width: 1px; } .ombox .mbox-invalid-type { text-align: center; } @media (min-width: 720px) { .ombox { margin: 4px 10%; } .ombox.mbox-small { /* @noflip */ clear: right; /* @noflip */ float: right; /* @noflip */ margin: 4px 0 4px 1em; width: 238px; } } /** T367463 */ body.skin--responsive table.ombox img { max-width: none !important; } @media screen { html.skin-theme-clientpref-night .ombox-speedy { background-color: #310402; /* Dark red, same hue/saturation as light */ } } @media screen and (prefers-color-scheme: dark) { html.skin-theme-clientpref-os .ombox-speedy { background-color: #310402; /* Dark red, same hue/saturation as light */ } } aknh3fw7498ygg9b1o85bvfqn56r48n 2813166 2813165 2025-08-29T06:04:06Z Izno 21264 fix bug for invalid type CSS 2813166 sanitized-css text/css /* {{pp|small=y}} */ .ombox { margin: 4px 0; border-collapse: collapse; border: 1px solid #a2a9b1; /* Default "notice" gray */ background-color: var(--background-color-neutral-subtle, #f8f9fa); box-sizing: border-box; color: var(--color-base, #202122); } /* For the "small=yes" option. */ .ombox.mbox-small { font-size: 88%; line-height: 1.25em; } .ombox-speedy { border: 2px solid #b32424; /* Red */ background-color: #fee7e6; /* Pink */ } .ombox-delete { border: 2px solid #b32424; /* Red */ } .ombox-content { border: 1px solid #f28500; /* Orange */ } .ombox-style { border: 1px solid #fc3; /* Yellow */ } .ombox-move { border: 1px solid #9932cc; /* Purple */ } .ombox-protection { border: 2px solid #a2a9b1; /* Gray-gold */ } .ombox .mbox-text { border: none; /* @noflip */ padding: 0.25em 0.9em; width: 100%; } .ombox .mbox-image { border: none; /* @noflip */ padding: 2px 0 2px 0.9em; text-align: center; } .ombox .mbox-imageright { border: none; /* @noflip */ padding: 2px 0.9em 2px 0; text-align: center; } /* An empty narrow cell */ .ombox .mbox-empty-cell { border: none; padding: 0; width: 1px; } /* keep synced with each other types of message box as this isn't qualified */ .mbox-invalid-type { text-align: center; } @media (min-width: 720px) { .ombox { margin: 4px 10%; } .ombox.mbox-small { /* @noflip */ clear: right; /* @noflip */ float: right; /* @noflip */ margin: 4px 0 4px 1em; width: 238px; } } /** T367463 */ body.skin--responsive table.ombox img { max-width: none !important; } @media screen { html.skin-theme-clientpref-night .ombox-speedy { background-color: #310402; /* Dark red, same hue/saturation as light */ } } @media screen and (prefers-color-scheme: dark) { html.skin-theme-clientpref-os .ombox-speedy { background-color: #310402; /* Dark red, same hue/saturation as light */ } } lggkfzvvlzfi1h2px1oqdnsp202iycp 2813167 2813166 2025-08-29T06:07:09Z Izno 21264 english 2813167 sanitized-css text/css /* {{pp|small=y}} */ .ombox { margin: 4px 0; border-collapse: collapse; border: 1px solid #a2a9b1; /* Default "notice" gray */ background-color: var(--background-color-neutral-subtle, #f8f9fa); box-sizing: border-box; color: var(--color-base, #202122); } /* For the "small=yes" option. */ .ombox.mbox-small { font-size: 88%; line-height: 1.25em; } .ombox-speedy { border: 2px solid #b32424; /* Red */ background-color: #fee7e6; /* Pink */ } .ombox-delete { border: 2px solid #b32424; /* Red */ } .ombox-content { border: 1px solid #f28500; /* Orange */ } .ombox-style { border: 1px solid #fc3; /* Yellow */ } .ombox-move { border: 1px solid #9932cc; /* Purple */ } .ombox-protection { border: 2px solid #a2a9b1; /* Gray-gold */ } .ombox .mbox-text { border: none; /* @noflip */ padding: 0.25em 0.9em; width: 100%; } .ombox .mbox-image { border: none; /* @noflip */ padding: 2px 0 2px 0.9em; text-align: center; } .ombox .mbox-imageright { border: none; /* @noflip */ padding: 2px 0.9em 2px 0; text-align: center; } /* An empty narrow cell */ .ombox .mbox-empty-cell { border: none; padding: 0; width: 1px; } /* keep synced with each other type of message box as this isn't qualified */ .mbox-invalid-type { text-align: center; } @media (min-width: 720px) { .ombox { margin: 4px 10%; } .ombox.mbox-small { /* @noflip */ clear: right; /* @noflip */ float: right; /* @noflip */ margin: 4px 0 4px 1em; width: 238px; } } /** T367463 */ body.skin--responsive table.ombox img { max-width: none !important; } @media screen { html.skin-theme-clientpref-night .ombox-speedy { background-color: #310402; /* Dark red, same hue/saturation as light */ } } @media screen and (prefers-color-scheme: dark) { html.skin-theme-clientpref-os .ombox-speedy { background-color: #310402; /* Dark red, same hue/saturation as light */ } } s3kd4o8l90hza7k55c82cm0ltkd3xp2 2813168 2813167 2026-06-06T08:30:20Z Jtneill 10242 16 revisions imported from [[:w:Module:Message_box/ombox.css]] 2813167 sanitized-css text/css /* {{pp|small=y}} */ .ombox { margin: 4px 0; border-collapse: collapse; border: 1px solid #a2a9b1; /* Default "notice" gray */ background-color: var(--background-color-neutral-subtle, #f8f9fa); box-sizing: border-box; color: var(--color-base, #202122); } /* For the "small=yes" option. */ .ombox.mbox-small { font-size: 88%; line-height: 1.25em; } .ombox-speedy { border: 2px solid #b32424; /* Red */ background-color: #fee7e6; /* Pink */ } .ombox-delete { border: 2px solid #b32424; /* Red */ } .ombox-content { border: 1px solid #f28500; /* Orange */ } .ombox-style { border: 1px solid #fc3; /* Yellow */ } .ombox-move { border: 1px solid #9932cc; /* Purple */ } .ombox-protection { border: 2px solid #a2a9b1; /* Gray-gold */ } .ombox .mbox-text { border: none; /* @noflip */ padding: 0.25em 0.9em; width: 100%; } .ombox .mbox-image { border: none; /* @noflip */ padding: 2px 0 2px 0.9em; text-align: center; } .ombox .mbox-imageright { border: none; /* @noflip */ padding: 2px 0.9em 2px 0; text-align: center; } /* An empty narrow cell */ .ombox .mbox-empty-cell { border: none; padding: 0; width: 1px; } /* keep synced with each other type of message box as this isn't qualified */ .mbox-invalid-type { text-align: center; } @media (min-width: 720px) { .ombox { margin: 4px 10%; } .ombox.mbox-small { /* @noflip */ clear: right; /* @noflip */ float: right; /* @noflip */ margin: 4px 0 4px 1em; width: 238px; } } /** T367463 */ body.skin--responsive table.ombox img { max-width: none !important; } @media screen { html.skin-theme-clientpref-night .ombox-speedy { background-color: #310402; /* Dark red, same hue/saturation as light */ } } @media screen and (prefers-color-scheme: dark) { html.skin-theme-clientpref-os .ombox-speedy { background-color: #310402; /* Dark red, same hue/saturation as light */ } } s3kd4o8l90hza7k55c82cm0ltkd3xp2 Template:Respond 10 329997 2813188 2023-03-04T14:26:24Z Terasail 2907175 Create 2813188 wikitext text/x-wiki <includeonly>[[File:{{#switch:{{{1|{{{icon|}}}}}} |greencheck=Symbol confirmed.svg |greencheck2=Yes check.svg |blackcheck=Black check.svg |yellowcheck=Yellow check.svg |bluecheck=Artículo 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2813188 2023-03-04T14:59:16Z Terasail 2907175 Move to doc subpage 2813189 wikitext text/x-wiki <includeonly>[[File:{{#switch:{{{1|{{{icon|}}}}}} |greencheck=Symbol confirmed.svg |greencheck2=Yes check.svg |blackcheck=Black check.svg |yellowcheck=Yellow check.svg |bluecheck=Artículo bueno-blue.svg |bluecheck2=Cornflower blue check.svg |xmark=Symbol unrelated.svg |xmark2=X mark.svg |minus=Symbol_declined.svg |grenplus=Symbol support vote.svg |blueplus=Symbol support2 vote.svg |bluebrokenplus=Symbol unsupport2 vote.svg |greybrokenplus=Symbol unsupport vote.svg |neutral=Symbol neutral vote.svg |deferred=Symbol deferred.svg |wait=Symbol_wait.svg |blueclock=Pictogram voting wait blue.svg |pinkclock=Pictogram voting wait pink.svg |yellowclock=Pictogram voting wait.svg |orangeclock=Pictogram voting wait orange.svg |greenclock=Pictogram voting wait green.svg |info=Symbol information vote.svg |orangeexclamation=Ambox important.svg |greyexclamation=Pictogram voting comment.svg |question=Pictogram voting question.svg |question2=Symbol question.svg |{{{1|{{{icon|Pictogram voting question.svg}}}}}}}}|{{{size|18px}}}|alt={{{alt|}}}]]&nbsp;'''{{{2|{{{text|Response}}}}}}'''</includeonly><noinclude> {{Documentation}} </noinclude> i2hwz0ay5gc9z9mplig87i032hz1tq0 2813190 2813189 2023-03-04T17:59:46Z MusikBot II 2893438 Protected "[[Template:Respond]]": [[Wikipedia:High-risk templates|High-risk template or module]]: 6294 transclusions ([[User:MusikBot II/TemplateProtector|more info]]) ([Edit=Require template editor access] (indefinite) [Move=Require template editor access] (indefinite)) 2813189 wikitext text/x-wiki <includeonly>[[File:{{#switch:{{{1|{{{icon|}}}}}} |greencheck=Symbol confirmed.svg |greencheck2=Yes check.svg |blackcheck=Black check.svg |yellowcheck=Yellow check.svg |bluecheck=Artículo bueno-blue.svg |bluecheck2=Cornflower blue check.svg |xmark=Symbol unrelated.svg |xmark2=X mark.svg |minus=Symbol_declined.svg |grenplus=Symbol support vote.svg |blueplus=Symbol support2 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<includeonly>[[File:{{#switch:{{{1|{{{icon|}}}}}} |blackcheck = Black check.svg |bluebrokenplus = Symbol unsupport2 vote.svg |bluecheck2 = Cornflower blue check.svg |bluecheck = Artículo bueno-blue.svg |blueclock = Pictogram voting wait blue.svg |blueplus = Symbol support2 vote.svg |deferred = Symbol deferred.svg |greencheck2 = Yes check.svg |greencheck = Symbol confirmed.svg |greenclock = Pictogram voting wait green.svg |greenplus = Symbol support vote.svg |greybrokenplus = Symbol unsupport vote.svg |greyexclamation = Pictogram voting comment.svg |info=Symbol information vote.svg |minus = Symbol declined.svg |neutral = Symbol neutral vote.svg |orangeclock = Pictogram voting wait orange.svg |orangeexclamation = Ambox important.svg |redexclamation = Ambox important red.svg |pinkclock = Pictogram voting wait pink.svg |question2 = Symbol question.svg |question = Pictogram voting question.svg |wait = Symbol_wait.svg |xmark2 = X mark.svg |xmark = Symbol unrelated.svg |yellowcheck = Yellow check.svg |yellowclock = Pictogram voting wait.svg |{{{1|{{{icon|Pictogram voting question.svg}}}}}}}}|{{{size|18px}}}|alt={{{alt|}}}]]&nbsp;'''{{{2|{{{text|Response}}}}}}'''</includeonly><noinclude> {{Documentation}} </noinclude> 340mugytd1w7nfqsacvtboi53q4nzod 2813195 2813194 2023-03-10T02:10:00Z Terasail 2907175 Add safesubst 2813195 wikitext text/x-wiki <includeonly>[[File:{{{{{|safesubst:}}}#switch:{{{1|{{{icon|}}}}}} |blackcheck = Black check.svg |bluebrokenplus = Symbol unsupport2 vote.svg |bluecheck2 = Cornflower blue check.svg |bluecheck = Artículo bueno-blue.svg |blueclock = Pictogram voting wait blue.svg |blueplus = Symbol support2 vote.svg |deferred = Symbol deferred.svg |greencheck2 = Yes check.svg |greencheck = Symbol confirmed.svg |greenclock = Pictogram voting wait green.svg |greenplus = Symbol support vote.svg |greybrokenplus = Symbol unsupport vote.svg |greyexclamation = Pictogram voting comment.svg |info=Symbol information vote.svg |minus = Symbol declined.svg |neutral = Symbol neutral vote.svg |orangeclock = Pictogram voting wait orange.svg |orangeexclamation = Ambox important.svg |redexclamation = Ambox important red.svg |pinkclock = Pictogram voting wait pink.svg |question2 = Symbol question.svg |question = Pictogram voting question.svg |wait = Symbol_wait.svg |xmark2 = X mark.svg |xmark = Symbol unrelated.svg |yellowcheck = Yellow check.svg |yellowclock = Pictogram voting wait.svg |{{{1|{{{icon|Pictogram voting question.svg}}}}}}}}|{{{size|18px}}}|alt={{{alt|}}}]]&nbsp;'''{{{2|{{{text|Response}}}}}}'''</includeonly><noinclude> {{Documentation}} </noinclude> kotx8mtbzl2oieezh86yjbtmz1932ar 2813196 2813195 2024-06-05T23:41:21Z Jonesey95 831143 Fix [[Special:LintErrors|Linter]] errors. Does not solve the problem of empty input. 2813196 wikitext text/x-wiki <includeonly>[[File:{{{{{|safesubst:}}}#switch:{{{1|{{{icon|}}}}}} |blackcheck = Black check.svg |bluebrokenplus = Symbol unsupport2 vote.svg |bluecheck2 = Cornflower blue check.svg |bluecheck = Artículo bueno-blue.svg |blueclock = Pictogram voting wait blue.svg |blueplus = Symbol support2 vote.svg |deferred = Symbol deferred.svg |greencheck2 = Yes check.svg |greencheck = Symbol confirmed.svg |greenclock = Pictogram voting wait green.svg |greenplus = Symbol support vote.svg |greybrokenplus = Symbol unsupport vote.svg |greyexclamation = Pictogram voting comment.svg |info=Symbol information vote.svg |minus = Symbol declined.svg |neutral = Symbol neutral vote.svg |orangeclock = Pictogram voting wait orange.svg |orangeexclamation = Ambox important.svg |redexclamation = Ambox important red.svg |pinkclock = Pictogram voting wait pink.svg |question2 = Symbol question.svg |question = Pictogram voting question.svg |wait = Symbol_wait.svg |xmark2 = X mark.svg |xmark = Symbol unrelated.svg |yellowcheck = Yellow check.svg |yellowclock = Pictogram voting wait.svg |{{{1|{{{icon|Pictogram voting question.svg}}}}}}}}|{{{size|18px}}}|alt={{{alt|}}}]]'''&nbsp;{{{2|{{{text|Response}}}}}}'''</includeonly><noinclude> {{Documentation}} </noinclude> hsyl8vi5m0h3is2h29ku385bbloiy75 2813197 2813196 2026-01-16T16:48:58Z Jonesey95 831143 fix non-working nbsp per TPER 2813197 wikitext text/x-wiki <includeonly><span class="nowrap">[[File:{{{{{|safesubst:}}}#switch:{{{1|{{{icon|}}}}}} |blackcheck = Black check.svg |bluebrokenplus = Symbol unsupport2 vote.svg |bluecheck2 = Cornflower blue check.svg |bluecheck = Artículo bueno-blue.svg |blueclock = Pictogram voting wait blue.svg |blueplus = Symbol support2 vote.svg |deferred = Symbol deferred.svg |greencheck2 = Yes check.svg |greencheck = Symbol confirmed.svg |greenclock = Pictogram voting wait green.svg |greenplus = Symbol support vote.svg |greybrokenplus = Symbol unsupport vote.svg |greyexclamation = Pictogram voting comment.svg |info=Symbol information vote.svg |minus = Symbol declined.svg |neutral = Symbol neutral vote.svg |orangeclock = Pictogram voting wait orange.svg |orangeexclamation = Ambox important.svg |redexclamation = Ambox important red.svg |pinkclock = Pictogram voting wait pink.svg |question2 = Symbol question.svg |question = Pictogram voting question.svg |wait = Symbol_wait.svg |xmark2 = X mark.svg |xmark = Symbol unrelated.svg |yellowcheck = Yellow check.svg |yellowclock = Pictogram voting wait.svg |{{{1|{{{icon|Pictogram voting question.svg}}}}}}}}|{{{size|18px}}}|alt={{{alt|}}}]]''' {{{2|{{{text|Response}}}}}}'''</span></includeonly><noinclude> {{Documentation}} </noinclude> 0hjrnyeogouw1pnqeuu2stpcftolp55 2813198 2813197 2026-01-16T19:14:48Z Andrybak 1869413 Add simple dark mode support to the monochrome [[File:Black check.svg]]. A possible alternative is something like [[Special:Diff/1242482846]]. 2813198 wikitext text/x-wiki <includeonly><span class="nowrap">[[File:{{{{{|safesubst:}}}#switch:{{{1|{{{icon|}}}}}} |blackcheck = Black check.svg{{!}}class=skin-invert |bluebrokenplus = Symbol unsupport2 vote.svg |bluecheck2 = Cornflower blue check.svg |bluecheck = Artículo bueno-blue.svg |blueclock = Pictogram voting wait blue.svg |blueplus = Symbol support2 vote.svg |deferred = Symbol deferred.svg |greencheck2 = Yes check.svg |greencheck = Symbol confirmed.svg |greenclock = Pictogram voting wait green.svg |greenplus = Symbol support vote.svg |greybrokenplus = Symbol unsupport vote.svg |greyexclamation = Pictogram voting comment.svg |info=Symbol information vote.svg |minus = Symbol declined.svg |neutral = Symbol neutral vote.svg |orangeclock = Pictogram voting wait orange.svg |orangeexclamation = Ambox important.svg |redexclamation = Ambox important red.svg |pinkclock = Pictogram voting wait pink.svg |question2 = Symbol question.svg |question = Pictogram voting question.svg |wait = Symbol_wait.svg |xmark2 = X mark.svg |xmark = Symbol unrelated.svg |yellowcheck = Yellow check.svg |yellowclock = Pictogram voting wait.svg |{{{1|{{{icon|Pictogram voting question.svg}}}}}}}}|{{{size|18px}}}|alt={{{alt|}}}]]''' {{{2|{{{text|Response}}}}}}'''</span></includeonly><noinclude> {{Documentation}} </noinclude> 0ycdi8ml3ns7pvwmwao6s0gzvxu7y82 2813199 2813198 2026-01-16T21:36:04Z Jonesey95 831143 update to adjust wrapping per TPER 2813199 wikitext text/x-wiki <includeonly><span class="nowrap">[[File:{{{{{|safesubst:}}}#switch:{{{1|{{{icon|}}}}}} |blackcheck = Black check.svg{{!}}class=skin-invert |bluebrokenplus = Symbol unsupport2 vote.svg |bluecheck2 = Cornflower blue check.svg |bluecheck = Artículo bueno-blue.svg |blueclock = Pictogram voting wait blue.svg |blueplus = Symbol support2 vote.svg |deferred = Symbol deferred.svg |greencheck2 = Yes check.svg |greencheck = Symbol confirmed.svg |greenclock = Pictogram voting wait green.svg |greenplus = Symbol support vote.svg |greybrokenplus = Symbol unsupport vote.svg |greyexclamation = Pictogram voting comment.svg |info=Symbol information vote.svg |minus = Symbol declined.svg |neutral = Symbol neutral vote.svg |orangeclock = Pictogram voting wait orange.svg |orangeexclamation = Ambox important.svg |redexclamation = Ambox important red.svg |pinkclock = Pictogram voting wait pink.svg |question2 = Symbol question.svg |question = Pictogram voting question.svg |wait = Symbol_wait.svg |xmark2 = X mark.svg |xmark = Symbol unrelated.svg |yellowcheck = Yellow check.svg |yellowclock = Pictogram voting wait.svg |{{{1|{{{icon|Pictogram voting question.svg}}}}}}}}|{{{size|18px}}}|alt={{{alt|}}}]] </span><b>{{{2|{{{text|Response}}}}}}</b></includeonly><noinclude> {{Documentation}} </noinclude> o6smoeltckwqdbjagseuc678htvxwwz 2813200 2813199 2026-01-18T00:57:44Z Jonesey95 831143 restore span to end of text; the change is causing too many Linter errors. People will just have to be concise. 2813200 wikitext text/x-wiki <includeonly><span class="nowrap">[[File:{{{{{|safesubst:}}}#switch:{{{1|{{{icon|}}}}}} |blackcheck = Black check.svg{{!}}class=skin-invert |bluebrokenplus = Symbol unsupport2 vote.svg |bluecheck2 = Cornflower blue check.svg |bluecheck = Artículo bueno-blue.svg |blueclock = Pictogram voting wait blue.svg |blueplus = Symbol support2 vote.svg |deferred = Symbol deferred.svg |greencheck2 = Yes check.svg |greencheck = Symbol confirmed.svg |greenclock = Pictogram voting wait green.svg |greenplus = Symbol support vote.svg |greybrokenplus = Symbol unsupport vote.svg |greyexclamation = Pictogram voting comment.svg |info=Symbol information vote.svg |minus = Symbol declined.svg |neutral = Symbol neutral vote.svg |orangeclock = Pictogram voting wait orange.svg |orangeexclamation = Ambox important.svg |redexclamation = Ambox important red.svg |pinkclock = Pictogram voting wait pink.svg |question2 = Symbol question.svg |question = Pictogram voting question.svg |wait = Symbol_wait.svg |xmark2 = X mark.svg |xmark = Symbol unrelated.svg |yellowcheck = Yellow check.svg |yellowclock = Pictogram voting wait.svg |{{{1|{{{icon|Pictogram voting question.svg}}}}}}}}|{{{size|18px}}}|alt={{{alt|}}}]] <b>{{{2|{{{text|Response}}}}}}</b></span></includeonly><noinclude> {{Documentation}} </noinclude> e46u9jzzi433zezrmzbfconxvbngmv9 2813201 2813200 2026-06-06T08:30:22Z Jtneill 10242 13 revisions imported from [[:w:Template:Respond]] 2813200 wikitext text/x-wiki <includeonly><span class="nowrap">[[File:{{{{{|safesubst:}}}#switch:{{{1|{{{icon|}}}}}} |blackcheck = Black check.svg{{!}}class=skin-invert |bluebrokenplus = Symbol unsupport2 vote.svg |bluecheck2 = Cornflower blue check.svg |bluecheck = Artículo bueno-blue.svg |blueclock = Pictogram voting wait blue.svg |blueplus = Symbol support2 vote.svg |deferred = Symbol deferred.svg |greencheck2 = Yes check.svg |greencheck = Symbol confirmed.svg |greenclock = Pictogram voting wait green.svg |greenplus = Symbol support vote.svg |greybrokenplus = Symbol unsupport vote.svg |greyexclamation = Pictogram voting comment.svg |info=Symbol information vote.svg |minus = Symbol declined.svg |neutral = Symbol neutral vote.svg |orangeclock = Pictogram voting wait orange.svg |orangeexclamation = Ambox important.svg |redexclamation = Ambox important red.svg |pinkclock = Pictogram voting wait pink.svg |question2 = Symbol question.svg |question = Pictogram voting question.svg |wait = Symbol_wait.svg |xmark2 = X mark.svg |xmark = Symbol unrelated.svg |yellowcheck = Yellow check.svg |yellowclock = Pictogram voting wait.svg |{{{1|{{{icon|Pictogram voting question.svg}}}}}}}}|{{{size|18px}}}|alt={{{alt|}}}]] <b>{{{2|{{{text|Response}}}}}}</b></span></includeonly><noinclude> {{Documentation}} </noinclude> e46u9jzzi433zezrmzbfconxvbngmv9 Template:Text diff/doc 10 329998 2813202 2021-02-27T17:34:04Z Levivich 2867064 +basic doc 2813202 wikitext text/x-wiki {{Documentation subpage}} <!-- Add categories where indicated at the bottom of this page and interwikis at Wikidata --> {{tsh|td}} == Usage == This template can be used to show proposed changes, e.g. on talk pages. Typing <code><nowiki>{{TextDiff|Old text.|New text.}}</nowiki></code> or <code><nowiki>{{td|Old text.|New text.}}</nowiki></code> produces a [[WP:diff|diff]] showing the changes: {{td|Old text|New text.}} <includeonly>{{sandbox other|| <!-- Categories below this line; interwikis at Wikidata --> }}</includeonly> 6sy8rew29b0f106ehptwuubsaecg63r 2813203 2813202 2021-05-03T18:19:31Z Terasail 2907175 + lua template 2813203 wikitext text/x-wiki {{Documentation subpage}} <!-- Add categories where indicated at the bottom of this page and interwikis at Wikidata --> {{tsh|td}} {{Lua|Module:Diff}} == Usage == This template can be used to show proposed changes, e.g. on talk pages. Typing <code><nowiki>{{TextDiff|Old text.|New text.}}</nowiki></code> or <code><nowiki>{{td|Old text.|New text.}}</nowiki></code> produces a [[WP:diff|diff]] showing the changes: {{td|Old text|New text.}} <includeonly>{{sandbox other|| <!-- Categories below this line; interwikis at Wikidata --> }}</includeonly> 6n7ent4dn6f165ibhw4785eqb8s6o9b 2813204 2813203 2022-09-15T20:52:34Z Qwerfjkl 3089802 /* Usage */ [[[w:en:User:Alexis Jazz/Factotum|Factotum]]] 2813204 wikitext text/x-wiki {{Documentation subpage}} <!-- Add categories where indicated at the bottom of this page and interwikis at Wikidata --> {{tsh|td}} {{Lua|Module:Diff}} == Usage == This template can be used to show proposed changes, e.g. on talk pages. Typing <code><nowiki>{{TextDiff|Old text.|New text.}}</nowiki></code> or <code><nowiki>{{td|Old text.|New text.}}</nowiki></code> produces a [[WP:diff|diff]] showing the changes: {{td|Old text|New text.}} == See also == * [[User:NguoiDungKhongDinhDanh/FormattedEditRequest]], a [[wp:user scripts|script]] for automatically producing diffs similar to the ones produced by this template <includeonly>{{sandbox other|| <!-- Categories below this line; interwikis at Wikidata --> }}</includeonly> ci8qqhuogek1p5twrjwe9ktbpfhn8r8 2813205 2813204 2022-09-30T07:24:06Z Olivaw-Daneel 2938752 /* See also */ +{{StringDiff}} 2813205 wikitext text/x-wiki {{Documentation subpage}} <!-- Add categories where indicated at the bottom of this page and interwikis at Wikidata --> {{tsh|td}} {{Lua|Module:Diff}} == Usage == This template can be used to show proposed changes, e.g. on talk pages. Typing <code><nowiki>{{TextDiff|Old text.|New text.}}</nowiki></code> or <code><nowiki>{{td|Old text.|New text.}}</nowiki></code> produces a [[WP:diff|diff]] showing the changes: {{td|Old text|New text.}} == See also == * [[User:NguoiDungKhongDinhDanh/FormattedEditRequest]], a [[wp:user scripts|script]] for automatically producing diffs similar to the ones produced by this template * {{tl|StringDiff}}, which produces character-level diffs <includeonly>{{sandbox other|| <!-- Categories below this line; interwikis at Wikidata --> }}</includeonly> 5vfaj2brq4dth2humpsskbkomtrfo7o 2813206 2813205 2022-10-01T00:53:37Z Olivaw-Daneel 2938752 simplify 2813206 wikitext text/x-wiki {{Documentation subpage}} <!-- Add categories where indicated at the bottom of this page and interwikis at Wikidata --> {{tsh|td}} {{Lua|Module:Diff}} This template can be used to show proposed changes, e.g. on talk pages. == Usage == {{demo|br=produces a [[WP:diff|diff]] showing the changes:|<nowiki>{{TextDiff|Old text.|New text.}}</nowiki>}} == See also == * [[User:NguoiDungKhongDinhDanh/FormattedEditRequest]], a [[wp:user scripts|script]] for automatically producing diffs similar to the ones produced by this template * {{tl|StringDiff}}, which produces character-level diffs <includeonly>{{sandbox other|| <!-- Categories below this line; interwikis at Wikidata --> }}</includeonly> r2j60yrw6my785ko94sw0n2imiil4f8 2813207 2813206 2022-10-01T00:54:43Z Olivaw-Daneel 2938752 line break 2813207 wikitext text/x-wiki {{Documentation subpage}} <!-- Add categories where indicated at the bottom of this page and interwikis at Wikidata --> {{tsh|td}} {{Lua|Module:Diff}} This template can be used to show proposed changes, e.g. on talk pages. == Usage == {{demo|br=produces a [[WP:diff|diff]] showing the changes:<br />|<nowiki>{{TextDiff|Old text.|New text.}}</nowiki>}} == See also == * [[User:NguoiDungKhongDinhDanh/FormattedEditRequest]], a [[wp:user scripts|script]] for automatically producing diffs similar to the ones produced by this template * {{tl|StringDiff}}, which produces character-level diffs <includeonly>{{sandbox other|| <!-- Categories below this line; interwikis at Wikidata --> }}</includeonly> mrsm1ooedo0pbhbmb7nsgpe8qf5trks 2813208 2813207 2023-05-23T23:55:25Z 47.25.251.161 /* Usage */ 2813208 wikitext text/x-wiki {{Documentation subpage}} <!-- Add categories where indicated at the bottom of this page and interwikis at Wikidata --> {{tsh|td}} {{Lua|Module:Diff}} This template can be used to show proposed changes, e.g. on talk pages. '''''Bold text''[[WUWS]]<ref> [[{{Did you know nominations/Ishūretsuzō}} ===== ===== Heading text ===== ==== === Heading text === == * Heading text # <nowiki>Numbered list item</nowiki><br> <big><small>Big text</small><sup><sub>Superscript text</sub> <gallery> #REDIRECT [[Example.jpg|Caption1 Example.jpg|Caption2]] </gallery> </sup></big> == ==== ===== ]] </ref>'''== Usage == {{demo|br=produces a [[WP:diff|diff]] showing the changes:<br />|<nowiki>{{TextDiff|Old text.|New text.}}</nowiki>}} == See also == * [[User:NguoiDungKhongDinhDanh/FormattedEditRequest]], a [[wp:user scripts|script]] for automatically producing diffs similar to the ones produced by this template * {{tl|StringDiff}}, which produces character-level diffs <includeonly>{{sandbox other|| <!-- Categories below this line; interwikis at Wikidata --> }}</includeonly> 95wzuqw0ntbs9ajdzj2pk3owi207jzo 2813209 2813208 2023-05-26T11:31:02Z Aidan9382 2958236 Undid revision 1156658993 by [[Special:Contributions/47.25.251.161|47.25.251.161]] ([[User talk:47.25.251.161|talk]]) 2813209 wikitext text/x-wiki {{Documentation subpage}} <!-- Add categories where indicated at the bottom of this page and interwikis at Wikidata --> {{tsh|td}} {{Lua|Module:Diff}} This template can be used to show proposed changes, e.g. on talk pages. == Usage == {{demo|br=produces a [[WP:diff|diff]] showing the changes:<br />|<nowiki>{{TextDiff|Old text.|New text.}}</nowiki>}} == See also == * [[User:NguoiDungKhongDinhDanh/FormattedEditRequest]], a [[wp:user scripts|script]] for automatically producing diffs similar to the ones produced by this template * {{tl|StringDiff}}, which produces character-level diffs <includeonly>{{sandbox other|| <!-- Categories below this line; interwikis at Wikidata --> }}</includeonly> mrsm1ooedo0pbhbmb7nsgpe8qf5trks 2813210 2813209 2023-08-08T07:04:07Z Ltbdl 2983760 adding templatedata 2813210 wikitext text/x-wiki {{Documentation subpage}} <!-- Add categories where indicated at the bottom of this page and interwikis at Wikidata --> {{tsh|td}} {{Lua|Module:Diff}} This template can be used to show proposed changes, e.g. on talk pages. == Usage == {{demo|br=produces a [[WP:diff|diff]] showing the changes:<br />|<nowiki>{{TextDiff|Old text.|New text.}}</nowiki>}} == TemplateData == {{templatedata header}} <templatedata> { "params": { "1": { "label": "Old text", "description": "Specify the old wikitext here.", "type": "content", "required": true }, "2": { "label": "New text", "description": "Specify the new wikitext here.", "type": "content", "required": true } }, "description": "This template compares two texts. The comparison is formatted similarly to a \"diff\" (see Help:Diff).", "format": "block" } </templatedata> == See also == * [[User:NguoiDungKhongDinhDanh/FormattedEditRequest]], a [[wp:user scripts|script]] for automatically producing diffs similar to the ones produced by this template * {{tl|StringDiff}}, which produces character-level diffs ih73xkoohvea1ooy4pqqq8pdaqky9mo 2813211 2813210 2023-11-23T11:14:34Z DB1729 2965319 Category:Diff templates 2813211 wikitext text/x-wiki {{Documentation subpage}} <!-- Add categories where indicated at the bottom of this page and interwikis at Wikidata --> {{tsh|td}} {{Lua|Module:Diff}} This template can be used to show proposed changes, e.g. on talk pages. == Usage == {{demo|br=produces a [[WP:diff|diff]] showing the changes:<br />|<nowiki>{{TextDiff|Old text.|New text.}}</nowiki>}} == TemplateData == {{templatedata header}} <templatedata> { "params": { "1": { "label": "Old text", "description": "Specify the old wikitext here.", "type": "content", "required": true }, "2": { "label": "New text", "description": "Specify the new wikitext here.", "type": "content", "required": true } }, "description": "This template compares two texts. The comparison is formatted similarly to a \"diff\" (see Help:Diff).", "format": "block" } </templatedata> == See also == * [[User:NguoiDungKhongDinhDanh/FormattedEditRequest]], a [[wp:user scripts|script]] for automatically producing diffs similar to the ones produced by this template * {{tl|StringDiff}}, which produces character-level diffs <includeonly>{{Sandbox other|| <!-- Categories below this line --> [[Category:Diff templates]] }}</includeonly> g6i528433hrcsrq7i30bizz9e8dfm9l 2813212 2813211 2023-12-03T19:27:49Z Jonesey95 831143 suitable only for inline text 2813212 wikitext text/x-wiki {{Documentation subpage}} <!-- Add categories where indicated at the bottom of this page and interwikis at Wikidata --> {{tsh|td}} {{Lua|Module:Diff}} This template can be used to show proposed changes, e.g. on talk pages. == Usage == {{demo|br=produces a [[WP:diff|diff]] showing the changes:<br />|<nowiki>{{TextDiff|Old text.|New text.}}</nowiki>}} ==Limitations== This template is suitable only for comparing a single in-line string of text. Use of this template to compare multi-line content (i.e. HTML block content, text including line breaks) will result in syntax errors. == TemplateData == {{templatedata header}} <templatedata> { "params": { "1": { "label": "Old text", "description": "Specify the old wikitext here.", "type": "content", "required": true }, "2": { "label": "New text", "description": "Specify the new wikitext here.", "type": "content", "required": true } }, "description": "This template compares two texts. The comparison is formatted similarly to a \"diff\" (see Help:Diff).", "format": "block" } </templatedata> == See also == * [[User:NguoiDungKhongDinhDanh/FormattedEditRequest]], a [[wp:user scripts|script]] for automatically producing diffs similar to the ones produced by this template * {{tl|StringDiff}}, which produces character-level diffs <includeonly>{{Sandbox other|| <!-- Categories below this line --> [[Category:Diff templates]] }}</includeonly> d3ukxlqoyb9ioinilfbnpgyfgtpla8j 2813213 2813212 2023-12-08T17:11:26Z HouseBlaster 2945622 HouseBlaster moved page [[Template:TextDiff/doc]] to [[Template:Text diff/doc]]: use plain english in title per [[WP:TPN]] 2813212 wikitext text/x-wiki {{Documentation subpage}} <!-- Add categories where indicated at the bottom of this page and interwikis at Wikidata --> {{tsh|td}} {{Lua|Module:Diff}} This template can be used to show proposed changes, e.g. on talk pages. == Usage == {{demo|br=produces a [[WP:diff|diff]] showing the changes:<br />|<nowiki>{{TextDiff|Old text.|New text.}}</nowiki>}} ==Limitations== This template is suitable only for comparing a single in-line string of text. Use of this template to compare multi-line content (i.e. HTML block content, text including line breaks) will result in syntax errors. == TemplateData == {{templatedata header}} <templatedata> { "params": { "1": { "label": "Old text", "description": "Specify the old wikitext here.", "type": "content", "required": true }, "2": { "label": "New text", "description": "Specify the new wikitext here.", "type": "content", "required": true } }, "description": "This template compares two texts. The comparison is formatted similarly to a \"diff\" (see Help:Diff).", "format": "block" } </templatedata> == See also == * [[User:NguoiDungKhongDinhDanh/FormattedEditRequest]], a [[wp:user scripts|script]] for automatically producing diffs similar to the ones produced by this template * {{tl|StringDiff}}, which produces character-level diffs <includeonly>{{Sandbox other|| <!-- Categories below this line --> [[Category:Diff templates]] }}</includeonly> d3ukxlqoyb9ioinilfbnpgyfgtpla8j 2813214 2813213 2024-01-07T20:40:58Z Jonesey95 831143 /* Limitations */ another limitation 2813214 wikitext text/x-wiki {{Documentation subpage}} <!-- Add categories where indicated at the bottom of this page and interwikis at Wikidata --> {{tsh|td}} {{Lua|Module:Diff}} This template can be used to show proposed changes, e.g. on talk pages. == Usage == {{demo|br=produces a [[WP:diff|diff]] showing the changes:<br />|<nowiki>{{TextDiff|Old text.|New text.}}</nowiki>}} ==Limitations== This template is suitable only for comparing a single in-line string of text. Use of this template to compare multi-line content (i.e. HTML block content, text including line breaks) will result in syntax errors. This template must be placed on a new line, i.e. with a line break before the template. Adding the template to a line that is indented with a colon or asterisk may result in syntax errors. == TemplateData == {{templatedata header}} <templatedata> { "params": { "1": { "label": "Old text", "description": "Specify the old wikitext here.", "type": "content", "required": true }, "2": { "label": "New text", "description": "Specify the new wikitext here.", "type": "content", "required": true } }, "description": "This template compares two texts. The comparison is formatted similarly to a \"diff\" (see Help:Diff).", "format": "block" } </templatedata> == See also == * [[User:NguoiDungKhongDinhDanh/FormattedEditRequest]], a [[wp:user scripts|script]] for automatically producing diffs similar to the ones produced by this template * {{tl|StringDiff}}, which produces character-level diffs <includeonly>{{Sandbox other|| <!-- Categories below this line --> [[Category:Diff templates]] }}</includeonly> qpaea65jbhf8y8hzxge61r4h5lfmz6w 2813215 2813214 2024-07-01T02:21:39Z 174.92.25.207 /* TemplateData */ Max 1 line per Limitations section 2813215 wikitext text/x-wiki {{Documentation subpage}} <!-- Add categories where indicated at the bottom of this page and interwikis at Wikidata --> {{tsh|td}} {{Lua|Module:Diff}} This template can be used to show proposed changes, e.g. on talk pages. == Usage == {{demo|br=produces a [[WP:diff|diff]] showing the changes:<br />|<nowiki>{{TextDiff|Old text.|New text.}}</nowiki>}} ==Limitations== This template is suitable only for comparing a single in-line string of text. Use of this template to compare multi-line content (i.e. HTML block content, text including line breaks) will result in syntax errors. This template must be placed on a new line, i.e. with a line break before the template. Adding the template to a line that is indented with a colon or asterisk may result in syntax errors. == TemplateData == {{templatedata header}} <templatedata> { "params": { "1": { "label": "Old text", "description": "Specify the old wikitext here.", "type": "line", "required": true }, "2": { "label": "New text", "description": "Specify the new wikitext here.", "type": "line", "required": true } }, "description": "This template compares two texts. The comparison is formatted similarly to a \"diff\" (see Help:Diff).", "format": "block" } </templatedata> == See also == * [[User:NguoiDungKhongDinhDanh/FormattedEditRequest]], a [[wp:user scripts|script]] for automatically producing diffs similar to the ones produced by this template * {{tl|StringDiff}}, which produces character-level diffs <includeonly>{{Sandbox other|| <!-- Categories below this line --> [[Category:Diff templates]] }}</includeonly> 7bou928kydwov2dwsjvqm21rczwafki 2813216 2813215 2025-09-09T16:22:35Z Wbm1058 290388 show new warning message 2813216 wikitext text/x-wiki {{Documentation subpage}} <!-- Add categories where indicated at the bottom of this page and interwikis at Wikidata --> {{tsh|td}} {{Lua|Module:Diff}} This template can be used to show proposed changes, e.g. on talk pages. == Usage == {{demo|br=produces a [[WP:diff|diff]] showing the changes:<br />|<nowiki>{{Text diff|Old text.|New text.}}</nowiki>}} {{demo|br=produces a warning mesage:<br />|<nowiki>{{Text diff|ORIGINAL_WIKITEXT|CHANGED_WIKITEXT}}</nowiki>}} ==Limitations== This template is suitable only for comparing a single in-line string of text. Use of this template to compare multi-line content (i.e. HTML block content, text including line breaks) will result in syntax errors. This template must be placed on a new line, i.e. with a line break before the template. Adding the template to a line that is indented with a colon or asterisk may result in syntax errors. == TemplateData == {{templatedata header}} <templatedata> { "params": { "1": { "label": "Old text", "description": "Specify the old wikitext here.", "type": "line", "required": true }, "2": { "label": "New text", "description": "Specify the new wikitext here.", "type": "line", "required": true } }, "description": "This template compares two texts. The comparison is formatted similarly to a \"diff\" (see Help:Diff).", "format": "block" } </templatedata> == See also == * [[User:NguoiDungKhongDinhDanh/FormattedEditRequest]], a [[wp:user scripts|script]] for automatically producing diffs similar to the ones produced by this template * {{tl|StringDiff}}, which produces character-level diffs <includeonly>{{Sandbox other|| <!-- Categories below this line --> [[Category:Diff templates]] }}</includeonly> 89s5fkdn7vbj8de22jyi38xcqvmbtni 2813217 2813216 2026-04-12T21:01:02Z W.andrea 461123 /* Usage */ Explain the warning message. 2813217 wikitext text/x-wiki {{Documentation subpage}} <!-- Add categories where indicated at the bottom of this page and interwikis at Wikidata --> {{tsh|td}} {{Lua|Module:Diff}} This template can be used to show proposed changes, e.g. on talk pages. == Usage == {{demo|br=produces a [[WP:diff|diff]] showing the changes:<br />|<nowiki>{{Text diff|Old text.|New text.}}</nowiki>}} === Warning === The template uses a [[sentinel value]] to detect whether it has been filled out properly when inserted via [[Template:Request edit button/preload]]. Using the sentinel value produces a warning mesage: {{demo|<nowiki>{{Text diff|ORIGINAL_WIKITEXT|CHANGED_WIKITEXT}}</nowiki>}} ==Limitations== This template is suitable only for comparing a single in-line string of text. Use of this template to compare multi-line content (i.e. HTML block content, text including line breaks) will result in syntax errors. This template must be placed on a new line, i.e. with a line break before the template. Adding the template to a line that is indented with a colon or asterisk may result in syntax errors. == TemplateData == {{templatedata header}} <templatedata> { "params": { "1": { "label": "Old text", "description": "Specify the old wikitext here.", "type": "line", "required": true }, "2": { "label": "New text", "description": "Specify the new wikitext here.", "type": "line", "required": true } }, "description": "This template compares two texts. The comparison is formatted similarly to a \"diff\" (see Help:Diff).", "format": "block" } </templatedata> == See also == * [[User:NguoiDungKhongDinhDanh/FormattedEditRequest]], a [[wp:user scripts|script]] for automatically producing diffs similar to the ones produced by this template * {{tl|StringDiff}}, which produces character-level diffs <includeonly>{{Sandbox other|| <!-- Categories below this line --> [[Category:Diff templates]] }}</includeonly> qm4h04lpc8if29r50stv5rib11mwz3d 2813218 2813217 2026-04-12T21:01:18Z W.andrea 461123 /* Usage */ manual line break not needed 2813218 wikitext text/x-wiki {{Documentation subpage}} <!-- Add categories where indicated at the bottom of this page and interwikis at Wikidata --> {{tsh|td}} {{Lua|Module:Diff}} This template can be used to show proposed changes, e.g. on talk pages. == Usage == {{demo|br=produces a [[WP:diff|diff]] showing the changes:|<nowiki>{{Text diff|Old text.|New text.}}</nowiki>}} === Warning === The template uses a [[sentinel value]] to detect whether it has been filled out properly when inserted via [[Template:Request edit button/preload]]. Using the sentinel value produces a warning mesage: {{demo|<nowiki>{{Text diff|ORIGINAL_WIKITEXT|CHANGED_WIKITEXT}}</nowiki>}} ==Limitations== This template is suitable only for comparing a single in-line string of text. Use of this template to compare multi-line content (i.e. HTML block content, text including line breaks) will result in syntax errors. This template must be placed on a new line, i.e. with a line break before the template. Adding the template to a line that is indented with a colon or asterisk may result in syntax errors. == TemplateData == {{templatedata header}} <templatedata> { "params": { "1": { "label": "Old text", "description": "Specify the old wikitext here.", "type": "line", "required": true }, "2": { "label": "New text", "description": "Specify the new wikitext here.", "type": "line", "required": true } }, "description": "This template compares two texts. The comparison is formatted similarly to a \"diff\" (see Help:Diff).", "format": "block" } </templatedata> == See also == * [[User:NguoiDungKhongDinhDanh/FormattedEditRequest]], a [[wp:user scripts|script]] for automatically producing diffs similar to the ones produced by this template * {{tl|StringDiff}}, which produces character-level diffs <includeonly>{{Sandbox other|| <!-- Categories below this line --> [[Category:Diff templates]] }}</includeonly> dzpaevku3sy3k6sbbj1nu8heymd29x0 2813219 2813218 2026-04-15T20:21:15Z Jonesey95 831143 typo 2813219 wikitext text/x-wiki {{Documentation subpage}} <!-- Add categories where indicated at the bottom of this page and interwikis at Wikidata --> {{tsh|td}} {{Lua|Module:Diff}} This template can be used to show proposed changes, e.g. on talk pages. == Usage == {{demo|br=produces a [[WP:diff|diff]] showing the changes:|<nowiki>{{Text diff|Old text.|New text.}}</nowiki>}} === Warning === The template uses a [[sentinel value]] to detect whether it has been filled out properly when inserted via [[Template:Request edit button/preload]]. Using the sentinel value produces a warning message: {{demo|<nowiki>{{Text diff|ORIGINAL_WIKITEXT|CHANGED_WIKITEXT}}</nowiki>}} ==Limitations== This template is suitable only for comparing a single in-line string of text. Use of this template to compare multi-line content (i.e. HTML block content, text including line breaks) will result in syntax errors. This template must be placed on a new line, i.e. with a line break before the template. Adding the template to a line that is indented with a colon or asterisk may result in syntax errors. == TemplateData == {{templatedata header}} <templatedata> { "params": { "1": { "label": "Old text", "description": "Specify the old wikitext here.", "type": "line", "required": true }, "2": { "label": "New text", "description": "Specify the new wikitext here.", "type": "line", "required": true } }, "description": "This template compares two texts. The comparison is formatted similarly to a \"diff\" (see Help:Diff).", "format": "block" } </templatedata> == See also == * [[User:NguoiDungKhongDinhDanh/FormattedEditRequest]], a [[wp:user scripts|script]] for automatically producing diffs similar to the ones produced by this template * {{tl|StringDiff}}, which produces character-level diffs <includeonly>{{Sandbox other|| <!-- Categories below this line --> [[Category:Diff templates]] }}</includeonly> 5kcd3whwgrye11flsszq3ew34v1e7rm 2813220 2813219 2026-06-06T08:30:24Z Jtneill 10242 18 revisions imported from [[:w:Template:Text_diff/doc]] 2813219 wikitext text/x-wiki {{Documentation subpage}} <!-- Add categories where indicated at the bottom of this page and interwikis at Wikidata --> {{tsh|td}} {{Lua|Module:Diff}} This template can be used to show proposed changes, e.g. on talk pages. == Usage == {{demo|br=produces a [[WP:diff|diff]] showing the changes:|<nowiki>{{Text diff|Old text.|New text.}}</nowiki>}} === Warning === The template uses a [[sentinel value]] to detect whether it has been filled out properly when inserted via [[Template:Request edit button/preload]]. Using the sentinel value produces a warning message: {{demo|<nowiki>{{Text diff|ORIGINAL_WIKITEXT|CHANGED_WIKITEXT}}</nowiki>}} ==Limitations== This template is suitable only for comparing a single in-line string of text. Use of this template to compare multi-line content (i.e. HTML block content, text including line breaks) will result in syntax errors. This template must be placed on a new line, i.e. with a line break before the template. Adding the template to a line that is indented with a colon or asterisk may result in syntax errors. == TemplateData == {{templatedata header}} <templatedata> { "params": { "1": { "label": "Old text", "description": "Specify the old wikitext here.", "type": "line", "required": true }, "2": { "label": "New text", "description": "Specify the new wikitext here.", "type": "line", "required": true } }, "description": "This template compares two texts. The comparison is formatted similarly to a \"diff\" (see Help:Diff).", "format": "block" } </templatedata> == See also == * [[User:NguoiDungKhongDinhDanh/FormattedEditRequest]], a [[wp:user scripts|script]] for automatically producing diffs similar to the ones produced by this template * {{tl|StringDiff}}, which produces character-level diffs <includeonly>{{Sandbox other|| <!-- Categories below this line --> [[Category:Diff templates]] }}</includeonly> 5kcd3whwgrye11flsszq3ew34v1e7rm Template:Warning sign 10 329999 2813252 2010-07-25T00:03:02Z Chzz 106879 I wish to use this and, sadly, I can't think of a better name for it; we have so many 'warn' thingys and 'inportant' thingys; meh 2813252 wikitext text/x-wiki [[Image:File:Nuvola apps important.svg|20px|link=|alt=]] '''Warning'''<noinclude> [[Category:Image with comment templates|{{PAGENAME}}]] </noinclude> sliguwdnwgje75pdcql8xwgeab231kr 2813253 2813252 2010-07-25T00:03:30Z Chzz 106879 image/file/make your mind up 2813253 wikitext text/x-wiki [[Image:Nuvola apps important.svg|20px|link=|alt=]] '''Warning'''<noinclude> [[Category:Image with comment templates|{{PAGENAME}}]] </noinclude> lu6omdwjy334cyqwaap11i85e2aldyn 2813254 2813253 2010-07-25T00:05:49Z Chzz 106879 docs?! 2813254 wikitext text/x-wiki [[Image:Nuvola apps important.svg|20px|link=|alt=Warning sign]] {{ucfirst:{{{1|Warning}}}}}<noinclude> [[Category:Image with comment templates|{{PAGENAME}}]] {{documentation}}</noinclude> gykjz676ff8cwvkjf886eyr11avj4zj 2813255 2813254 2010-07-25T00:17:44Z Chzz 106879 bold 2813255 wikitext text/x-wiki [[Image:Nuvola apps important.svg|20px|link=|alt=Warning sign]] {{ucfirst:{{{1|'''Warning'''}}}}}<noinclude> [[Category:Image with comment templates|{{PAGENAME}}]] {{documentation}}</noinclude> sr69tbrtqujr9fcakhw8ismhkeyge6n 2813256 2813255 2010-11-13T20:19:15Z WOSlinker 351635 update image 2813256 wikitext text/x-wiki [[Image:Ambox warning pn.svg|20px|link=|alt=Warning sign]] {{ucfirst:{{{1|'''Warning'''}}}}}<noinclude> [[Category:Image with comment templates|{{PAGENAME}}]] {{documentation}}</noinclude> b1r8t577c64mb0qhhxkrn6yhwzs0l2p 2813257 2813256 2013-03-16T17:23:19Z Magog the Ogre 26224 safesubst 2813257 wikitext text/x-wiki [[Image:Ambox warning pn.svg|20px|link=|alt=Warning sign]] {{{{{|safesubst:}}}ucfirst:{{{1|'''Warning'''}}}}}<noinclude> [[Category:Image with comment templates|{{PAGENAME}}]] {{documentation}}</noinclude> orsu9fbrv21xg5ux1aly8hm9tese5hr 2813258 2813257 2013-07-29T18:45:18Z ChristTrekker 459343 2813258 wikitext text/x-wiki [[Image:Ambox warning pn.svg|20px|link=|alt=⚠]] {{{{{|safesubst:}}}ucfirst:{{{1|'''Warning'''}}}}}<noinclude> [[Category:Image with comment templates|{{PAGENAME}}]] {{documentation}}</noinclude> 7d2dt44l7fj5mjdebbpncc3qe5x0wf9 2813259 2813258 2013-09-03T00:58:32Z 59.167.25.9 2813259 wikitext text/x-wiki [[Image:http://www.wikiprogress.org/images//Ambox_warning_pn.svg|20px|link=|alt=⚠]] {{{{{|safesubst:}}}ucfirst:{{{1|'''Warning'''}}}}}<noinclude> [[Category:Image with comment templates|{{PAGENAME}}]] {{documentation}}</noinclude> e4ckey46yfzx9yelfm9t2szmg9ekasj 2813260 2813259 2013-09-03T07:07:55Z Klortho 296249 [[Help:Reverting|Reverted]] edits by [[Special:Contributions/59.167.25.9|59.167.25.9]] ([[User talk:59.167.25.9|talk]]) to last version by ChristTrekker 2813260 wikitext text/x-wiki [[Image:Ambox warning pn.svg|20px|link=|alt=⚠]] {{{{{|safesubst:}}}ucfirst:{{{1|'''Warning'''}}}}}<noinclude> [[Category:Image with comment templates|{{PAGENAME}}]] {{documentation}}</noinclude> 7d2dt44l7fj5mjdebbpncc3qe5x0wf9 2813261 2813260 2014-02-19T22:54:12Z Anon126 1078576 allow bolded custom message 2813261 wikitext text/x-wiki [[Image:Ambox warning pn.svg|20px|link=|alt=⚠]] '''{{{{{|safesubst:}}}ucfirst:{{{1|Warning}}}}}'''<noinclude> [[Category:Image with comment templates|{{PAGENAME}}]] {{documentation}}</noinclude> fox02aa2xffw3d1pknezgg7fmoaaavq 2813262 2813261 2014-08-19T20:00:09Z Funandtrvl 47226 fx code 2813262 wikitext text/x-wiki [[File:Ambox warning pn.svg|20px|link=|alt=⚠]]&nbsp;'''{{{{{|safesubst:}}}ucfirst:{{{1|Warning}}}}}'''<noinclude> {{Documentation}} <!--Please add this template's categories to the /doc subpage, not here - thanks!--> </noinclude> 1pssf1jjsub1k8rvvma596xy33lwgkp 2813263 2813262 2017-11-13T23:10:01Z AlexiusHoratius 2087767 Protected "[[Template:Warnsign]]": [[WP:High-risk templates|Highly visible template]] ([Edit=Require template editor access] (indefinite) [Move=Require template editor access] (indefinite)) 2813262 wikitext text/x-wiki [[File:Ambox warning pn.svg|20px|link=|alt=⚠]]&nbsp;'''{{{{{|safesubst:}}}ucfirst:{{{1|Warning}}}}}'''<noinclude> {{Documentation}} <!--Please add this template's categories to the /doc subpage, not here - thanks!--> </noinclude> 1pssf1jjsub1k8rvvma596xy33lwgkp 2813264 2813263 2020-10-20T22:38:28Z Jonesey95 831143 fix alt text 2813264 wikitext text/x-wiki [[File:Ambox warning pn.svg|20px|link=|alt=red-outlined triangle containing exclamation point]]&nbsp;'''{{{{{|safesubst:}}}ucfirst:{{{1|Warning}}}}}'''<noinclude> {{Documentation}} <!--Please add this template's categories to the /doc subpage, not here - thanks!--> </noinclude> d5qrzriz7mjoplnhw283nabckh3ynbn 2813265 2813264 2020-10-20T22:39:04Z Jonesey95 831143 rm empty link= 2813265 wikitext text/x-wiki [[File:Ambox warning pn.svg|20px|alt=red-outlined triangle containing exclamation point]]&nbsp;'''{{{{{|safesubst:}}}ucfirst:{{{1|Warning}}}}}'''<noinclude> {{Documentation}} <!--Please add this template's categories to the /doc subpage, not here - thanks!--> </noinclude> 9o601nqu40t2p0v6v7udbuutvllyier 2813266 2813265 2021-01-13T14:19:01Z Sdkb 526932 delinking warning sign file, since that's not what anyone is seeking 2813266 wikitext text/x-wiki [[File:Ambox warning pn.svg|20px|link={{{link|}}}|alt=red-outlined triangle containing exclamation point]]&nbsp;'''{{{{{|safesubst:}}}ucfirst:{{{1|Warning}}}}}'''<noinclude> {{Documentation}} <!--Please add this template's categories to the /doc subpage, not here - thanks!--> </noinclude> 46anubbj5564549damcz67hvn0fxy5u 2813267 2813266 2023-03-09T01:55:22Z Terasail 2907175 Use [[Template:Respond]] 2813267 wikitext text/x-wiki {{Respond|Ambox warning pn.svg|{{{{{|safesubst:}}}ucfirst:{{{1|Warning}}}}}|alt=red-outlined triangle containing exclamation point}}<noinclude> {{Documentation}} <!--Please add this template's categories to the /doc subpage, not here - thanks!--> </noinclude> dz6iifo3cg6ihgjxxk348elxinlz0vn 2813268 2813267 2024-06-29T13:49:47Z Hex 758220 Scott moved page [[Template:Warnsign]] to [[Template:Warning sign]]: Plain English title improves searchability 2813267 wikitext text/x-wiki {{Respond|Ambox warning pn.svg|{{{{{|safesubst:}}}ucfirst:{{{1|Warning}}}}}|alt=red-outlined triangle containing exclamation point}}<noinclude> {{Documentation}} <!--Please add this template's categories to the /doc subpage, not here - thanks!--> </noinclude> dz6iifo3cg6ihgjxxk348elxinlz0vn 2813269 2813268 2026-01-31T04:41:12Z TechnoSquirrel69 2965499 Implementing [[Template talk:Warning sign#Template-protected edit request, January 26, 2026|template-protected edit request]] from yours truly 2813269 wikitext text/x-wiki {{Respond|Ambox warning pn.svg|{{{1|Warning}}}}}<noinclude> {{Documentation}} <!--Please add this template's categories to the /doc subpage, not here - thanks!--> </noinclude> h4i25xjf0dpk4k143boqzheakwiujk0 2813270 2813269 2026-06-06T08:30:25Z Jtneill 10242 18 revisions imported from [[:w:Template:Warning_sign]] 2813269 wikitext text/x-wiki {{Respond|Ambox warning pn.svg|{{{1|Warning}}}}}<noinclude> {{Documentation}} <!--Please add this template's categories to the /doc subpage, not here - thanks!--> </noinclude> h4i25xjf0dpk4k143boqzheakwiujk0 File:VLSI.Arith.2A.CLA.20260606.pdf 6 330000 2813272 2026-06-06T08:43:36Z Young1lim 21186 {{Information |Description=Carry Lookahead Adders 2A traditional (20260606 - 20260605) |Source={{own|Young1lim}} |Date=2026-06-06 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2813272 wikitext text/x-wiki == Summary == {{Information |Description=Carry Lookahead Adders 2A traditional (20260606 - 20260605) |Source={{own|Young1lim}} |Date=2026-06-06 |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}} b722sr3kedyzxjrfg41krr87owglm02 File:VLSI.Arith.2B.CLA.20260606.pdf 6 330001 2813273 2026-06-06T08:44:19Z Young1lim 21186 {{Information |Description=Carry Lookahead Adders 2B simplified (20260606 - 20260605) |Source={{own|Young1lim}} |Date=2026-06-06 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2813273 wikitext text/x-wiki == Summary == {{Information |Description=Carry Lookahead Adders 2B simplified (20260606 - 20260605) |Source={{own|Young1lim}} |Date=2026-06-06 |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}} 4xpp6m2sy9nk8j4knx0tjop35bw8w6d File:C04.SA0.PtrOperator.1A.20260606.pdf 6 330002 2813275 2026-06-06T09:24:07Z Young1lim 21186 {{Information |Description=C04.SA0: Address and Dereference Operators (20260606 - 20260605) |Source={{own|Young1lim}} |Date=2026-06-06 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2813275 wikitext text/x-wiki == Summary == {{Information |Description=C04.SA0: Address and Dereference Operators (20260606 - 20260605) |Source={{own|Young1lim}} |Date=2026-06-06 |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}} 6edv3n9reen53rk9zl5auzhjpubsi2r File:Laurent.5.Permutation.6C.20260606.pdf 6 330003 2813277 2026-06-06T09:31:59Z Young1lim 21186 {{Information |Description=Laurent.5: Permutation 6C (20260606 - 20260605) |Source={{own|Young1lim}} |Date=2026-06-06 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2813277 wikitext text/x-wiki == Summary == {{Information |Description=Laurent.5: Permutation 6C (20260606 - 20260605) |Source={{own|Young1lim}} |Date=2026-06-06 |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}} aa5wpf7alv4t9c68f2kfmswkfvl4c27