Wikiversity enwikiversity https://en.wikiversity.org/wiki/Wikiversity:Main_Page MediaWiki 1.47.0-wmf.11 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 4 28 2818495 2818281 2026-07-18T21:41:19Z ~2026-40514-81 3101893 /* July 2026 Wikimedia Café meetups regarding Wikimedia governance and options for reform */ Reply 2818495 wikitext text/x-wiki {{Wikiversity:Colloquium/Header}} <!-- MESSAGES GO BELOW --> == 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) ::A few days shy of 30, it seems obvious that this is not going to pass. So I '''withdraw''' as presumptively '''failed'''. ―[[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> 07:14, 9 June 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]] ^{[[User talk:Mu301|talk]]} 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) * {{oppose}} per above. Wikiversity<math>\not=</math> Wikinews - not a good idea to mix the scope of projects. --[[User:Bert Niehaus|Bert Niehaus]] ([[User talk:Bert Niehaus|discuss]] • [[Special:Contributions/Bert Niehaus|contribs]]) 12:03, 8 June 2026 (UTC) * {{abstain}} I will abstain since I'm not an active Wikiversity contributor. But I just feel like Wikinews had a very clear and specific goal of providing news, and Wikiversity is just a different project with different goals. For me, it would be odd to rehost Wikinews here. But please do not count my vote, this is only a comment. --[[User:Antimundo|Antimundo]] ([[User talk:Antimundo|discuss]] • [[Special:Contributions/Antimundo|contribs]]) 13:19, 6 June 2026 (UTC) * {{oppose}} Although I think it's a pity that Wikinews is closed. --[[User:Dick Bos|Dick Bos]] ([[User talk:Dick Bos|discuss]] • [[Special:Contributions/Dick Bos|contribs]]) 19:06, 8 June 2026 (UTC) *{{support}} In 2018 I initiated [[:Category:Videoconferences on media and democracy]] as a platform for disseminating public affairs events. In 2021 I officially initiated a podcast series on "Media & Democracy" syndicated for the [[w:List of Pacifica Radio stations and affiliates|Pacifica radio network]]. In 2024 I converted it from irregular to fortnightly. I think this is all educational and supports the Wikiversity education mission, and I think that "rehost Wikinews here" would be appropriate. (I had some experience with Wikinews a few years ago. I felt it was too tightly controlled: Article submissions went stale, because I could not get official permission to publish and I could not get the information needed to understand what I was supposed to do to obtain the official permission. I would be opposed to rehosting Wikinews here if the policy similarly made it unreasonably difficult for volunteer contributor to get the information needed to meet the journalistic standards imposed by the overworked editors.) {{unsigned|DavidMCEddy}} ===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;">^{Talk page}</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;">^{Talk page}</span></b>]] 22:43, 5 June 2026 (UTC) *:::Thats a good point. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:09, 9 June 2026 (UTC) == Create an autopatrolled user group? == {{tracked|T428269|resolved}} 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) : @[[User:Jtneill|Jtneill]] and @[[User:Koavf|Koavf]]: the autopatroller user group has been implemented here. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:14, 8 June 2026 (UTC) ::Thanks. ―[[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> 07:13, 9 June 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) :Recently agreed [[Wikiversity:Artificial intelligence|policy]] welcome users to tag AI generated pages. Me personally I am not against the use of AI. What is the difference in abstract schematic image created by a human and the same by an AI. If the users does not have finances to pay digital artest and you dont want to let them use AI, would you pay the artest for them? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:07, 8 June 2026 (UTC) ::Wikimedia has a lot of ''volunteer'' artists who can illustrate if asked. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 08:11, 9 June 2026 (UTC) :::Interesting! That's good to know. Where can we find the volunteer artists for illustrating? [[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 20:11, 9 June 2026 (UTC) ::::Wikimedia commons has [[commons:Commons:Graphic Lab/Illustration workshop]] [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 02:18, 10 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) : Juandev has posted some comments on the [[Wikiversity talk:Inactivity policy|talk page]]. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:30, 12 June 2026 (UTC) == Proposed user group and/or possible policy changes == {{tracked|T430416|fixed}} 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) :#Yes, I agree. :#Thats a good point, but I dont know. At least I dont think its a good idea that both groups i.e. crats and custodiants can do that, it may create chaos. :#Another good point. It seems to me that the current situation is somewhat unclear and should be clarified. I understand the original status of [[Wikiversity:Probationary custodians|Probationary custodians]] as a historicall and invalid, but at the same time I consider myself a probationary custodian, because on the Wikiversity:Custodianship page in the ''[[Wikiversity:Custodianship#How does one become a custodian?|How does one become a custodian?]]'' section it says, I quote, ''"II ...then you will be approved as a probationary custodian for a period of at least four weeks"''. :::Mentors should definitely be kept, but for certain applicants the probation and mentorship should be abolished. For example, if someone was an active custodian for 5 years, then loses their rights or gives them up for a year and then wants to resume their custodial activities, there is no reason for them to undergo a training period. It burdens both the mentors and the community with double voting. The only exception could be a situation where policies or tools for custodians change significantly during that year, or the candidate wants to. :[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 06:08, 9 June 2026 (UTC) == New user what do I do here == I love wikipedia and the wikiversity project seems super interesting. However I know very little about wikiversity and would like to know how i can best contribute to the project. Also if there are forums or discord or reddit that would be very helpful. (One last thing is it normal that my userboxes don't work here) {{unsigned|AUBSTRAWBS}} :Hey {{ping|AUBSTRAWBS}} Welcome to Wikiversity! I've left a welcome message on your talk page so that should provide you a plethora of useful links for you to look at so you can familiarize yourself with the project. Also, feel free to create the userboxes you need. Wikiversity doesn't have as many userboxes as Wikipedia. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 21:45, 8 June 2026 (UTC) :Thank you very much :) hope to contribute a lot. [[User:AUBSTRAWBS|AUBSTRAWBS]] ([[User talk:AUBSTRAWBS|discuss]] • [[Special:Contributions/AUBSTRAWBS|contribs]]) 21:50, 8 June 2026 (UTC) == Towards an Ethics policy == In connection with the [[Wikiversity:Community Review/Removal of Wikidebates|discussion of Wikidebates]], I said that it would be good to establish a policy on ethics, or rather a boundary between ethical and unethical content, so that we don't have to discuss individual cases. In addition, today we also have some global policies that prohibit, for example, attacks on members of the Wikimedia movement or undermining other projects. However, at the very beginning, I would start by collecting your opinions. What content or what research should not be allowed on Wikiversity? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 05:52, 9 June 2026 (UTC) :One ethical issue that I think should be non-controversial is related to good faith in the learning modules. So, learning materials should not be hoaxes or encourage behavior or methods that don't work or that misrepresent the facts or the likelihood of something occurring, etc. and authors should also not plagiarize or misrepresent authorship, etc. That was quite a run-on, but I hope that others can tease out what I mean here. ―[[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> 07:39, 9 June 2026 (UTC) ::I look at it from a practical perspective. We can give that to the policy, but I see the problem in that we are not able to check it except plagiarism. ::Plagiarism can be partially detected during patrolling. I see a new text, I put part of it in Google and I check if it is copied from the web. It is a problem with copying from books or other offline sources, but sometimes it happens that someone finds out that something is copied from somewhere and it can be deleted. ::The biggest issue we have here is that we are missing Wikipedia's control mechanism: references. Only some types of resources on Wikiversity require references. In-line references are not often used in courses, exercises, lectures, etc. We are thus deprived of one of the excellent control mechanisms and the only option is for the increase in the number of members with various qualifications to check it for their colleagues. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 07:59, 9 June 2026 (UTC) :::Having a policy and enforcing that policy are indeed two different things. If we are only concerned with issues that we can definitively enforce, then that will definitely change this conversation. ―[[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> 08:06, 9 June 2026 (UTC) ::::ok [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 15:55, 13 June 2026 (UTC) :AI generated content should not be allowed as it is inherently plagiarism. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 08:14, 9 June 2026 (UTC) ::And if the user mention it was generated by an AI? Note that there is something called as public domain, that is the author wave its rights. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 09:53, 9 June 2026 (UTC) :::Plagiarism isn’t copyright violation. Crediting the AI is not crediting the authors the AI stole from without credit. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 10:18, 9 June 2026 (UTC) ::::I see, now I understand your point. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 15:56, 13 June 2026 (UTC) == Deployment of Legal and Safety Contacts Link in the Footer of Your Wiki == Hello community, The Wikimedia Foundation has provided [[foundation:Legal:Wikimedia Foundation Legal and Safety Contact Information|a single legal and safety contact page]], to be linked in the footer of your wiki, to ensure access to accurate legal information. This is a regulatory requirement. We have already rolled out links to English, German, Italian, Spanish Wikipedias and other wikis and we will deploy to your wiki soon. Please [[m:Wikimedia Foundation Legal and Safety Contacts FAQ|read more on the project page]] and leave any comments in this thread or on [[m:Talk:Wikimedia Foundation Legal and Safety Contacts FAQ|the talk page]]. –– [[User:STei (WMF)|STei (WMF)]] ([[User talk:STei (WMF)|discuss]] • [[Special:Contributions/STei (WMF)|contribs]]) 18:12, 9 June 2026 (UTC) :Thanks for the notice. In case anyone is not clear, we cannot locally change the text at the footer, as it [[:mw:Manual:Footer|requires access to the server settings]]. If we locally needed to change it, we would have to file a ticket at [[:phab:]]. Since the above was sent by someone from the WMF, I think they are on it and it will be updated without any action from anyone here. ―[[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> 18:24, 9 June 2026 (UTC) == Image not displaying == Can anyone work out why this image isn't displaying?<br> [[Educational Media Awareness Campaign/Physics/POTD 10]] -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:45, 11 June 2026 (UTC) :Not sure, but it was an issue with the file itself and either way, it should be (and I have since done this) replaced with the SVG [[:File:Telescope-schematic.svg]]. ―[[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> 13:59, 11 June 2026 (UTC) == New nomination template(s) == I created {{tlx|Nomination}} when someone requests curator or custodian permissions, which often at least require mentorship. On the other hand, I might create {{tlx|Nomination 2}}, in which the latter does not have a section about mentorship (often used for bureaucrat or interface administrator nominations). [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:29, 12 June 2026 (UTC) == June 2026 Wikimedia Café meetups regarding the English Wikipedia Editor Reflections project == <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: 10px; padding-right: 10px; padding-left: 10px; padding-bottom: 10px;">[[File:Wikimedia Café logo in plain SVG format.svg|60px|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 June. Both sessions will focus on the [https://en.wikipedia.org/wiki/Wikipedia:Editor_reflections English Wikipedia Editor Reflections project]. The featured guest in the Café will be [https://en.wikipedia.org/wiki/User:Clovermoss User:Clovermoss]. Participants may attend either or both sessions. #'''27 June 2026 15:00 UTC''' ([https://zonestamp.toolforge.org/1782572400 timestamp converter]), at a time friendly to the Americas, Africa, and Europe #'''28 June 2026 03:00 UTC''' ([https://zonestamp.toolforge.org/1782615600 timestamp converter]), at a time friendly to Asia and the Pacific Please see the Café page for more information, including [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> 04:00, 15 June 2026 (UTC) == Mobile friendly main page == Hello, I have recently been using wikiversity on mobile and unlike wikipedia some images and boxes stick out instead of all having a set width which means you can scroll a little side to side, which makes the site feel a bit unfinished. Its just a suggestion but I think it will wake the user experience much better {{unsigned|AUBSTRAWBS}} :{{Ping|AUBSTRAWBS}} I don't use a smartphone. Can you give me more details or even take some screenshots? You can upload them at [[:c:Category:English Wikiversity screenshots]]. ―[[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> 13:30, 18 June 2026 (UTC) ::Hi i uploaded an image of the problem. Since some of the images are larger than the screen and not adjusted to fit they stick out and makes the page larger which lets you scroll right and have a big white rectangle on the side [[User:AUBSTRAWBS|AUBSTRAWBS]] ([[User talk:AUBSTRAWBS|discuss]] • [[Special:Contributions/AUBSTRAWBS|contribs]]) 14:03, 18 June 2026 (UTC) :::Thanks. I agree that this is an issue, but it's a pretty minor-to-moderate one to me and I don't think I will be able to dedicate time to fix it myself. Showing it to others here is useful in case someone else wants to tinker with the CSS to resolve it. Thanks for bringing it to the community's attention. ―[[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> 15:42, 18 June 2026 (UTC) ::::I do know CSS as I like to maintain a blog online so I could try and fix it but I don't know if I have the access to do that, would i need to be a curator/ custodian. Alternatively i could edit a sandbox version of the main page and then send it to someone. [[User:AUBSTRAWBS|AUBSTRAWBS]] ([[User talk:AUBSTRAWBS|discuss]] • [[Special:Contributions/AUBSTRAWBS|contribs]]) 20:00, 18 June 2026 (UTC) :::::Oh great. There are a lot of draft versions of the main page like [[Wikiversity:Main Page/Draft version 0.2]], so you can make [[Wikiversity:Main Page/Sandbox]] if you want and edit there. If you can tinker it to your liking, I can edit the main page. ―[[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> 20:14, 18 June 2026 (UTC) ::::::thank you, i'll check it out [[User:AUBSTRAWBS|AUBSTRAWBS]] ([[User talk:AUBSTRAWBS|discuss]] • [[Special:Contributions/AUBSTRAWBS|contribs]]) 22:16, 18 June 2026 (UTC) == Main page titles == Currently, the title says "Wikiversity:Main Page", but in my opinion, it's too basic. I would like to propose changing it with the following options (you may only pick one): # Option 1: Set both [[MediaWiki:Mainpage-title]] and [[MediaWiki:Mainpage-title-loggedin]] to blank, giving the main page a portal-like design (as with English Wikipedia, English Wikibooks, etc.) # Option 2: Modify [[MediaWiki:Mainpage-title]] to <code>Welcome to Wikiversity</code> (for unregistered users), and [[MediaWiki:Mainpage-title-loggedin]] to <code><nowiki>Welcome to Wikiversity, $1!</nowiki></code>; the latter would display to me as <code>Welcome to Wikiversity, Codename Noreste!</code> Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:34, 18 June 2026 (UTC) : Pinging @[[User:Jtneill|Jtneill]] and @[[User:Koavf|Koavf]] for input above. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:03, 24 June 2026 (UTC) :I'm afraid that I don't have strong feelings on this. Changing to either or staying with the status quo are all fine to 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> 16:10, 24 June 2026 (UTC) : I like the option of being consistent with Wikipedia and Wikibooks -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 00:04, 25 June 2026 (UTC) : {{done}}. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:34, 30 June 2026 (UTC) : Thankyou - looks good. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 03:29, 2 July 2026 (UTC) == Wiki x AI preconference day @ Wikimania == There will be a preconference day at Wikimania about [[meta:Artificial_intelligence/2026_Wiki_AI | Wiki AI]]. It will be mostly offline, but there will be at least one hybrid session for demos of community-developed AI tools and workflows. * If you've built something cool, that is a chance to show it off, list it on the gallery of tools in progress, and get feedback. * If you could ask the people shaping AI on the wikis (WMF, tool builders, model trainers, GLAM and policy folks) a question, what would it be? Cheers, <span style="padding:0 2px 0 2px;background-color:white;color:#bbb;">&ndash;[[User:Sj|SJ]][[User Talk:Sj|<span style="color:#ff9900;">+</span>]]</span> 23:12, 20 June 2026 (UTC) and Alaexis<br>{{comment|1=Copied from https://en.wikiversity.org/w/index.php?title=Talk%3AMotivation_and_emotion%2FAssessment%2FUsing_generative_AI&diff=2816357&oldid=2807052}} == RFC about AI-generated content in Wikimedia Commons == You are invited to participate in a [[c:Commons:Requests for comment/Policy update for AI content|request for comment on Wikimedia Commons about a policy update for AI content]]. This may affect files that are uploaded to Wikimedia Commons for use on this project. Thank you. [[m:User:Codename Noreste|Codename Noreste]] ([[m:User talk:Codename Noreste|discuss]]) 17:12, 23 June 2026 (UTC) <!-- Message sent by User:Codename Noreste@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Global_message_delivery&oldid=30513860 --> == Deployment of Legal and Safety Contacts Link in the Footer of Your Wiki == <section begin="Message"/> '''Legal & Safety Contacts''' Hello community, the Wikimedia Foundation has provided a [[wmf:Special:MyLanguage/Legal:Wikimedia Foundation Legal and Safety Contact Information|single legal and safety contact page]], to be linked in the footer of your wiki, to ensure access to accurate legal information. This is a regulatory requirement. We have already rolled out links to English, German, Italian, Spanish and other wikis and we will deploy to your wiki soon. [[m:Special:MyLanguage/Wikimedia_Foundation_Legal_and_Safety_Contacts_FAQ|Please read more on the project page]] and leave any comments in this thread or on the [[m:Special:MyLanguage/Talk:Wikimedia Foundation Legal and Safety Contacts FAQ|talk page]]. <section end="Message"/> -- [[User:Sannita (WMF)|User:Sannita (WMF)]] ([[User talk:Sannita (WMF)|talk]]) 13:31, 25 June 2026 (UTC) <!-- Message sent by User:Sannita (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=User:Sannita_(WMF)/Mass_sending_test&oldid=30731267 --> == Preparing manuscript for submission to the ''WikiJournal of Humanities'' == I am working on preparing an article in Wikipedia for a dual goal of submitting it for a featured article candidacy in Wikipedia and submitting it to the ''[[WikiJournal of Humanities]]''. I have an open request for pre-submission peer review at [[en:Wikipedia:Wikipedia:Peer review/Rei Ayanami/archive2|Wikipedia:Peer review/Rei Ayanami/archive2]], and I am asking for someone experienced with submitting journals to WikiJournals. The article is not ready for submission, and I would like to know where I can get assistance from users who submitted articles to the journal, but did not necessarily review them. Furthermore, I said there that submitting to the ''WikiJournal of Humanities'' depends on whether the article attains featured article status in Wikipedia, as I would like to use the featured article as a manuscript for a journal article. [[User:Z. Patterson|Z. Patterson]] ([[User talk:Z. Patterson|discuss]] • [[Special:Contributions/Z. Patterson|contribs]]) 12:04, 30 June 2026 (UTC) == I could probably save many kilobytes by compressing my LLM chat history == I could probably save many kilobytes by compressing my LLM chat history. Would that be ok? I like how my new "method" looks: [[User:ThinkingScience/All General AI Prompt History Archive]] very compressed and neat. I like how to find new ways to make people using LLMs not become "secondary citizens". [[User:ThinkingScience|ThinkingScience]] ([[User talk:ThinkingScience|discuss]] • [[Special:Contributions/ThinkingScience|contribs]]) 05:05, 7 July 2026 (UTC) :{{replyto|ThinkingScience}} I do not see why not. People should be able to look in your page's history for LLM chats before your compression. [[User:Z. Patterson|Z. Patterson]] ([[User talk:Z. Patterson|discuss]] • [[Special:Contributions/Z. Patterson|contribs]]) 02:34, 14 July 2026 (UTC) ::That's good. Then an admin if they have been instructed to compress...then they can delete the edit history perhaps and save space, then it's up to the admin and I don't need to worry about taking up too many resources. That's good! [[User:ThinkingScience|ThinkingScience]] ([[User talk:ThinkingScience|discuss]] • [[Special:Contributions/ThinkingScience|contribs]]) 07:50, 14 July 2026 (UTC) == July 2026 Wikimedia Café meetups regarding Wikimedia governance and options for reform == <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: 10px; padding-right: 10px; padding-left: 10px; padding-bottom: 10px;">[[File:Wikimedia Café logo in plain SVG format.svg|60px|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 in July. Both sessions will focus on Wikimedia governance, including possible follow-ups to the [https://meta.wikimedia.org/wiki/Movement_Charter Movement Charter] and options for reform. Participants may attend either or both Café sessions. This month, to deconflict the Café meetups from Wikimania, the meetups will be held one day later than usual. #'''26 July 2026 15:00 UTC''' ([https://zonestamp.toolforge.org/1785078000 timestamp converter]), at a time friendly to the Americas, Africa, and Europe #'''27 July 2026 03:00 UTC''' ([https://zonestamp.toolforge.org/1785121200 timestamp converter]), at a time friendly to Asia and the Pacific Please see the Café page for more information, including [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> 03:51, 13 July 2026 (UTC) :@[[User:Pine|Pine]], Some wikimedians who may contribute less popular, but hopefully productive, comments, are those who are currently blocked on Meta. I myself was blocked infinitely on Meta several years ago, but was allowed to contInue contributing on other wikimedia projects, such as this one. There are many long term contributors, some who have contrIbuted tens of thousands of edits to wikimedia projects who are are now blocked on specific projects. :Those who happen to be blocked on Meta have, to all intents and purposes, been removed from the wikimedia movement. :Ottawahitech [[Special:Contributions/&#126;2026-40514-81|&#126;2026-40514-81]] ([[User talk:&#126;2026-40514-81|talk]]) 21:41, 18 July 2026 (UTC) 0yvd6gadj700428ssbas246dph435ds 102 4704 2818481 2815925 2026-07-18T14:49:06Z Grimes2 2895877 +1 2818481 wikitext text/x-wiki [[File:Musical notes.svg|right|372x372px]] Music is a self-expressed art form that organizes sound and silence through rhythm, tempo, pitch, melody, harmony, timbre, texture, dynamics, and other acoustic or electronic means, including instruments, synthesis, sampling, and environmental sound. Whether a work is recognized as music often depends on cultural context and the shared understanding of performers and listeners. Music may be created individually or collectively through ensembles, orchestras, groups, and other forms of participation. While atonal music exists, many musical traditions are characterized by melody and harmony. Rhythm encompasses tempo, meter, and articulation, while dynamics describe variations in loudness. Timbre and texture refer to the distinctive qualities and combinations of sounds, sometimes described as the "color" of music. ==Music and aural theory== ===[[Fundamentals of Music|Fundamentals of music]]=== Music is organized sound, combining pitch, rhythm, timbre, dynamics, texture, and form. These elements provide us a common vocabulary for describing how music is structured and experienced. Fundamentals of Music serves as a foundation for later study in theory, performance, composition, and musicology, introducing the basic concepts used to analyze and discuss music regardless of genre or cultural origin. * [[Introduction to music]] (To be absorbed into Fundamentals of music) ===Western music theory=== The goal of this section is to equip the student with the tools and skills necessary to compose, arrange and analyze music. Upon studies of these resources, students will possess the skills and knowledge of western theory, creative writing, arranging, as well as having a portfolio of original works. {{colbegin|3}}* [[Music Theory I: Basics]] ** [[Music Notation]] ** [[Scales (music)]] ** [[Key Signatures]] ** [[Meter and Time Signatures]] * [[Harmony]] * [[Music Theory II: Harmony 1]] ** [[Triads]] ** [[Cadence]] ** [[Inversions (music)]] * [[Chords (music)|Chords]] * [[Music Theory III: Harmony 2]] ** [[Seventh chords]] ** [[Modulation]] ** [[Modal mixture]] ** [[Chromatic Harmony]] * [[Music Theory IV: Advanced Analysis]] ** [[Form and Analysis|Form and analysis]] ** [[Counterpoint]] ** [[Fugue]] * [[Music Theory V: Developing Frontiers]] ** [[Non-functional Harmony]] ** [[Xenharmonic Music Theory]] ** [[Xenrhythm]] * [[Music Appreciation|Music appreciation]] * [[w:Glossary of music terminology|Glossary of music terminology]]{{colend}} === Non-western music theory=== Some Non-European cultures have different music composition, arrangement and analysis traditions, less commonly known in western cultural spheres.{{colbegin|3}} * [[Gamelan music theory]] * [[Carnatic music theory]] * [[Babylonian Tonal System]] * [[Andalusi classical music theory]] * [[Persian classical music theory]] * [[Arabic classical music theory]] * [[Ottoman classical music theory]] * [[Hindustani classical music theory]]{{colend}} ===Aural theory and ear training=== Ear training is learning/training your ears to recognize what you hear and put it down onto paper. These are basic learning guides, exercises and projects to help you understand in a meaningful way the flurry of sound in music.{{colbegin|3}} * [[Ear training - Intervals and Harmony|Ear training - Intervals and Harmony (pitch oriented)]] * [[Ear training - rhythm]] * [[Sight singing]] * [[Transcription (music)]] {{colend}} ===Genres=== Some genres of Western music have genre-specific music theory.{{colbegin|3}} * [[Basic Blues & Rock]] * [[Country music]] * [[Folk music]] * [[Jazz]]{{colend}} ==Composition== {{colbegin|3}} * [[Beginning composition]] * [[Advanced composition]] * [[Lyrical composition|Lyrical composition]] * [[Arranging]] * [[Orchestration]] * [[Film scoring for Musicians|Film scoring]] (in conjunction with the [[Course:Practical narrative film editing|Film editing course]]) * [[Final Theory Project]] ===[[Music Technology]]=== *[[DAWs]] * [[MIDI]] {{colend}} ==Performance== {{colbegin|3}} * [[Solo performance]] * [[Ensemble performance]] * [[Improvisation (music)]] * [[Conducting]] * [[Music pedagogy]] {{colend}} ==Musicology== {{colbegin|3}} ===General Musicology=== * [[Music Appreciation|Music appreciation and history]] * [[Survey of Musical Genres|Survey of musical genres]] * [[Music in Film|Music in film]] * [[The Symphony and the Opera|Symphony and opera]] ===Historical Musicology=== ====Western Music History==== * [[Brief History of Western Music]] * [[Music of the Medieval Era]] * [[Music of the Renaissance]] * [[Music of the Baroque Era]] * [[Music of the Classical Era]] * [[Music of the Romantic Era]] * [[Music of the 20th Century]] ====Nonwestern Music History==== * [[Gamelan music history]] * [[Carnatic music history]] * [[Andalusi classical music history]] * [[Persian classical music history]] * [[Arabic classical music history]] * [[Ottoman classical music history]] * [[Hindustani classical music ]] ===Ethnomusicology=== * [[Folk Music]] * [[Indigenous Music]] * [[Comparative Ethnomusicology]] {{colend}} ==Music instruments== {{MultiCol}} === [[String instruments]] === * [[Violin]] * [[Viola]] * [[Violoncello]] * [[Double bass]] * [[Fiddle]] * [[Harp]] * [[Guitar]] ** [[Classical guitar|Classic guitar (or ''"Acoustic"'' guitar)]] ** [[Electric Guitar|Electric guitar]] ** [[Bass guitar|Bass guitar]] * [[Ukulele]] * [[Banjo]] * [[Mandolin]] * [[Lute]] {{ColBreak}} === [[Woodwind instruments]] === * [[Flute]] * [[Oboe]] * [[Clarinet]] * [[Bassoon]] * [[Saxophone]] ** [[Soprano Saxophone|Soprano saxophone]] ** [[Alto Saxophone|Alto saxophone]] ** [[Tenor Saxophone|Tenor saxophone]] ** [[Baritone Saxophone|Baritone saxophone]] * [[Recorder]] * [[Ocarina]] ===[[Brass instruments]]=== * [[Trumpet]] * [[French horn]] * [[Trombone]] * [[Tuba]] * [[Euphonium]] {{ColBreak}} === [[Percussion instruments]] === * [[Concert Percussion|Concert percussion]] (Snare drum, crash cymbals, timpani, etc.) * [[Drum set]] * [[Mallet Instruments|Mallet instruments]] (Marimba, xylophone, vibraphone, chimes, etc.) * [[Tabla]] (an Indian pair of drums) * [[Pipe and tabor]] === [[Keyboard instruments]] === * [[Piano]] * [[Organ]] === [[Topic: Voice | Voice]] === * [[Soprano]] * [[Contralto]] * [[Countertenor]] * [[Tenor]] * [[Baritone]] * [[Bass (voice)|Bass]] {{EndMultiCol}} ==Music resources== [[wikibooks:Subject:Music|Wikibooks - Music]]{{MultiCol}} === Hands on === * [[Blues basics]] * [[Rock basics]] * [[Wikiversity the Movie/music|Wikiversity the movie : music]] * [[Jamming Online|Jamming online]] * [[Experimental music]] * [[Film scoring for Musicians|Practical lessons in film scoring]] (in conjunction with the [[Course:Practical narrative film editing|film editing course]]) * [[Digital Audio Workstation]] === Textbooks === * [[b:Music|Wikibooks Music theory]] * [[b:Western Music History|Western music history]] * [[b:Sound Recording|Sound recording]] {{ColBreak}} === Open-Source software === ;For all operating systems * [http://openmetronome.sourceforge.net/ Metronome] * [http://sourceforge.net/projects/vtone/ Vtones] (Basic Midi editor) * [http://audacity.sourceforge.net/ Audacity] (Sound editor) * [http://ardour.org/ Ardour] (Digital Audio Workstation; A great program for multi-track recording, mixing, mastering, etc.) * [http://musescore.org/ Musescore] (Music Notation software) * [https://otuner.sourceforge.net/ Tuning Software] * [https://supercollider.github.io/ SuperCollider] (Programming Language and Environment for sound synthesis and algorithmic composition) {{ColBreak}} ;For Linux * [https://github.com/calf-studio-gear/calf/ Calf Plugins] Sound Plugins including compressor, multichorus, reverb,etc. * [http://www.antcom.de/gtick/ GTick] (very nice and useful metronome for Gnome desktop) * [[:w:LMMS|Linux MultiMedia Studio (''LMMS'')]] === External links === * [http://music.wikia.com/wiki/Music_Hub Music topics on Wikia] * [[w:Wikipedia:Sound/list|Musical works available for download]] * [[w:History of music|'History of music' on Wikipedia]] {{EndMultiCol}} ==Active participants== ''If you are an active participant in this school, you can list your name below. (this can help small schools grow and the participants communicate better)'' Please leave a timestamp - if it is more than a year old, there is potential for nomination to the inactive participants list. *[[User:Kirby_-_Electrotechnics|Kirby]] (he/him), Banjo, May 2026 ==Inactive participants== *[[User:CQ|CQ]] * Since 20 February 2012‎. Reviewed [[Portal:Pentatonic Impressionism (China Wu Sheng) in the view of Neo-classical Piano Techniques-training]] for Main Page News about 8 August 2019. --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 19:58, 16 January 2020 (UTC) * [[User:SelfieCity|SelfieCity]] 12 July 2021 *[[User:HappyCamper|HappyCamper]] *[[User:Thierry613|Thierry613]] *[[User:Bibeyjj|Bibeyjj]] 0bzabi5nkepcacv3514a9ysoufqbd93 0 47035 2818490 2721030 2026-07-18T20:16:59Z OutsideNormality 3006966 Undid revision [[Special:Diff/2590440|2590440]] by [[Special:Contributions/103.87.50.3|103.87.50.3]] ([[User talk:103.87.50.3|talk]]): cross-wiki spam 2818490 wikitext text/x-wiki {{TOCright}} '''Customer relationship management (CRM)''' is a term applied to processes implemented by a company to handle its contact with its customers. CRM software is used to support these processes, storing information on current and prospective customers. Information in the system can be accessed and entered by employees in different departments, such as sales, marketing, customer service, training, professional development, performance management, human resource development, and compensation. Details on any customer contacts can also be stored in the system. The rationale behind this approach is to improve services provided directly to customers and to use the information in the system for targeted marketing and sales purposes. While the term is generally used to refer to a software-based approach to handling customer relationships, most CRM software vendors stress that a successful CRM strategy requires a holistic approach. CRM initiatives often fail because implementation was limited to software installation without providing the appropriate motivations for employees to learn, provide input, and take full advantage of the information systems ==Overview== From the outside, customers interacting with a company perceive the business as a single entity, despite often interacting with a variety of employees in different roles and departments. CRM is a combination of policies, processes, and strategies implemented by a company that unify its customer interaction and provides a mechanism for tracking customer information. ==Types of CRM== CRM includes many aspects which relate directly to one another: '''Front office operations''' — Direct interaction with customers, e.g. face to face meetings, phone calls, e-mail, online services etc. Back office operations — Operations that ultimately affect the activities of the front office (e.g., billing, maintenance, planning, marketing, advertising, finance, manufacturing, etc.) '''Business relationships''' — Interaction with other companies and partners, such as suppliers/vendors and retail outlets/distributors, industry networks (lobbying groups, trade associations). This external network supports front and back office activities. Analysis — Key CRM data can be analyzed in order to plan target-marketing campaigns, conceive business strategies, and judge the success of CRM activities (e.g., market share, number and types of customers, revenue, profitability, etc.). There are several different approaches to CRM, with different software packages focusing on different aspects. In general, Campaign Management and Sales Force Automation form the core of the system (with SFA being the most popular[citation needed]). ==Operational CRM== Operational CRM provides support to "front office" business processes, e.g. to sales, marketing and service staff. Interactions with customers are generally stored in customers' contact histories, and staff can retrieve customer information as necessary. The contact history provides staff members with immediate access to important information on the customer (products owned, prior support calls etc.), eliminating the need to individually obtain this information directly from the customer. Consequently, many call centers use some form of CRM software. '''Operational CRM processes customer data for a variety of purposes''': 'Managing Campaigns' Enterprise Marketing Automation Sales Force Automation ==Sales Force Automation (SFA)== Sales Force Automation automates sales force-related activities such as: '''Tracking leads''' Scheduling sales calls or mailings Tracking responses Generating reports ==Analytical CRM== Analytical CRM analyzes customer data for a variety of purposes: Designing and executing targeted marketing campaigns Designing and executing campaigns, e.g. customer acquisition, cross-selling, up-selling Analysing customer behavior in order to make decisions relating to products and services (e.g. pricing, product development) Management decisions (e.g. financial forecasting and customer profitability analysis) Analytical CRM generally makes heavy use of data mining. ==Sales Intelligence CRM== Sales Intelligence CRM is similar to Analytical CRM, but is intended as a more direct sales tool. Features include alerts sent to sales staff regarding: Cross-selling/Up-selling/Switch-selling opportunities Customer drift Sales performance Customer trends Customer margins ==Campaign Management== Campaign management combines elements of Operational and Analytical CRM. Campaign management functions include: Target groups formed from the client base according to selected criteria Sending campaign-related material (e.g. on special offers) to selected recipients using various channels (e.g. e-mail, telephone, post) Tracking, storing, and analyzing campaign statistics, including tracking responses and analyzing trends ==Collaborative CRM== Collaborative CRM covers aspects of a company's dealings with customers that are handled by various departments within a company, such as sales, technical support and marketing. Staff members within the departments can share information collected when interacting with customers. For example, feedback received by customer support agents can provide other staff members with information on the services and features requested by customers. Collaborative CRM's ultimate goal is to use information collected by all departments to improve the quality of services provided by the company.[2] ==Geographic CRM== Geographic CRM (GCRM) combines geographic information system and traditional CRM. Geographic data can be analysed to provide a snapshot of potential customers in a region or to plan routes for customer visits. ==Strategy== Several commercial CRM software packages are available, and they vary in their approach to CRM. However, as mentioned above, CRM is not just a technology but rather a comprehensive, customer-centric approach to an organization's philosophy of dealing with its customers. This includes policies and processes, front-of-house customer service, employee training, marketing, systems and information management. Hence, it is important that any CRM implementation considerations stretch beyond technology toward the broader organizational requirements. The objectives of a CRM strategy must consider a company’s specific situation and its customers' needs and expectations. Information gained through CRM initiatives can support the development of marketing strategy by developing the organization's knowledge in areas such as identifying customer segments, improving customer retention, improving product offerings (by better understanding customer needs), and by identifying the organization's most profitable customers.[3] CRM strategies can vary in size, complexity, and scope. Some companies consider a CRM strategy only to focus on the management of a team of salespeople. However, other CRM strategies can cover customer interaction across the entire organization. Many commercial CRM software packages that are available provide features that serve the sales, marketing, event management, project management, and finance industries. ==Successes== While there are numerous reports of "failed" implementations of various types of CRM projects,[4] these are often the result of unrealistic high expectations and exaggerated claims by CRM vendors. Many of these "failures" are also related to data quality and availability. Data cleaning is a major issue. If a company's CRM strategy is to track life-cycle revenues, costs, margins, and interactions between individual customers, this must be reflected in all business processes. Data must be extracted from multiple sources (e.g., departmental/divisional databases such as sales, manufacturing, supply chain, logistics, finance, service etc.), which requires an integrated, comprehensive system in place with well-defined structures and high data quality. Data from other systems can be transferred to CRM systems using appropriate interfaces. A well [[Technical writing specification|specified system]] is of vital importance before starting any implementation, as it can lead to a significant reduction in the time and cost of implementation, as well as highlighting any unrealistic expectations. Failure of CRM system starts when the prime users - sales reps are not comfortable with the system. Too many data entry points will make the usage painful. The data entry points should be minimal and the integration between various modules involving the customers should be efficient enough to pre populate values based on a unique customer code. Data integration from other systems, like, daily currency exchange values inducted into the CRM system are more beneficiary to the end users. The system should be closely linked to frequently used software like MS Word / MS excel etc. The design method should be bottom up to make the system user friendly than management friendly. Users should be periodically and extensively educated with the updates and added functionality to increase the adoption rate - as old habits die hard. == Case studies == _{'''Sprint To Help Customers Understand Their Phones'''} Sprint Nextel Corp. is making a big push to help customers understand their phones, creating a formal program to make store employees available to explain their products and set them up for buyers. The campaign to be announced Tuesday is the first official program for in-person help by a cell carrier, but is similar to moves in the wider consumer electronics industry to demystify gadgets through one-on-one contact. Sprint closed all of its 1,219 stores on Sunday Aug. 17 to train its employees for the "Ready Now" program. The goal is that customers should leave stores with their phones "completely set up and personalized," said Kim Dixon, Sprint's senior vice president of stores. Customers "have got these really great devices ... but they just don't know how to set it all up," Dixon said. Employees will now set up e-mail access, move over contacts from an old phone, connect Bluetooth headsets and explain other functions. Customers who don't have time for the 10-minute to half-hour sessions can make appointments for later visits. These free sessions will be available to existing subscribers as well, even if they aren't buying a new phone or accessory, Dixon said. The company tested the system in St. Louis and Pittsburgh in July, and customer satisfaction improved so much that the company rushed it into a nationwide launch, she said. "Not only will the customer be more satisfied, but as a business, we expect to reduce the number of returns," Dixon said. Over the holidays, as many as 20 percent of "smart" phones Sprint sold were returned because the customer didn't understand them. Apart from reduced return rates, the company saw signs that the uptake of data services like e-mail and mobile Web access improved in the trial cities. As competition has cut into margins on the voice side, cell carriers are looking to data services as the new way to improve profits. The program is reminiscent of the "Genius Bars" of Apple Inc.'s stores, where customers go to get help with their Macs and iPods. On Friday, Microsoft Corp. said it plans to deploy its own customer-service representatives, called "Gurus," at retailers like Best Buy Co. and Circuit City Stores Inc. to help people with their PC purchases, but they won't offer technical support to people who have already bought. Best Buy Mobile has run campaigns called "Walk Out Working," under which store employees set up select gadgets for customers. It currently applies to smart phones. ==See also== * [[School:Business]] * [https://lenzcreative.net School:Business] [[Category:Business]] [[Category:Marketing]] s1jqkob9caidz2tb6vyzsde0t1afevk 0 68978 2818508 2791901 2026-07-19T04:10:51Z OutsideNormality 3006966 Undid 3 revisions from [[Special:Diff/2766100|2766100]] until [[Special:Diff/2766104|2766104]]: cross-wiki spam 2818508 wikitext text/x-wiki {{engineering}} '''Construction engineering''' concerns the planning and management of the construction of structures such as highways, bridges, airports, rail roads, [[building]]s, dams, and reservoirs. Construction of such projects requires knowledge of engineering and management principles and business procedures, economics, and human behavior. Construction engineers engage in the design of structures temporary, cost estimating, planning and scheduling, materials procurement, selection of equipment, and cost control. [[Image:Sky scraper construction.jpg|thumb|250px|In large '''construction''' projects, such as skyscrapers, cranes are essential.]] Construction engineering is differentiated from [[Construction management]] from the standpoint of the use of mathematics, science and engineering to analyze problems and design a construction process. A good familiarity with reading [[Introduction to engineering drawings|blueprints]] is necessary because Construction engineers build many of the things that people use everyday. Construction engineering involves many aspects of construction including: commercial, residential, bridges, airports, tunnels, and dams. It is an extremely large industry that provides jobs to many and continues to grow. Currently there are nearly 6 million people working on construction in the United States [http://www.ndsu.edu/ndsu/academic/factsheets/eng_arch/consteng.shtml]. Construction engineers are in high demand so it is easy for a CE to get a job in any part of the country. ==Careers== Construction is the largest industry in the United States. It provides jobs to millions ranging in all types of education. Construction makes 14% of our countries Gross National Product [http://www.ndsu.edu/ndsu/academic/factsheets/eng_arch/consteng.shtml]. Construction engineering is an important part to the construction industry because it provides much of the design aspect to the field. Construction engineers follow the plans of architects and sometimes design the actual structure. After the structure has been designed the engineers make sure it has been built correctly by testing and overseeing the construction. '''Tasks''' - Construction engineers have a lot of responsibilities in their job. Certain tasks have to be completed everyday in order to get the job done correctly. Analyzing reports is a main part of their job description. They must analyze maps, drawings, blueprints, aerial photography and other topographical information. Construction engineers also have to use computer software to design hydraulic systems and structures while following construction codes. They have to calculate load and grade requirements, liquid flow rates and material stress points to ensure that the structure can withstand stress. Keeping a safe workplace is key to having a successful construction company. It is the construction engineer's job to make sure that everything is conducted correctly. In addition to safety, the construction engineer has to make sure that the site stays clean and sanitary . Surveying the land before construction begins is also a job of the construction engineer. They have to make sure that there are no impediments in the way of where the structure will be built and if there are any they must move them. They also must estimate costs and keep the project under budget. Construction engineers have to test the soils and materials used for adequate strength. Finally, construction engineers have to provide construction information, including repairs and cost changes, to the managers. '''Knowledge''' - Construction engineers build structures that are used by people everyday so they have to be safe and be able to withstand the elements. To complete the job properly construction engineers have to have the knowledge of many different aspects. Those aspects include engineering, technology, design, math, construction, English, customer service, management, transportation, public safety, and computers. They understand technical [[Engineering Drawing|drawings]] related to the industry and use the engineering, technology, and math aspects to make sure they build the structure to the set standards. They use English, customer service, and management knowledge to deal with the people that could possibly buy the structure. They also use this knowledge to inform the management on how the project is coming along and if any changes are needed. '''Skills''' - Most construction engineers have a love for math and science. In addition to these abilities there are many other skills needed to be a construction engineer. Critical thinking, listening, learning, problem solving, monitoring, and decision making are all very important in construction engineering. Construction engineers have to be able to think about all aspects of a problem and listen to other’s ideas so that they can learn everything about a project before it begins. After they have begun a project they must solve the problems that they encounter using math and science. They also have to monitor the workers on the job site for safety and to make sure that the project is on time and done correctly. Whenever a problem occurs it is up to the construction engineer to make the decision on how to fix it. '''Abilities''' - Construction engineers have many different kinds of abilities they use to do their job. They have the abilities to reason, express themselves orally, sense a problem, comprehend (oral and written), order information, speak clearly, and visualize. Construction engineers use these abilities to communicate with other workers and to solve problems. They also have to use there abilities to know what kinds of materials to order and how to get those materials while staying under the budget. '''Work Activities''' - Construction engineers have many activities that they have to do everyday. Those activities include drafting, decision making, computer interaction, communication, documenting, creative thinking, organizing, information collecting, estimating, and analyzing. Construction engineers use drafting to design structures and to show others how to build them. They have to analyze information and make the best decision and solve problems. Computers are an important tool used by construction engineers. They use them to write programs and solve equations. Communication is used everyday to interact with coworkers and supervisors. They have to communicate in person, by telephone, and through e-mail. Documentation is used to record important information that needs to be passed on to management. Most documenting is done in electronic form. Creative thinking is used to come up with new ideas and solve problems. Construction engineers have to be organized to accomplish goals and prioritize jobs. They have to gather information on the task at hand before they can start a project. This will help ensure that the job is completed correctly. In order to keep a project under budget, construction engineers have to estimate costs of materials and workers. Finally, they have to analyze data to find answers to problems they are having on the job site. '''Educational Requirements''' - Construction Engineers are educated to design and build structures that are necessary for everyday life. There are only a handful of schools that offer a major in Construction engineering. Some of the more popular colleges are Purdue University, Iowa State University, Bradley University, California State Polytechnic University-Pomona, and National University [http://www.usnews.com/usnews/edu/college/majors/brief/major_14-33_brief.php]. Bradley University offers a Bachelor of Science degree in Construction while Cal Poly Pomona and National University offer a Bachelor of Science degree in Construction Engineering Technology. Cal Poly Pomona, Bradley University and National University are not accredited by ABET to offer degrees in Construction Engineering. However, Cal Poly Pomona is accredited by the Technology Accreditation Committee (T.A.C.) of ABET and their graduates are eligible to sit for the California E.I.T. exam. The list of accredited ABET programs in Construction Engineering can be found here [http://www.abet.org/schoolareaeac.asp]. The University of Southern California offers a Bachelor of Science in Civil Engineering with an option in Construction Engineering. For most Construction engineering jobs a Bachelor of Science is required and some construction experience. In order to work on projects that will be used by the public a construction engineer has to get a professional engineers license. The Fundamentals of Engineering and Principles and Practice exams must be passed, and other requirements met, for a construction engineer to receive a Professional Engineers license. {{commonscat|Construction}} {{wiktionary|construction}} ==See also== * [[Architecture]] * [[Earthquake engineering]] * [[Project management]] {{Building structures}} [[Category:Construction engineering| ]] [[Category:Buildings]] k2dp6ssq190ij24ch5w7vu2zbugdt2x 2 93152 2818520 2796315 2026-07-19T11:34:31Z Solstag 64708 2818520 wikitext text/x-wiki [[File:Virada do ano 2011 com meu pai usando o Empathy.png|thumb|300px|New year's eve of 2011, chatting with dad over Empathy]] Welcome! My friends call me Ale. My enemies never call me, so hopefully I have none. ;D Beyond this page, you're welcome to visit my pages in other language Wikiversities through the language selection menu, or [https://orcid.org/0000-0002-4849-4631 my ORCiD]. To get in touch with me, leave a message on [[User talk:Solstag|my discussion page]] or through my [https://hubzilla.com.br/channel/aleabdo personal social channel]. The lists below are ''semi-exhaustive'' and not always ''up-to-date'' ;-) ==Research activities== [https://gitlab.com/cortext-usp/team-hub/-/raw/main/research-artifacts/slides/2026/CleanCode_Cortext_LISIS_Seminar.pdf?ref_type=heads&inline=true Re-engineering software for open science]: an action-research software engineering project [[:fr:Recherche:Atelier_du_Vivant|Atelier du Vivant]]: an inquiry on “the living” and its expressions in research and policy [[:fr:Recherche:Un_agenda_pour_la_recherche_participative|ARPS]]: Sketching an inclusive agenda for participatory research [https://gitlab.com/solstag/argentina169 Argentina169]: investigating the interaction of social movements and academia on the issue of pesticides in Argentina [https://github.com/amchagas/open-hardware-supply/ OSH-QA] - measuring the openness qualities of scientific publications of Open Hardware. [https://gitlab.com/solstag/waysout/ Mapping WAYS-OUT]: how the problematic of destabilization and phase-out appears throughout the social sciences. === On hold === [[CyberGen]], a mixed-methods branch of the larger project. === Concluded === ''Only projects whose content was hosted on English language Wikiversity.'' [[The dynamics and social organization of innovation in the field of oncology]] [[Open Global Health/ContentMine|ContentMine in Open Global Health]] - A short experimental project awarded a ContentMine fellowship. [[E-participation in the EuroDIG]] - A short project executed during the final phase of the IGCB Programme of the Diplo Foundation === Abandoned === [[Collaboration in the development of Open Science Hardware|COSH: Collaboration in the development of Open Science Hardware]] [https://gitlab.com/solstag/aireg/ AIREG]: mapping policy discourse around the regulation of artificial intelligence. == Presentations == ''See [[User:Solstag/Presentations]].'' == Proposals == === Accepted === [[Open Global Health/ContentMine Fellowship Application|Open Global Health ContentMine Fellowship Application]] [[User:Solstag/OCSDNet Proposal with Sarita and Jenny|OCSDNet Proposal with Sarita and Jenny]] - Social development from open and collaborative science communities. [[/E-participation and Internet Governance|E-participation and Internet Governance]] - A research proposal prepared for the final phase of the IGCB Programme of the Diplo Foundation. [[/Spreading health information and behaviour through social media seeded by a national survey in Brazil|Spreading health information and behaviour through social media seeded by a national survey in Brazil]] === Abandoned === [[User:Solstag/OCSDNet Proposal|OCSDNet Proposal]] - Understanding and strengthening the development impact of community based knowledge production spaces by combining ethnographic and data analysis of their evolution as a collaboration network. [[/Portuguese Wikipedia Qualitative Researcher|Portuguese Wikipedia Qualitative Researcher]] - A research proposal in response to Wikimedia Foundation's RFP. [[User:Solstag/Refusing Intellectual Property|Refusing Intellectual Property]] - How could the State recognize and empower free-as-in-freedom authors, and how could these authors make it happen? == Research groups == [[:fr:LISIS/GTI|LISIS GTI]] [[:fr:Cortext|Cortext]] [https://cindalab.ibict.br/ Cindalab] [https://garoa.net.br/ Garoa HC] === Past === [[ContentMine]] [[Open Global Health]] == Commentary == [[/Trust transitivity in social networks|Trust transitivity in social networks]] - minor notes about [http://arxiv.org/abs/1012.1358 a paper in arXiv] from Oliver Richters and Tiago P. Peixoto. [[pt:Utilizador:Solstag]] [[fr:Utilisateur:Solstag]] 8bjvsrsf2wzhpzoao94oggzbzma01x6 2 102099 2818487 1978408 2026-07-18T16:53:02Z Charles Jeffrey Danoff 186052 added a reference and references section 2818487 wikitext text/x-wiki {{Notes}} {{TOC right}} inspiration for the potential of this work comes from [[w:Bill_James#The_Bill_James_Baseball_Abstracts|Bill James's early Baseball Abstracts]]. == Overview == This document is an attempt to collect my thoughts (and hopefully those of many others) on the answer to this question: :: '''How do you build a NBA Dynasty?''' In October 2010 as part of my work with the Peer to Peer University ([[w:Peer_to_Peer_University|P2PU]]) Fall 2010 Course [http://archive.p2pu.org/general/diy-math/document/re-booting-modeling-wins-nba-work-progress.html DIY Math] I wrote a [https://web.archive.org/web/20101105161302/http://p2pu.org/general/node/5571/document/10218 post outlining this research], its background<ref>I also want to take into account this [[w:Association_for_Professional_Basketball_Research|famous forum]] and [https://apbr.org/metrics/viewtopic.php?p=33990&hilit=danoff#p33990 two] of my [https://apbr.org/metrics/viewtopic.php?p=33203&hilit=danoff#p33203 posts]</ref> and where I might want to go next. :Indented line == Roster Forecasting == [[User:Charles Jeffrey Danoff/How-to Build an NBA Dynasty Notes/Roster Forecasting]] == Relevant Ideas == * Brady said a lot of organizations do not have the competetive stamina the Patriots do, they want 'Victory Monday' off following a win, they don't want to do 2 hour practices on Friday, the patriots don't take any days off. That's what Matt Patricia is trying to do in Detroit, build up those calluses so at times like this (in 4th quarter) it's easy. - Parahprased from Al Micheals and Cris Collinsworth Sunday night broadcast of Pats vs Lions game -- [[User:Charles Jeffrey Danoff|Charles Jeffrey Danoff]] ([[User talk:Charles Jeffrey Danoff|discuss]] • [[Special:Contributions/Charles Jeffrey Danoff|contribs]]) 02:57, 24 September 2018 (UTC) * Ken Dryden said part of reason for Canadiens continued championships was 'conspiracy of expectations' -- [[User:Charles Jeffrey Danoff|Charles Jeffrey Danoff]] ([[User talk:Charles Jeffrey Danoff|discuss]] • [[Special:Contributions/Charles Jeffrey Danoff|contribs]]) 02:57, 24 September 2018 (UTC) *''Kelly was asked how the Eagles' offense could be better without Jackson on the team, and he said, "I don't think our offense has ever been predicated on one player. We used three quarterbacks last year. We used multiple running backs. Our tight ends are integral to what we do. I think we have an outstanding offensive line. It's never been about just one guy. It's why I think we have diversity" on offense, he added. "I mean, we were first in rushing and I think ninth in passing. I think there is balance in terms of how we do things. You want great players at every position, and I think that's what we're always striving for in terms of where we can put people into position. It's never been predicated on just one guy, because if it is, if he's out for a water break or hurt for an extended period of time, then you're going to be in trouble."'' via http://espn.go.com/nfl/story/_/id/10675583/chip-kelly-noncommittal-desean-jackson-future-philadelphia-eagles --[[User:Charles Jeffrey Danoff|Charles Jeffrey Danoff]] ([[User talk:Charles Jeffrey Danoff|discuss]] • [[Special:Contributions/Charles Jeffrey Danoff|contribs]]) 16:01, 26 March 2014 (UTC) * http://espn.go.com/blog/chicago/bulls/post/_/id/17509/bulls-building-winning-program?ex_cid=espnapi_public Bulls building winning program February, 2, 2014 12:00PM CT By Nick Friedell ""I think the successful teams in the pros, they run a year-round program," Thibodeau said before Saturday's game against the New Orleans Pelicans. "So from that standpoint I think it's very similar to the top college programs. They're year-round programs. I don't think it's any one particular thing that you do, I think it's how you do everything. It's how you approach things." ""It's not if you trap the pick-and-roll, or you force the pick-and-roll to the baseline, or you blitz and catch and shoot," Thibodeau said. "It's how you conduct your business. What you expect from your players, the professionalism, how you approach each and every day. To put everything you have into it, to play for the team. To put the team first. When you look at the teams that have been able to build over a period of time, they have those characteristics." " --[[User:Charles Jeffrey Danoff|Charles Jeffrey Danoff]] ([[User talk:Charles Jeffrey Danoff|discuss]] • [[Special:Contributions/Charles Jeffrey Danoff|contribs]]) 06:32, 3 February 2014 (UTC) * when you have great players, you need to have others who will accept not getting shots and focus on doing the dirty work ... I don't know if Pierce and Garnett will do the dirty work - Charles Barkley before 12 Dec 2013 Nets vs. Clippers game on TNT --[[User:Charles Jeffrey Danoff|Charles Jeffrey Danoff]] ([[User talk:Charles Jeffrey Danoff|discuss]] • [[Special:Contributions/Charles Jeffrey Danoff|contribs]]) 09:36, 13 December 2013 (UTC) * In the NBA it's more about philosophy: what do I do in a 2 on 1; when do I shoot; when do I pass? Because you don't have time with the 24 second shot clock to call plays. In college with the longer shotclock you can run plays, but in the NBA you need to have guys who understand the team philosophy and can just act with the shot clock winding down. Kenny Smith on (21 Nov 2013; following Bulls v Nuggets) Inside the NBA --[[User:Charles Jeffrey Danoff|Charles Jeffrey Danoff]] ([[User talk:Charles Jeffrey Danoff|discuss]] • [[Special:Contributions/Charles Jeffrey Danoff|contribs]]) 06:35, 22 November 2013 (UTC) * [http://sports.nationalpost.com/2013/10/28/toronto-raptors-gm-masai-ujiri-taking-his-job-one-step-at-a-time/ Toronto Raptors GM Masai Ujiri taking his job one step at a time] by Bruce Arthur --[[User:Charles Jeffrey Danoff|Charles Jeffrey Danoff]] ([[User talk:Charles Jeffrey Danoff|discuss]] • [[Special:Contributions/Charles Jeffrey Danoff|contribs]]) 06:59, 2 November 2013 (UTC) :: ''What are the odds you can win a title without superstars?“The chances are, for me, they’re slim,” Ujiri says. “I totally agree that, yes, you have to have one or a couple of those guys, but how do you create it where — I can’t come into Toronto or come into Denver and give the excuse that, ‘Hey, I can’t get any of those guys.’ “So I have to create the environment here. I have to create that environment. It starts with the young players, it starts with the culture you create there, it starts with drafting well. You bring those players, and you attract those players.”'' * [http://blogs.charlotte.com/inside_the_nba/2013/11/charlotte-bobcats-owner-michael-jordan-says-hes-learn-to-be-quicker-in-admitting-mistakes.html Charlotte Bobcats owner Michael Jordan says he's learned to be quicker in admitting mistakes] by Rick Bonnell :: ''-- Jordan on Michael Kidd-Gilchrist: “He is a hard-working kid. He’s learning on the run. It’s not that he doesn’t have talent. But it’s hard to learn in this (NBA) culture because there are so many games. (Bismack) Biyombo is the same way.”'' :: ''-- He says the biggest thing he’s learned as an owner is to be quicker to admit and correct mistakes. He mentioned the one-season tenure of Mike Dunlap as coach as an example. “I liked Mike, I absolutely did. I said I could have played for him. But that’s not today’s kid. The difference between Steve (Clifford) and Mike is the kids articulate what (Clifford) is saying.”'' * Sports/Economics --[[User:Charles Jeffrey Danoff|Charles Jeffrey Danoff]] ([[User talk:Charles Jeffrey Danoff|discuss]] • [[Special:Contributions/Charles Jeffrey Danoff|contribs]]) 19:32, 27 September 2013 (UTC) ** http://espn.go.com/blog/truehoop/post/_/id/62298/whats-up-with-the-milwaukee-bucks ** http://espn.go.com/blog/truehoop/post/_/id/62334/economists-vs-tanking ** http://espn.go.com/blog/truehoop/post/_/id/40055/does-tanking-even-work ** http://www.suu.edu/faculty/berri/RottenbergJPE1956.pdf ** http://web.archive.org/web/20100827184656/http://www.rodneyfort.com/index.htm ** http://www.pistonpowered.com/2013/09/pistons-rank-last-in-projected-value-of-future-draft-picks/ * A young player will be more likely to improve if surrounded by better veterans. Could go the other way, where a player's path to playing time is blocked so he does not develop. Overall more likely a 2 players of equivalent talent, one given keys to playing time, other surrounded by better veterans and forced to earn playing time will be better. ... Prove this statistically? --[[User:Charles Jeffrey Danoff|Charles Jeffrey Danoff]] ([[User talk:Charles Jeffrey Danoff|discuss]] • [[Special:Contributions/Charles Jeffrey Danoff|contribs]]) 04:26, 16 September 2013 (UTC) * "In terms of the Titans, the onus was probably on them to realize that Johnson might struggle to reach these heights again going forward. They re-signed Johnson, who was holding out at the time, to a four-year contract extension before the 2011 season, which was a full year after his monster campaign. General manager Ruston Webster is admittedly not a numbers guy, and it's not always important for a general manager to be in tune with the numbers, but this was a case where understanding outliers and outlier seasons would have saved the Titans a bunch of money. There's no way they or anybody else would have handed Johnson this contract if they had a second chance. That understanding of outliers isn't a Titans-only issue or even a running back–only problem. It's about mistaking one year of performance as more meaningful than it should be. Statistics reflect performance and help identify the outliers in this context, but having big numbers isn't necessary to join this conversation, to which Dallas's massive contract for left tackle turned right turnstile Doug Free can attest. ... And in terms of Johnson, it's about properly calibrating our expectations. ... That we still wait for Johnson to hit his prior strides speaks to our human nature and the way we interpret growth in young athletes. The reality is that when we expect some players to be hitting their stride and beginning a steady growth process, they might actually be peaking" http://www.grantland.com/story/_/id/9572848/bill-barnwell-looks-numbers-see-chris-johnson-return-record-setting-ways --[[User:Charles Jeffrey Danoff|Charles Jeffrey Danoff]] ([[User talk:Charles Jeffrey Danoff|discuss]] • [[Special:Contributions/Charles Jeffrey Danoff|contribs]]) 22:49, 17 August 2013 (UTC) * "Not about the stars that are out, just becomes about next guy doing his job." - Steve Kerr / "This series is going to be about who imposes their will upon whom and they definitely did that tonight. No question." - [http://www.nba.com/video/playoffs/2013/eastseries5/# Eric Spoelstra] / "All playoff games have challenges ... Question is how quickly can you adapt to whatever challenge is in front of you." - Tom Thibodeau (all following Chicago Bulls game 1 victory over Miami Heat in Easter Conference Seminals 2013) --[[User:Charles Jeffrey Danoff|Charles Jeffrey Danoff]] ([[User talk:Charles Jeffrey Danoff|discuss]] • [[Special:Contributions/Charles Jeffrey Danoff|contribs]]) 03:27, 7 May 2013 (UTC) * from Bill Simmons' [http://www.grantland.com/story/_/id/9030601/the-worst-contracts-nbamost overpaid NBA contracts] article:<blockquote>4. Eric Gordon: four years, $58.4 million His missed games starting with his rookie year in 2008: 4, 20, 26, 57, 37 (and counting). You'd have to be doing drugs to deal for that monstrosity of a contract; I'd need to see one full healthy season from him before I considered it. Could the Chris Paul trade have worked out any worse for New Orleans? They ended up with a possible lemon (Gordon), a few months of Chris Kaman (long gone), a year and a half of Al-Farouq Aminu (leaving after this season), and a lottery pick they turned into Austin Rivers (who generated some "historically bad" buzz over the holidays before breaking his hand last week). That's Reason No. 479 why you should ALWAYS swap a bunch of promising young guys, draft picks and whatever else for a Guaranteed Sure Thing, whether it's real sports or fantasy sports. If you're turning quarters and dimes into a dollar, you do it. And you do it without blinking.</blockquote> --[[User:Charles Jeffrey Danoff|Charles Jeffrey Danoff]] ([[User talk:Charles Jeffrey Danoff|discuss]] • [[Special:Contributions/Charles Jeffrey Danoff|contribs]]) 05:44, 11 March 2013 (UTC) * From/inspired by Sloan conference [http://www.slate.com/articles/sports/sports_nut/2013/02/nba_stats_gurus_can_t_work_together_anymore_that_s_a_problem.single.htmlNet Loss How the NBA’s love for advanced stats has slowed our understanding of the game] via [http://www.grantland.com/story/_/id/9014162/sports-analytics-mit-sloan-conference Katie Baker's recap] ... [http://www.sloansportsconference.com/?p=10561 hockey analysis @ the conference] --[[User:Charles Jeffrey Danoff|Charles Jeffrey Danoff]] ([[User talk:Charles Jeffrey Danoff|discuss]] • [[Special:Contributions/Charles Jeffrey Danoff|contribs]]) 06:15, 8 March 2013 (UTC) * advancedballstats.com ---> Hubie & Mike Tirico discussing how advanced stats have integrated NBA 3 Mar 2013 Clippers vs OKC telecast -----> Hubie: Need to make it so everyone in organization understands them, incl. players --[[User:Charles Jeffrey Danoff|Charles Jeffrey Danoff]] ([[User talk:Charles Jeffrey Danoff|discuss]] • [[Special:Contributions/Charles Jeffrey Danoff|contribs]]) 06:44, 4 March 2013 (UTC) * "they're the gold standard. it's a proven system and hopefully we're on that track. we like to think we are, but you have to do it year after year." - Tom Thibodeau prior to his team's 11 Feb 2013 loss to a Spurs team w/o Duncan, Ginobli & Parker ... Spurs system embarassed the bulls on that night. --[[User:Charles Jeffrey Danoff|Charles Jeffrey Danoff]] ([[User talk:Charles Jeffrey Danoff|discuss]] • [[Special:Contributions/Charles Jeffrey Danoff|contribs]]) 06:34, 12 February 2013 (UTC) * "The most important thing is to know what your job is. The next is to do your job. But you can’t do your job if you don’t know what it is. So it’s important for everyone to learn and know multiple positions." - via my Dad http://blogs.bulls.com/2013/01/bulls-stop-bucks-here-in-milwaukee/ --[[User:Charles Jeffrey Danoff|Charles Jeffrey Danoff]] ([[User talk:Charles Jeffrey Danoff|discuss]] • [[Special:Contributions/Charles Jeffrey Danoff|contribs]]) 23:01, 2 February 2013 (UTC) * http://hoopshype.com/articles/narducci/memphis-committed-to-analytics wonderful article on the history, limits of and benefits from using statistical analysis in the NBA --[[User:Charles Jeffrey Danoff|Charles Jeffrey Danoff]] ([[User talk:Charles Jeffrey Danoff|discuss]] • [[Special:Contributions/Charles Jeffrey Danoff|contribs]]) 00:48, 1 February 2013 (UTC) * "That's the thing about the NBA. The better the team, the more you have to do your homework, and we're prepared to do that." - Kobe on Lakers' upcoming game w/Heat this week, following his team's win over Bucks --[[User:Charles Jeffrey Danoff|Charles Jeffrey Danoff]] ([[User talk:Charles Jeffrey Danoff|talk]]) 06:37, 16 January 2013 (UTC) * Helpful GM Links (sidenote, impressive how many years apart Krause won his NBA Executive awards + how many brother/people with last name who have been NBA GMs not including Billy King & Billy Knight). --[[User:Charles Jeffrey Danoff|Charles Jeffrey Danoff]] ([[User talk:Charles Jeffrey Danoff|talk]]) 06:55, 14 January 2013 (UTC) ** Wikipedia Template of Current NBA GMs http://thissongissick.com/blog/2013/aylen-bootleg-pack-2013-bootleg-mashup-download/#more-22138 ** http://www.basketball-reference.com/executives/ ** http://sportsillustrated.cnn.com/2007/writers/kelly_dwyer/06/25/gm.rankings/index.html ** http://sports.yahoo.com/nba/blog/ball_dont_lie/post/The-top-10-NBA-general-managers-of-the-last-deca;_ylt=AnDtXIkjx2yjECLM8Y38YSy8vLYF?urn=nba,191489 * Van Gundy commenting on coaches being fired when teams go bad ... and that person picking the players is most important in organization because they decide the talent and character of the players. (during Lakers vs. OKC on 11 Jan 2013) --[[User:Charles Jeffrey Danoff|Charles Jeffrey Danoff]] ([[User talk:Charles Jeffrey Danoff|talk]]) 06:08, 12 January 2013 (UTC) * In terms of who would win in a series Kenny Smith said "who can minimize their mistakes ... all teams have them" ... later in a discussion where Shaq asked Charles to clarify what makes somebody a great player Chuck said "making those around you better ... some guys are floaters, if they don't start well they'll float through the game, so I'd make sure to get them going early, that way they'd play better defense, better offense" following Clippers vs Celtics 27 Dec. '12 matchup --[[User:Charles Jeffrey Danoff|Charles Jeffrey Danoff]] ([[User talk:Charles Jeffrey Danoff|talk]]) 07:00, 28 December 2012 (UTC) * "If you have depth it takes longer to develop chemistry." Nuggets/Grizzlies ESPN broadcast 14 Dec 2012 -- Charlie Danoff via [[Special:Contributions/108.200.241.51|108.200.241.51]] 04:08, 15 December 2012 (UTC) * On MNF Gruden commended Belicheck saying all of his players are versatile, able to adjust to what the opponent brings. Additionally he does an insane amount of research finding undrafted players to play roles on his team, including, according to Gruden, a Ohio State rugby player whose doing special teams. --[[User:Charles Jeffrey Danoff|Charles Jeffrey Danoff]] ([[User talk:Charles Jeffrey Danoff|talk]]) 03:47, 11 December 2012 (UTC) * Pat Riley "Disease of more (money, shots, etc.) that comes from winning a champtionship. Via David Aldrich --[[User:Charles Jeffrey Danoff|Charles Jeffrey Danoff]] ([[User talk:Charles Jeffrey Danoff|talk]]) 23:13, 8 December 2012 (UTC) * Create a GM index, comparing the Winshares of a player relative to his draft class. Drafted number 17, win shares are number 12 in his class, so GM gets a +5 or something. Also include wins, playoff wins and how guys have done after they've been acquired in a trade ... have like a 10 year window for evaluation --[[User:Charles Jeffrey Danoff|Charles Jeffrey Danoff]] ([[User talk:Charles Jeffrey Danoff|talk]]) 20:26, 8 December 2012 (UTC) * Kobe on leadership, via Facebook https://www.facebook.com/Kobe/posts/10151211862620419 ** "Sometimes you must prioritize the success of the team ahead of how your own image is perceived. The ability to elevate those around you is more than simply sharing the ball or making teammates feel a certain level of comfort. It's pushing them to find their inner beast, even if they end up resenting you for it at the time. ... I'd rather be perceived as a winner than a good teammate. I wish they both went hand in hand all the time but that's just not reality." --[[User:Charles Jeffrey Danoff|Charles Jeffrey Danoff]] ([[User talk:Charles Jeffrey Danoff|talk]]) 20:26, 8 December 2012 (UTC) * Bill Simmons interviewing Bill Russell and commenting on Kobe, ** "But that would have failed unless everyone embraced their role, and that's the thing — everyone has to have a role." from Russell about working with teammates ** Top players of all time "That list currently looks like this: Jordan, then Russell, then Kareem, then Magic/Bird, then Wilt, then Kobe/Duncan (or Duncan/Kobe), then West/Oscar, then Hakeem/Shaq/Moses. With LeBron lurking in there somewhere. We just don't know where yet." from http://www.grantland.com/story/_/id/8724362/the-kobe-question --[[User:Charles Jeffrey Danoff|Charles Jeffrey Danoff]] ([[User talk:Charles Jeffrey Danoff|talk]]) 20:26, 8 December 2012 (UTC) * Bill Simmons on dynasties "In the NBA's seven-decade history, we only witnessed four true dynasties: Russell's Celtics (13 years, 11 titles), Jordan's Bulls (eight years, six titles), Magic's Lakers (12 years, nine Finals appearances, five titles)2 and Mikan's Lakers (six years, five titles)" http://www.grantland.com/story/_/id/8486795/the-harden-dilemma --[[User:Charles Jeffrey Danoff|Charles Jeffrey Danoff]] ([[User talk:Charles Jeffrey Danoff|talk]]) 12:03, 12 October 2012 (UTC) * 22 Feb 2012 Tom Thibodeau news conference said something to this effect: "You play 66 games and then hopefully you make the playoffs. Once there whomever's healthiest and playing the best will win. I don't think you want to approach it as 'getting through something' because its not the goal, whatever you're going through, you want to keep the ultimate goal in mind." --[[User:Charles Jeffrey Danoff|Charles Jeffrey Danoff]] 03:37, 23 February 2012 (UTC) * "The day of the trade deadline was good. I did what I always do. Worked out, did some rehab… You can’t control what’s going to happen so you have to do what you have to do. Talking to your agent is not going to change anything." - [[w:Caron Butler]] from a June 2011 [http://blogs.hoopshype.com/blogs/butler/2011/03/02/im-getting-closer/ blog post] by him. * "I probably look at this more simplistic than you want. It doesn't matter whether you're a former player or you come up through the video room, whether you come from college. You have to earn that trust from NBA players every day. It doesn't stop unless, of course, you win 10 rings. But even at that point, you have to prove that there's competency, there's a work ethic, there's a plan, there's an organization to your thoughts. Then, at some point, they believe you can help them achieve what they want to achieve." " - [[w:Erick Spoelstra]] from a June 2011 ESPN.com [http://sports.espn.go.com/nba/truehoop/miamiheat/columns/story?page=Spoelstra-110601 story] on him. ** [http://blogs.hoopshype.com/blogs/zarrabi/2011/06/02/for-shaq-life-was-first/ Yin] to Spoelestra's work-ethic Yang? * On [[w:Inside the NBA|Inside the NBA]] tonight [[w:Charles Barkley|Charles]] said "Every great team needs a 'dude' and a 'crazy guy'". (e.g. 2011 Playoffs Mavericks have Dirk & DeShaun Stevenson; LA has Kobe & Ron Artest) ----[[User:Charles Jeffrey Danoff|Charles Jeffrey Danoff]] 02:46, 3 May 2011 (UTC) * NBA dynasty ... its about making moves to give yourself a possibility to be great (thus acquiring great players?) --[[User:Charles Jeffrey Danoff|Charles Jeffrey Danoff]] 17:14, 10 April 2011 (UTC) * i was thinking, while re-building is it more important to accumulate talent (e.g. krause w/pre rebuilding bulls) or to play solid basketball every year (jazz post stockton/malone) ... need both, but maybe better to focus on always playing good basketball on some level and trying to make playoffs regardless of personnell (to degrees, saving cap space) than just getting as many good young players at 1 time? --[[User:Charles Jeffrey Danoff|Charles Jeffrey Danoff]] 01:09, 28 March 2011 (UTC) * for some players injuries can be big problems vs. others whose "best ability is availability" from jeff van gundy (he said he heard it from nfl live) on por vs. oakland 27 mar 2011 game --[[User:Charles Jeffrey Danoff|Charles Jeffrey Danoff]] 00:16, 28 March 2011 (UTC) * Bill Simmons, "Basketball is relatively simple. Teams win championships conventionally (an alpha dog superstar, an accomplished sidekick, a third elite player, than five or six role players) and unconventionally (three or four elite starters who complement each other, then four or five role players), but never radically. The Heat are thinking radically, even if it might not seem like it. They believe two alpha dog superstars -- both perimeter players, both creators who need the ball in their hands, both franchise players who developed particular habits carrying inferior teammates these past seven years -- can reinvent themselves as co-CEOs of a basketball team." from [http://sports.espn.go.com/espn/page2/story?page=simmons/10129&sportCat=nba LeBron and Wade: Can it work?] 29 Oct 2010 * Beckley Mason, "The best NBA coaches have a track record of enhancing their players’ awareness. Phil Jackson’s Zen philosophy of being “in the moment” is really about being fully cognizant of teammates and opponents. In Utah, Jerry Sloan’s persistent use of the flex offense has the interesting effect of increasing his team’s collective awareness level. Because they devote so much energy and practice time to one offense, Jazz players are intensely aware of the near infinite permutations of each set, and are focused on reading and reacting. With an internalized, cohesive offensive strategy, Jazz players are able to consistently rise above offensive expectations. See, being aware isn’t just for the greats, it’s also how bad players become average and mediocre players become Kevin Martins. In short, it’s how a player maximizes his potential." from [http://hoopspeak.com/2010/11/awareness-greatness-kobe-bryant-kevin-garnett/ On Awareness: The Intangible Skill That Makes All The Difference] November 19th, 2010 * must a dynasty necessarily include something new, that couldn't be predicted by a mathematical equation. like a new pop hit needs to have something fresh to catch that is inspired by what came b4, but also not able to be predicted. e.g. jordan being first 2-guard led dynasty. \ * godel's theorum needs to fit into this analysis * [http://on.wsj.com/geTmsK WSJ article] about the amount of scouting/research a color analyst (Bill Raftery) does to prepare for games ([http://online.wsj.com/article/SB10001424052748703555804576102461442595674.html?mod=wsj_share_twitter alternative link]) <blockquote> * Basketball color analyst Bill Raftery carries a stack of DVDs in his briefcase wherever he goes. * It can take Mr. Raftery several hours to get through a single, 40-minute college basketball game. His fingers are constantly on the rewind and fast-forward buttons so he can make sure his diagrams match the plays he is watching. * During games Mr. Raftery keeps a notebook open to a page where he has written each player's season average for minutes played, rebounds, assists, points, three-pointers, field goal percentage, steals and blocks. Writing down the numbers himself helps him commit them to memory. </blockquote> * Steve Young commented about the Packers following their Super Bowl 45 victory that they had a self-policing lockeroom. I think that's a really important part of dynasties, players can't rely on coaches or management to police them and make sure they toe the line, they have to do it themselves and hold eachother accountable. Maybe. * Is an effective measure of a "truly great player" (a true "1" in terms of a leader, not the point guard position) number of all star games started? * Announcer (Thurl Bailey?) from 28 Feb 2011 Celtics vs. Jazz game on NBA TV said that the Celtics's greatest strength is their mix of young and old players ... another announcer added how some NBA teams have problems b/c they have too many of the same kind of player --[[User:Charles Jeffrey Danoff|Charles Jeffrey Danoff]] 02:53, 1 March 2011 (UTC) * Tom Thibodeau after 28 Feb game vs. Wizards ... every team in NBA is good & poor records usually result of scheduling/injuries/etc ... so if you don't prepare for the game the right way you could trip --[[User:Charles Jeffrey Danoff|Charles Jeffrey Danoff]] 02:53, 1 March 2011 (UTC) * do a GM power ranking? adjusted periodically for recent transactions/results? ... include hoops analyst's gm reports? --[[User:Charles Jeffrey Danoff|Charles Jeffrey Danoff]] 02:57, 1 March 2011 (UTC) * measure shooters by the 40-50-90 "magic #" http://www.thedaily.com/page/2010/11/07/030211-sports-nash-1-4/ --[[User:Charles Jeffrey Danoff|Charles Jeffrey Danoff]] 20:39, 3 March 2011 (UTC) * Ben Golliver "[http://blogs.hoopshype.com/blogs/golliver/2011/03/10/walk-like-a-gm-talk-like-a-gm/ Walk like a GM, talk like a GM]" March 10, 2011 @ 6:29 pm "We have looked at three thoughts. First, that the NBA trade market is not nearly as efficient as it should be, but that analytics may help improve that. Second, that the short-term pressure on NBA GMs to win big is enormous and that some GMs have shifted course to focus on risk assessment so that they can swing for the fences when necessary. Third, that NBA executives are feeling limitations in their ability to use analytics to gain an advantage during the draft process and that non-quantifiables such as work ethic and dedication remain undervalued assets. Taken together, we’ve reached a point where we can officially declare that the honeymoon period is over for the cult of advanced statistics in basketball. ... The message from the 2011 Sloan Conference, then, wasn’t the same as previous years: It wasn’t a declaration of arrival, a celebration of newfound success or even an admission of early struggles. Rather, it leaned more towards a high school reunion, where the like-minded from around the country, now fully content and secure with their place at the table, gathered to swap stories and best practices in front of a rapidly growing crowd of followers, anxious to learn how they might grab their own seat. * kendall gill said something about how the key to defense in the NBA is the 2nd man? --[[User:Charles Jeffrey Danoff|Charles Jeffrey Danoff]] 03:33, 15 March 2011 (UTC) === styles of play === ==== Havoc ==== Coach Smart's teams play a style of basketball known as "[[havoc]]". Smart described his "havoc" defensive philosophy during his introductory press conference as, "We are going to wreak havoc on our opponent's psyche and their plan of attack." On the court, the "havoc" defensive mindset is visible through the heavy use of the [[full court press]] and pressing after made baskets to disrupt opponent's timing of offensive sets. In his second season, VCU led the Colonial Athletic Association in steals. Offensively, Shaka Smart-coached teams play uptempo and push the ball after misses and makes with what Smart describes as, "The freedom to make plays in the open court". [[w:Shaka Smart|From Wikipedia]]. --[[User:Charles Jeffrey Danoff|Charles Jeffrey Danoff]] 03:03, 19 March 2011 (UTC) ==== Princeton Offense==== * Princeton ** Basketball's Princeton-Style Offense: A Simplified Approach for High School Coaches by Derek Sheridan (ISBN-10: [http://en.wikipedia.org/w/index.php?title=Special%3ABookSources&isbn=1930546920 1930546920]) * Northwestern under Bill Carmody * Georgetown under John Thompson III * Richmond? ==== Triangle Offense==== * Tex Winter designed for Phil Jackson Bulls & Lakers teams ** The Triple Post Offense Book by [[w:Tex Winter|Tex Winter]] ([[w:Amazon Standard Identification Number|ASIN]]: [http://www.amazon.com/Triple-Post-Offense-Book-Winter/dp/B000A7OVR6/ref=reg_hu-rd_add_1_dp_T2 B000A7OVR6]) ==== Pick & Roll Offense ==== * Jerry Sloan coached Utah Jazz Teams === statistics === * "[http://agiletesting.blogspot.com/2005/04/sparkplot-creating-sparklines-with.html sparkplot: creating sparklines with matplotlib]" by [http://www.linkedin.com/in/ggheorghiu Grig Gheorghiu] found via [http://www.edwardtufte.com/bboard/q-and-a-fetch-msg?msg_id=0001lh E.T.] it has an example of sparklines used to quickly analyze a lakers season that i think would be helpful to have in a big quantity to evaluate teams. == APBRmetrics in the NBA == * Google Book Search for "nba statistics" http://www.google.com/search?q=%22NBA%20statistics%22&tbs=bks:1 --[[User:Charles Jeffrey Danoff|Charles Jeffrey Danoff]] 07:48, 20 December 2010 (UTC) * [http://www.silverscreenandroll.com/2010/10/28/1779160/the-question-of-lakers-vs-heat-is-a-defining-battle-in-the#storyjump The question of Lakers vs. Heat is a defining battle in the statistical revolution] by C.A. Clark on Oct 28, 2010 8:30 AM PDT in Editorial ** The basketball statistical revolution is coming. Slowly but surely, advanced statistics are creeping into the game. The signs are everywhere: John Hollinger is a prominent national basketball writer for ESPN, primarily on the basis of his statistical model for player evaluation, PER. Daryl Morey and Rich Cho are two of the youngest GMs in the league, and both have foundations less rooted in basketball than in statistics. More than half the league's teams employ at least one full-time analyst devoted to statistical work. Regardless of how you feel about advanced stats, it is impossible to ignore the growing impact they have on the modernizing NBA. *** via [http://tpsnation.com/2010/10/29/the-10-man-rotation-starring-tyrus-thomas-dancin-away/ The 10-man rotation, starring Tyrus Thomas dancin’ away] via pingback from [http://hoopsanalyst.com/blog/?p=572 NBA Predictions 2010-11] * Hollinger got a job w/Wizards. Go back and read early Baskteball Prospectuses? --[[Special:Contributions/108.200.241.51|108.200.241.51]] 09:14, 15 December 2012 (UTC) == ''cf.'' == * The October 26th, 2010 version of [http://danoff.org/ledive/index.php?title=Re-Booting_Modeling_Wins_in_the_NBA:_A_Work_in_Progress_%E2%80%A6_Notepad&oldid=36839 Re-Booting Modeling Wins in the NBA: A Work in Progress … Notepad] a document I created for notes on the How-to paper published that day. * My original write-up of an Independent Study I did as a Colgate University undergrad on the Determinants of Success in the NBA: [http://p2pu.org/node/5571/document/10218 Re-Booting Modeling Wins in the NBA: A Work in Progress …] * [http://freedarko.blogspot.com/ Freedarko] for a philosophical take on the question. == Resources == * [http://basketballprospectus.com/ Basketball Prospectus] * [http://knickerblogger.net/statpage/ Knickerblogger.net Stats Page] * [http://stackoverflow.com/questions/643753/does-anybody-know-of-a-real-time-sports-scores-stats-web-service Does anybody know of a real time sports scores/stats web service? (closed)] - way to get text based box scores for data/archival purposes? == to-read == * Second Wind by Bill Russell * Basketball on Paper * Wages & Wins ** [http://www.wagesofwins.com/WOWarticles.html Academic articles from Wages of Wins] * [http://danrosenbaum.blogspot.com/ Dan Rosenbaum's NBA Blog] * [http://www.bepress.com/jqas/ Journal of Quantitative Analysis in Sports] * [http://www.amstat.org/sections/sis/ Statistics in Sports: A Section of the American Statistical Association] * Presentations from [http://ncssors.wikidot.com/program 2010 NORTHERN CALIFORNIA SYMPOSIUM ON STATISTICS AND OPERATIONS RESEARCH IN SPORTS] == to-do == * fit this research into http://www.aeaweb.org/journal/jel_class_system.php * explore panels and papers from this conference http://www.sloansportsconference.com/panels-2 ==References== {{Reflist|40em}}/ 18tawsqe7gqk511mtr3b180fd8klasv 2818488 2818487 2026-07-18T16:53:37Z Charles Jeffrey Danoff 186052 /* References */ removed random slash 2818488 wikitext text/x-wiki {{Notes}} {{TOC right}} inspiration for the potential of this work comes from [[w:Bill_James#The_Bill_James_Baseball_Abstracts|Bill James's early Baseball Abstracts]]. == Overview == This document is an attempt to collect my thoughts (and hopefully those of many others) on the answer to this question: :: '''How do you build a NBA Dynasty?''' In October 2010 as part of my work with the Peer to Peer University ([[w:Peer_to_Peer_University|P2PU]]) Fall 2010 Course [http://archive.p2pu.org/general/diy-math/document/re-booting-modeling-wins-nba-work-progress.html DIY Math] I wrote a [https://web.archive.org/web/20101105161302/http://p2pu.org/general/node/5571/document/10218 post outlining this research], its background<ref>I also want to take into account this [[w:Association_for_Professional_Basketball_Research|famous forum]] and [https://apbr.org/metrics/viewtopic.php?p=33990&hilit=danoff#p33990 two] of my [https://apbr.org/metrics/viewtopic.php?p=33203&hilit=danoff#p33203 posts]</ref> and where I might want to go next. :Indented line == Roster Forecasting == [[User:Charles Jeffrey Danoff/How-to Build an NBA Dynasty Notes/Roster Forecasting]] == Relevant Ideas == * Brady said a lot of organizations do not have the competetive stamina the Patriots do, they want 'Victory Monday' off following a win, they don't want to do 2 hour practices on Friday, the patriots don't take any days off. That's what Matt Patricia is trying to do in Detroit, build up those calluses so at times like this (in 4th quarter) it's easy. - Parahprased from Al Micheals and Cris Collinsworth Sunday night broadcast of Pats vs Lions game -- [[User:Charles Jeffrey Danoff|Charles Jeffrey Danoff]] ([[User talk:Charles Jeffrey Danoff|discuss]] • [[Special:Contributions/Charles Jeffrey Danoff|contribs]]) 02:57, 24 September 2018 (UTC) * Ken Dryden said part of reason for Canadiens continued championships was 'conspiracy of expectations' -- [[User:Charles Jeffrey Danoff|Charles Jeffrey Danoff]] ([[User talk:Charles Jeffrey Danoff|discuss]] • [[Special:Contributions/Charles Jeffrey Danoff|contribs]]) 02:57, 24 September 2018 (UTC) *''Kelly was asked how the Eagles' offense could be better without Jackson on the team, and he said, "I don't think our offense has ever been predicated on one player. We used three quarterbacks last year. We used multiple running backs. Our tight ends are integral to what we do. I think we have an outstanding offensive line. It's never been about just one guy. It's why I think we have diversity" on offense, he added. "I mean, we were first in rushing and I think ninth in passing. I think there is balance in terms of how we do things. You want great players at every position, and I think that's what we're always striving for in terms of where we can put people into position. It's never been predicated on just one guy, because if it is, if he's out for a water break or hurt for an extended period of time, then you're going to be in trouble."'' via http://espn.go.com/nfl/story/_/id/10675583/chip-kelly-noncommittal-desean-jackson-future-philadelphia-eagles --[[User:Charles Jeffrey Danoff|Charles Jeffrey Danoff]] ([[User talk:Charles Jeffrey Danoff|discuss]] • [[Special:Contributions/Charles Jeffrey Danoff|contribs]]) 16:01, 26 March 2014 (UTC) * http://espn.go.com/blog/chicago/bulls/post/_/id/17509/bulls-building-winning-program?ex_cid=espnapi_public Bulls building winning program February, 2, 2014 12:00PM CT By Nick Friedell ""I think the successful teams in the pros, they run a year-round program," Thibodeau said before Saturday's game against the New Orleans Pelicans. "So from that standpoint I think it's very similar to the top college programs. They're year-round programs. I don't think it's any one particular thing that you do, I think it's how you do everything. It's how you approach things." ""It's not if you trap the pick-and-roll, or you force the pick-and-roll to the baseline, or you blitz and catch and shoot," Thibodeau said. "It's how you conduct your business. What you expect from your players, the professionalism, how you approach each and every day. To put everything you have into it, to play for the team. To put the team first. When you look at the teams that have been able to build over a period of time, they have those characteristics." " --[[User:Charles Jeffrey Danoff|Charles Jeffrey Danoff]] ([[User talk:Charles Jeffrey Danoff|discuss]] • [[Special:Contributions/Charles Jeffrey Danoff|contribs]]) 06:32, 3 February 2014 (UTC) * when you have great players, you need to have others who will accept not getting shots and focus on doing the dirty work ... I don't know if Pierce and Garnett will do the dirty work - Charles Barkley before 12 Dec 2013 Nets vs. Clippers game on TNT --[[User:Charles Jeffrey Danoff|Charles Jeffrey Danoff]] ([[User talk:Charles Jeffrey Danoff|discuss]] • [[Special:Contributions/Charles Jeffrey Danoff|contribs]]) 09:36, 13 December 2013 (UTC) * In the NBA it's more about philosophy: what do I do in a 2 on 1; when do I shoot; when do I pass? Because you don't have time with the 24 second shot clock to call plays. In college with the longer shotclock you can run plays, but in the NBA you need to have guys who understand the team philosophy and can just act with the shot clock winding down. Kenny Smith on (21 Nov 2013; following Bulls v Nuggets) Inside the NBA --[[User:Charles Jeffrey Danoff|Charles Jeffrey Danoff]] ([[User talk:Charles Jeffrey Danoff|discuss]] • [[Special:Contributions/Charles Jeffrey Danoff|contribs]]) 06:35, 22 November 2013 (UTC) * [http://sports.nationalpost.com/2013/10/28/toronto-raptors-gm-masai-ujiri-taking-his-job-one-step-at-a-time/ Toronto Raptors GM Masai Ujiri taking his job one step at a time] by Bruce Arthur --[[User:Charles Jeffrey Danoff|Charles Jeffrey Danoff]] ([[User talk:Charles Jeffrey Danoff|discuss]] • [[Special:Contributions/Charles Jeffrey Danoff|contribs]]) 06:59, 2 November 2013 (UTC) :: ''What are the odds you can win a title without superstars?“The chances are, for me, they’re slim,” Ujiri says. “I totally agree that, yes, you have to have one or a couple of those guys, but how do you create it where — I can’t come into Toronto or come into Denver and give the excuse that, ‘Hey, I can’t get any of those guys.’ “So I have to create the environment here. I have to create that environment. It starts with the young players, it starts with the culture you create there, it starts with drafting well. You bring those players, and you attract those players.”'' * [http://blogs.charlotte.com/inside_the_nba/2013/11/charlotte-bobcats-owner-michael-jordan-says-hes-learn-to-be-quicker-in-admitting-mistakes.html Charlotte Bobcats owner Michael Jordan says he's learned to be quicker in admitting mistakes] by Rick Bonnell :: ''-- Jordan on Michael Kidd-Gilchrist: “He is a hard-working kid. He’s learning on the run. It’s not that he doesn’t have talent. But it’s hard to learn in this (NBA) culture because there are so many games. (Bismack) Biyombo is the same way.”'' :: ''-- He says the biggest thing he’s learned as an owner is to be quicker to admit and correct mistakes. He mentioned the one-season tenure of Mike Dunlap as coach as an example. “I liked Mike, I absolutely did. I said I could have played for him. But that’s not today’s kid. The difference between Steve (Clifford) and Mike is the kids articulate what (Clifford) is saying.”'' * Sports/Economics --[[User:Charles Jeffrey Danoff|Charles Jeffrey Danoff]] ([[User talk:Charles Jeffrey Danoff|discuss]] • [[Special:Contributions/Charles Jeffrey Danoff|contribs]]) 19:32, 27 September 2013 (UTC) ** http://espn.go.com/blog/truehoop/post/_/id/62298/whats-up-with-the-milwaukee-bucks ** http://espn.go.com/blog/truehoop/post/_/id/62334/economists-vs-tanking ** http://espn.go.com/blog/truehoop/post/_/id/40055/does-tanking-even-work ** http://www.suu.edu/faculty/berri/RottenbergJPE1956.pdf ** http://web.archive.org/web/20100827184656/http://www.rodneyfort.com/index.htm ** http://www.pistonpowered.com/2013/09/pistons-rank-last-in-projected-value-of-future-draft-picks/ * A young player will be more likely to improve if surrounded by better veterans. Could go the other way, where a player's path to playing time is blocked so he does not develop. Overall more likely a 2 players of equivalent talent, one given keys to playing time, other surrounded by better veterans and forced to earn playing time will be better. ... Prove this statistically? --[[User:Charles Jeffrey Danoff|Charles Jeffrey Danoff]] ([[User talk:Charles Jeffrey Danoff|discuss]] • [[Special:Contributions/Charles Jeffrey Danoff|contribs]]) 04:26, 16 September 2013 (UTC) * "In terms of the Titans, the onus was probably on them to realize that Johnson might struggle to reach these heights again going forward. They re-signed Johnson, who was holding out at the time, to a four-year contract extension before the 2011 season, which was a full year after his monster campaign. General manager Ruston Webster is admittedly not a numbers guy, and it's not always important for a general manager to be in tune with the numbers, but this was a case where understanding outliers and outlier seasons would have saved the Titans a bunch of money. There's no way they or anybody else would have handed Johnson this contract if they had a second chance. That understanding of outliers isn't a Titans-only issue or even a running back–only problem. It's about mistaking one year of performance as more meaningful than it should be. Statistics reflect performance and help identify the outliers in this context, but having big numbers isn't necessary to join this conversation, to which Dallas's massive contract for left tackle turned right turnstile Doug Free can attest. ... And in terms of Johnson, it's about properly calibrating our expectations. ... That we still wait for Johnson to hit his prior strides speaks to our human nature and the way we interpret growth in young athletes. The reality is that when we expect some players to be hitting their stride and beginning a steady growth process, they might actually be peaking" http://www.grantland.com/story/_/id/9572848/bill-barnwell-looks-numbers-see-chris-johnson-return-record-setting-ways --[[User:Charles Jeffrey Danoff|Charles Jeffrey Danoff]] ([[User talk:Charles Jeffrey Danoff|discuss]] • [[Special:Contributions/Charles Jeffrey Danoff|contribs]]) 22:49, 17 August 2013 (UTC) * "Not about the stars that are out, just becomes about next guy doing his job." - Steve Kerr / "This series is going to be about who imposes their will upon whom and they definitely did that tonight. No question." - [http://www.nba.com/video/playoffs/2013/eastseries5/# Eric Spoelstra] / "All playoff games have challenges ... Question is how quickly can you adapt to whatever challenge is in front of you." - Tom Thibodeau (all following Chicago Bulls game 1 victory over Miami Heat in Easter Conference Seminals 2013) --[[User:Charles Jeffrey Danoff|Charles Jeffrey Danoff]] ([[User talk:Charles Jeffrey Danoff|discuss]] • [[Special:Contributions/Charles Jeffrey Danoff|contribs]]) 03:27, 7 May 2013 (UTC) * from Bill Simmons' [http://www.grantland.com/story/_/id/9030601/the-worst-contracts-nbamost overpaid NBA contracts] article:<blockquote>4. Eric Gordon: four years, $58.4 million His missed games starting with his rookie year in 2008: 4, 20, 26, 57, 37 (and counting). You'd have to be doing drugs to deal for that monstrosity of a contract; I'd need to see one full healthy season from him before I considered it. Could the Chris Paul trade have worked out any worse for New Orleans? They ended up with a possible lemon (Gordon), a few months of Chris Kaman (long gone), a year and a half of Al-Farouq Aminu (leaving after this season), and a lottery pick they turned into Austin Rivers (who generated some "historically bad" buzz over the holidays before breaking his hand last week). That's Reason No. 479 why you should ALWAYS swap a bunch of promising young guys, draft picks and whatever else for a Guaranteed Sure Thing, whether it's real sports or fantasy sports. If you're turning quarters and dimes into a dollar, you do it. And you do it without blinking.</blockquote> --[[User:Charles Jeffrey Danoff|Charles Jeffrey Danoff]] ([[User talk:Charles Jeffrey Danoff|discuss]] • [[Special:Contributions/Charles Jeffrey Danoff|contribs]]) 05:44, 11 March 2013 (UTC) * From/inspired by Sloan conference [http://www.slate.com/articles/sports/sports_nut/2013/02/nba_stats_gurus_can_t_work_together_anymore_that_s_a_problem.single.htmlNet Loss How the NBA’s love for advanced stats has slowed our understanding of the game] via [http://www.grantland.com/story/_/id/9014162/sports-analytics-mit-sloan-conference Katie Baker's recap] ... [http://www.sloansportsconference.com/?p=10561 hockey analysis @ the conference] --[[User:Charles Jeffrey Danoff|Charles Jeffrey Danoff]] ([[User talk:Charles Jeffrey Danoff|discuss]] • [[Special:Contributions/Charles Jeffrey Danoff|contribs]]) 06:15, 8 March 2013 (UTC) * advancedballstats.com ---> Hubie & Mike Tirico discussing how advanced stats have integrated NBA 3 Mar 2013 Clippers vs OKC telecast -----> Hubie: Need to make it so everyone in organization understands them, incl. players --[[User:Charles Jeffrey Danoff|Charles Jeffrey Danoff]] ([[User talk:Charles Jeffrey Danoff|discuss]] • [[Special:Contributions/Charles Jeffrey Danoff|contribs]]) 06:44, 4 March 2013 (UTC) * "they're the gold standard. it's a proven system and hopefully we're on that track. we like to think we are, but you have to do it year after year." - Tom Thibodeau prior to his team's 11 Feb 2013 loss to a Spurs team w/o Duncan, Ginobli & Parker ... Spurs system embarassed the bulls on that night. --[[User:Charles Jeffrey Danoff|Charles Jeffrey Danoff]] ([[User talk:Charles Jeffrey Danoff|discuss]] • [[Special:Contributions/Charles Jeffrey Danoff|contribs]]) 06:34, 12 February 2013 (UTC) * "The most important thing is to know what your job is. The next is to do your job. But you can’t do your job if you don’t know what it is. So it’s important for everyone to learn and know multiple positions." - via my Dad http://blogs.bulls.com/2013/01/bulls-stop-bucks-here-in-milwaukee/ --[[User:Charles Jeffrey Danoff|Charles Jeffrey Danoff]] ([[User talk:Charles Jeffrey Danoff|discuss]] • [[Special:Contributions/Charles Jeffrey Danoff|contribs]]) 23:01, 2 February 2013 (UTC) * http://hoopshype.com/articles/narducci/memphis-committed-to-analytics wonderful article on the history, limits of and benefits from using statistical analysis in the NBA --[[User:Charles Jeffrey Danoff|Charles Jeffrey Danoff]] ([[User talk:Charles Jeffrey Danoff|discuss]] • [[Special:Contributions/Charles Jeffrey Danoff|contribs]]) 00:48, 1 February 2013 (UTC) * "That's the thing about the NBA. The better the team, the more you have to do your homework, and we're prepared to do that." - Kobe on Lakers' upcoming game w/Heat this week, following his team's win over Bucks --[[User:Charles Jeffrey Danoff|Charles Jeffrey Danoff]] ([[User talk:Charles Jeffrey Danoff|talk]]) 06:37, 16 January 2013 (UTC) * Helpful GM Links (sidenote, impressive how many years apart Krause won his NBA Executive awards + how many brother/people with last name who have been NBA GMs not including Billy King & Billy Knight). --[[User:Charles Jeffrey Danoff|Charles Jeffrey Danoff]] ([[User talk:Charles Jeffrey Danoff|talk]]) 06:55, 14 January 2013 (UTC) ** Wikipedia Template of Current NBA GMs http://thissongissick.com/blog/2013/aylen-bootleg-pack-2013-bootleg-mashup-download/#more-22138 ** http://www.basketball-reference.com/executives/ ** http://sportsillustrated.cnn.com/2007/writers/kelly_dwyer/06/25/gm.rankings/index.html ** http://sports.yahoo.com/nba/blog/ball_dont_lie/post/The-top-10-NBA-general-managers-of-the-last-deca;_ylt=AnDtXIkjx2yjECLM8Y38YSy8vLYF?urn=nba,191489 * Van Gundy commenting on coaches being fired when teams go bad ... and that person picking the players is most important in organization because they decide the talent and character of the players. (during Lakers vs. OKC on 11 Jan 2013) --[[User:Charles Jeffrey Danoff|Charles Jeffrey Danoff]] ([[User talk:Charles Jeffrey Danoff|talk]]) 06:08, 12 January 2013 (UTC) * In terms of who would win in a series Kenny Smith said "who can minimize their mistakes ... all teams have them" ... later in a discussion where Shaq asked Charles to clarify what makes somebody a great player Chuck said "making those around you better ... some guys are floaters, if they don't start well they'll float through the game, so I'd make sure to get them going early, that way they'd play better defense, better offense" following Clippers vs Celtics 27 Dec. '12 matchup --[[User:Charles Jeffrey Danoff|Charles Jeffrey Danoff]] ([[User talk:Charles Jeffrey Danoff|talk]]) 07:00, 28 December 2012 (UTC) * "If you have depth it takes longer to develop chemistry." Nuggets/Grizzlies ESPN broadcast 14 Dec 2012 -- Charlie Danoff via [[Special:Contributions/108.200.241.51|108.200.241.51]] 04:08, 15 December 2012 (UTC) * On MNF Gruden commended Belicheck saying all of his players are versatile, able to adjust to what the opponent brings. Additionally he does an insane amount of research finding undrafted players to play roles on his team, including, according to Gruden, a Ohio State rugby player whose doing special teams. --[[User:Charles Jeffrey Danoff|Charles Jeffrey Danoff]] ([[User talk:Charles Jeffrey Danoff|talk]]) 03:47, 11 December 2012 (UTC) * Pat Riley "Disease of more (money, shots, etc.) that comes from winning a champtionship. Via David Aldrich --[[User:Charles Jeffrey Danoff|Charles Jeffrey Danoff]] ([[User talk:Charles Jeffrey Danoff|talk]]) 23:13, 8 December 2012 (UTC) * Create a GM index, comparing the Winshares of a player relative to his draft class. Drafted number 17, win shares are number 12 in his class, so GM gets a +5 or something. Also include wins, playoff wins and how guys have done after they've been acquired in a trade ... have like a 10 year window for evaluation --[[User:Charles Jeffrey Danoff|Charles Jeffrey Danoff]] ([[User talk:Charles Jeffrey Danoff|talk]]) 20:26, 8 December 2012 (UTC) * Kobe on leadership, via Facebook https://www.facebook.com/Kobe/posts/10151211862620419 ** "Sometimes you must prioritize the success of the team ahead of how your own image is perceived. The ability to elevate those around you is more than simply sharing the ball or making teammates feel a certain level of comfort. It's pushing them to find their inner beast, even if they end up resenting you for it at the time. ... I'd rather be perceived as a winner than a good teammate. I wish they both went hand in hand all the time but that's just not reality." --[[User:Charles Jeffrey Danoff|Charles Jeffrey Danoff]] ([[User talk:Charles Jeffrey Danoff|talk]]) 20:26, 8 December 2012 (UTC) * Bill Simmons interviewing Bill Russell and commenting on Kobe, ** "But that would have failed unless everyone embraced their role, and that's the thing — everyone has to have a role." from Russell about working with teammates ** Top players of all time "That list currently looks like this: Jordan, then Russell, then Kareem, then Magic/Bird, then Wilt, then Kobe/Duncan (or Duncan/Kobe), then West/Oscar, then Hakeem/Shaq/Moses. With LeBron lurking in there somewhere. We just don't know where yet." from http://www.grantland.com/story/_/id/8724362/the-kobe-question --[[User:Charles Jeffrey Danoff|Charles Jeffrey Danoff]] ([[User talk:Charles Jeffrey Danoff|talk]]) 20:26, 8 December 2012 (UTC) * Bill Simmons on dynasties "In the NBA's seven-decade history, we only witnessed four true dynasties: Russell's Celtics (13 years, 11 titles), Jordan's Bulls (eight years, six titles), Magic's Lakers (12 years, nine Finals appearances, five titles)2 and Mikan's Lakers (six years, five titles)" http://www.grantland.com/story/_/id/8486795/the-harden-dilemma --[[User:Charles Jeffrey Danoff|Charles Jeffrey Danoff]] ([[User talk:Charles Jeffrey Danoff|talk]]) 12:03, 12 October 2012 (UTC) * 22 Feb 2012 Tom Thibodeau news conference said something to this effect: "You play 66 games and then hopefully you make the playoffs. Once there whomever's healthiest and playing the best will win. I don't think you want to approach it as 'getting through something' because its not the goal, whatever you're going through, you want to keep the ultimate goal in mind." --[[User:Charles Jeffrey Danoff|Charles Jeffrey Danoff]] 03:37, 23 February 2012 (UTC) * "The day of the trade deadline was good. I did what I always do. Worked out, did some rehab… You can’t control what’s going to happen so you have to do what you have to do. Talking to your agent is not going to change anything." - [[w:Caron Butler]] from a June 2011 [http://blogs.hoopshype.com/blogs/butler/2011/03/02/im-getting-closer/ blog post] by him. * "I probably look at this more simplistic than you want. It doesn't matter whether you're a former player or you come up through the video room, whether you come from college. You have to earn that trust from NBA players every day. It doesn't stop unless, of course, you win 10 rings. But even at that point, you have to prove that there's competency, there's a work ethic, there's a plan, there's an organization to your thoughts. Then, at some point, they believe you can help them achieve what they want to achieve." " - [[w:Erick Spoelstra]] from a June 2011 ESPN.com [http://sports.espn.go.com/nba/truehoop/miamiheat/columns/story?page=Spoelstra-110601 story] on him. ** [http://blogs.hoopshype.com/blogs/zarrabi/2011/06/02/for-shaq-life-was-first/ Yin] to Spoelestra's work-ethic Yang? * On [[w:Inside the NBA|Inside the NBA]] tonight [[w:Charles Barkley|Charles]] said "Every great team needs a 'dude' and a 'crazy guy'". (e.g. 2011 Playoffs Mavericks have Dirk & DeShaun Stevenson; LA has Kobe & Ron Artest) ----[[User:Charles Jeffrey Danoff|Charles Jeffrey Danoff]] 02:46, 3 May 2011 (UTC) * NBA dynasty ... its about making moves to give yourself a possibility to be great (thus acquiring great players?) --[[User:Charles Jeffrey Danoff|Charles Jeffrey Danoff]] 17:14, 10 April 2011 (UTC) * i was thinking, while re-building is it more important to accumulate talent (e.g. krause w/pre rebuilding bulls) or to play solid basketball every year (jazz post stockton/malone) ... need both, but maybe better to focus on always playing good basketball on some level and trying to make playoffs regardless of personnell (to degrees, saving cap space) than just getting as many good young players at 1 time? --[[User:Charles Jeffrey Danoff|Charles Jeffrey Danoff]] 01:09, 28 March 2011 (UTC) * for some players injuries can be big problems vs. others whose "best ability is availability" from jeff van gundy (he said he heard it from nfl live) on por vs. oakland 27 mar 2011 game --[[User:Charles Jeffrey Danoff|Charles Jeffrey Danoff]] 00:16, 28 March 2011 (UTC) * Bill Simmons, "Basketball is relatively simple. Teams win championships conventionally (an alpha dog superstar, an accomplished sidekick, a third elite player, than five or six role players) and unconventionally (three or four elite starters who complement each other, then four or five role players), but never radically. The Heat are thinking radically, even if it might not seem like it. They believe two alpha dog superstars -- both perimeter players, both creators who need the ball in their hands, both franchise players who developed particular habits carrying inferior teammates these past seven years -- can reinvent themselves as co-CEOs of a basketball team." from [http://sports.espn.go.com/espn/page2/story?page=simmons/10129&sportCat=nba LeBron and Wade: Can it work?] 29 Oct 2010 * Beckley Mason, "The best NBA coaches have a track record of enhancing their players’ awareness. Phil Jackson’s Zen philosophy of being “in the moment” is really about being fully cognizant of teammates and opponents. In Utah, Jerry Sloan’s persistent use of the flex offense has the interesting effect of increasing his team’s collective awareness level. Because they devote so much energy and practice time to one offense, Jazz players are intensely aware of the near infinite permutations of each set, and are focused on reading and reacting. With an internalized, cohesive offensive strategy, Jazz players are able to consistently rise above offensive expectations. See, being aware isn’t just for the greats, it’s also how bad players become average and mediocre players become Kevin Martins. In short, it’s how a player maximizes his potential." from [http://hoopspeak.com/2010/11/awareness-greatness-kobe-bryant-kevin-garnett/ On Awareness: The Intangible Skill That Makes All The Difference] November 19th, 2010 * must a dynasty necessarily include something new, that couldn't be predicted by a mathematical equation. like a new pop hit needs to have something fresh to catch that is inspired by what came b4, but also not able to be predicted. e.g. jordan being first 2-guard led dynasty. \ * godel's theorum needs to fit into this analysis * [http://on.wsj.com/geTmsK WSJ article] about the amount of scouting/research a color analyst (Bill Raftery) does to prepare for games ([http://online.wsj.com/article/SB10001424052748703555804576102461442595674.html?mod=wsj_share_twitter alternative link]) <blockquote> * Basketball color analyst Bill Raftery carries a stack of DVDs in his briefcase wherever he goes. * It can take Mr. Raftery several hours to get through a single, 40-minute college basketball game. His fingers are constantly on the rewind and fast-forward buttons so he can make sure his diagrams match the plays he is watching. * During games Mr. Raftery keeps a notebook open to a page where he has written each player's season average for minutes played, rebounds, assists, points, three-pointers, field goal percentage, steals and blocks. Writing down the numbers himself helps him commit them to memory. </blockquote> * Steve Young commented about the Packers following their Super Bowl 45 victory that they had a self-policing lockeroom. I think that's a really important part of dynasties, players can't rely on coaches or management to police them and make sure they toe the line, they have to do it themselves and hold eachother accountable. Maybe. * Is an effective measure of a "truly great player" (a true "1" in terms of a leader, not the point guard position) number of all star games started? * Announcer (Thurl Bailey?) from 28 Feb 2011 Celtics vs. Jazz game on NBA TV said that the Celtics's greatest strength is their mix of young and old players ... another announcer added how some NBA teams have problems b/c they have too many of the same kind of player --[[User:Charles Jeffrey Danoff|Charles Jeffrey Danoff]] 02:53, 1 March 2011 (UTC) * Tom Thibodeau after 28 Feb game vs. Wizards ... every team in NBA is good & poor records usually result of scheduling/injuries/etc ... so if you don't prepare for the game the right way you could trip --[[User:Charles Jeffrey Danoff|Charles Jeffrey Danoff]] 02:53, 1 March 2011 (UTC) * do a GM power ranking? adjusted periodically for recent transactions/results? ... include hoops analyst's gm reports? --[[User:Charles Jeffrey Danoff|Charles Jeffrey Danoff]] 02:57, 1 March 2011 (UTC) * measure shooters by the 40-50-90 "magic #" http://www.thedaily.com/page/2010/11/07/030211-sports-nash-1-4/ --[[User:Charles Jeffrey Danoff|Charles Jeffrey Danoff]] 20:39, 3 March 2011 (UTC) * Ben Golliver "[http://blogs.hoopshype.com/blogs/golliver/2011/03/10/walk-like-a-gm-talk-like-a-gm/ Walk like a GM, talk like a GM]" March 10, 2011 @ 6:29 pm "We have looked at three thoughts. First, that the NBA trade market is not nearly as efficient as it should be, but that analytics may help improve that. Second, that the short-term pressure on NBA GMs to win big is enormous and that some GMs have shifted course to focus on risk assessment so that they can swing for the fences when necessary. Third, that NBA executives are feeling limitations in their ability to use analytics to gain an advantage during the draft process and that non-quantifiables such as work ethic and dedication remain undervalued assets. Taken together, we’ve reached a point where we can officially declare that the honeymoon period is over for the cult of advanced statistics in basketball. ... The message from the 2011 Sloan Conference, then, wasn’t the same as previous years: It wasn’t a declaration of arrival, a celebration of newfound success or even an admission of early struggles. Rather, it leaned more towards a high school reunion, where the like-minded from around the country, now fully content and secure with their place at the table, gathered to swap stories and best practices in front of a rapidly growing crowd of followers, anxious to learn how they might grab their own seat. * kendall gill said something about how the key to defense in the NBA is the 2nd man? --[[User:Charles Jeffrey Danoff|Charles Jeffrey Danoff]] 03:33, 15 March 2011 (UTC) === styles of play === ==== Havoc ==== Coach Smart's teams play a style of basketball known as "[[havoc]]". Smart described his "havoc" defensive philosophy during his introductory press conference as, "We are going to wreak havoc on our opponent's psyche and their plan of attack." On the court, the "havoc" defensive mindset is visible through the heavy use of the [[full court press]] and pressing after made baskets to disrupt opponent's timing of offensive sets. In his second season, VCU led the Colonial Athletic Association in steals. Offensively, Shaka Smart-coached teams play uptempo and push the ball after misses and makes with what Smart describes as, "The freedom to make plays in the open court". [[w:Shaka Smart|From Wikipedia]]. --[[User:Charles Jeffrey Danoff|Charles Jeffrey Danoff]] 03:03, 19 March 2011 (UTC) ==== Princeton Offense==== * Princeton ** Basketball's Princeton-Style Offense: A Simplified Approach for High School Coaches by Derek Sheridan (ISBN-10: [http://en.wikipedia.org/w/index.php?title=Special%3ABookSources&isbn=1930546920 1930546920]) * Northwestern under Bill Carmody * Georgetown under John Thompson III * Richmond? ==== Triangle Offense==== * Tex Winter designed for Phil Jackson Bulls & Lakers teams ** The Triple Post Offense Book by [[w:Tex Winter|Tex Winter]] ([[w:Amazon Standard Identification Number|ASIN]]: [http://www.amazon.com/Triple-Post-Offense-Book-Winter/dp/B000A7OVR6/ref=reg_hu-rd_add_1_dp_T2 B000A7OVR6]) ==== Pick & Roll Offense ==== * Jerry Sloan coached Utah Jazz Teams === statistics === * "[http://agiletesting.blogspot.com/2005/04/sparkplot-creating-sparklines-with.html sparkplot: creating sparklines with matplotlib]" by [http://www.linkedin.com/in/ggheorghiu Grig Gheorghiu] found via [http://www.edwardtufte.com/bboard/q-and-a-fetch-msg?msg_id=0001lh E.T.] it has an example of sparklines used to quickly analyze a lakers season that i think would be helpful to have in a big quantity to evaluate teams. == APBRmetrics in the NBA == * Google Book Search for "nba statistics" http://www.google.com/search?q=%22NBA%20statistics%22&tbs=bks:1 --[[User:Charles Jeffrey Danoff|Charles Jeffrey Danoff]] 07:48, 20 December 2010 (UTC) * [http://www.silverscreenandroll.com/2010/10/28/1779160/the-question-of-lakers-vs-heat-is-a-defining-battle-in-the#storyjump The question of Lakers vs. Heat is a defining battle in the statistical revolution] by C.A. Clark on Oct 28, 2010 8:30 AM PDT in Editorial ** The basketball statistical revolution is coming. Slowly but surely, advanced statistics are creeping into the game. The signs are everywhere: John Hollinger is a prominent national basketball writer for ESPN, primarily on the basis of his statistical model for player evaluation, PER. Daryl Morey and Rich Cho are two of the youngest GMs in the league, and both have foundations less rooted in basketball than in statistics. More than half the league's teams employ at least one full-time analyst devoted to statistical work. Regardless of how you feel about advanced stats, it is impossible to ignore the growing impact they have on the modernizing NBA. *** via [http://tpsnation.com/2010/10/29/the-10-man-rotation-starring-tyrus-thomas-dancin-away/ The 10-man rotation, starring Tyrus Thomas dancin’ away] via pingback from [http://hoopsanalyst.com/blog/?p=572 NBA Predictions 2010-11] * Hollinger got a job w/Wizards. Go back and read early Baskteball Prospectuses? --[[Special:Contributions/108.200.241.51|108.200.241.51]] 09:14, 15 December 2012 (UTC) == ''cf.'' == * The October 26th, 2010 version of [http://danoff.org/ledive/index.php?title=Re-Booting_Modeling_Wins_in_the_NBA:_A_Work_in_Progress_%E2%80%A6_Notepad&oldid=36839 Re-Booting Modeling Wins in the NBA: A Work in Progress … Notepad] a document I created for notes on the How-to paper published that day. * My original write-up of an Independent Study I did as a Colgate University undergrad on the Determinants of Success in the NBA: [http://p2pu.org/node/5571/document/10218 Re-Booting Modeling Wins in the NBA: A Work in Progress …] * [http://freedarko.blogspot.com/ Freedarko] for a philosophical take on the question. == Resources == * [http://basketballprospectus.com/ Basketball Prospectus] * [http://knickerblogger.net/statpage/ Knickerblogger.net Stats Page] * [http://stackoverflow.com/questions/643753/does-anybody-know-of-a-real-time-sports-scores-stats-web-service Does anybody know of a real time sports scores/stats web service? (closed)] - way to get text based box scores for data/archival purposes? == to-read == * Second Wind by Bill Russell * Basketball on Paper * Wages & Wins ** [http://www.wagesofwins.com/WOWarticles.html Academic articles from Wages of Wins] * [http://danrosenbaum.blogspot.com/ Dan Rosenbaum's NBA Blog] * [http://www.bepress.com/jqas/ Journal of Quantitative Analysis in Sports] * [http://www.amstat.org/sections/sis/ Statistics in Sports: A Section of the American Statistical Association] * Presentations from [http://ncssors.wikidot.com/program 2010 NORTHERN CALIFORNIA SYMPOSIUM ON STATISTICS AND OPERATIONS RESEARCH IN SPORTS] == to-do == * fit this research into http://www.aeaweb.org/journal/jel_class_system.php * explore panels and papers from this conference http://www.sloansportsconference.com/panels-2 ==References== {{Reflist|40em}} i67ypnyu6i2fertpzipu1rvafl3916s 0 139384 2818472 2818451 2026-07-18T13:50:48Z Young1lim 21186 /* Adder */ 2818472 wikitext text/x-wiki == Adder == * Binary Adder Architecture Exploration ( [[Media:Adder.20131113.pdf|pdf]] ) {| class="wikitable" |- ! Adder type !! Overview !! Analysis !! VHDL Level Design !! CMOS Level Design |- | '''1. Ripple Carry Adder''' || [[Media:VLSI.Arith.1A.RCA.20250522.pdf|A]]|| || [[Media:Adder.rca.20140313.pdf|pdf]] || [[Media:VLSI.Arith.1D.RCA.CMOS.20211108.pdf|pdf]] |- | '''2. Carry Lookahead Adder''' || [[Media:VLSI.Arith.2A.CLA.20260718.pdf|A]], [[Media:VLSI.Arith.2B.CLA.20260718.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]] p2mzyo4hlj896hlem7av4mxe8saixhv 0 171005 2818477 2818456 2026-07-18T14:05:30Z Young1lim 21186 /* Geometric Series Examples */ 2818477 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.20260718.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]] qtx092x7ojazu0572f9jxklu5401bdu 0 285380 2818475 2818454 2026-07-18T14:02:07Z Young1lim 21186 /* Applications */ 2818475 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.20260718.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> a15vaiotmp5qrd7664dfduepiqao3iv 4 285491 2818509 2818134 2026-07-19T06:21:10Z Alexis Jazz 791434 Updating gadget usage statistics from [[Special:GadgetUsage]] ([[phab:T121049]]) 2818509 wikitext text/x-wiki {{#ifexist:Project:GUS2Wiki/top|{{/top}}|This page provides a historical record of [[Special:GadgetUsage]] through its page history. To get the data in CSV format, see wikitext. To customize this message or add categories, create [[/top]].}} The following data is cached, and was last updated 2026-07-16T11:03:52Z. A maximum of {{PLURAL:5000|one result is|5000 results are}} available in the cache. {| class="sortable wikitable" ! Gadget !! data-sort-type="number" | Number of users !! data-sort-type="number" | Active users |- |CleanDeletions || 75 || 0 |- |EnhancedTalk || 1399 || 3 |- |HideFundraisingNotice || 839 || 7 |- |HotCat || 922 || 13 |- |LintHint || 118 || 1 |- |Round Corners || 1188 || 2 |- |contribsrange || 388 || 3 |- |dark-mode || 137 || 2 |- |dark-mode-toggle || 190 || 3 |- |edittop || 519 || 7 |- |popups || 881 || 4 |- |purge || 738 || 8 |- |sidebartranslate || 560 || 1 |- |usurper-count || 119 || 1 |} * [[Special:GadgetUsage]] * [[m:Meta:GUS2Wiki/Script|GUS2Wiki]] <!-- data in CSV format: CleanDeletions,75,0 EnhancedTalk,1399,3 HideFundraisingNotice,839,7 HotCat,922,13 LintHint,118,1 Round Corners,1188,2 contribsrange,388,3 dark-mode,137,2 dark-mode-toggle,190,3 edittop,519,7 popups,881,4 purge,738,8 sidebartranslate,560,1 usurper-count,119,1 --> qm5k1fb19p2zj1mal8ylpea881jolf5 2 289273 2818504 2816685 2026-07-19T02:55:30Z Dc.samizdat 2856930 /* Special relativity describes Euclidean 4-space */ 2818504 wikitext text/x-wiki = Real Euclidean four-dimensional space R⁴ = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - June 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as a Euclidean space of four orthogonal spatial dimensions. Space itself has a fourth orthogonal dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.</blockquote> == Summary == We observe that physical space has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is translating through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions. == A theory of the Euclidean cosmos == The physical universe is properly visualized as a [[w:Four-dimensional_space|Euclidean space of four orthogonal spatial dimensions]]. Space itself has a fourth orthogonal dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move through Euclidean 4-space at velocity <math>c</math> as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical spiral galaxy such as ours is a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our galaxy appear from our viewpoint to be distributed in a cloud of elliptical spirals occupying a flattened ellipsoid region of 3-dimensional space, but they are not so confined: they are distributed within a spherical region of 4-dimensional space. The galaxy's actual shape is spherical, not a flattened ellipsoid, but it is rounder than round can be in our ordinary experience: it occupies a hyperspherical region of space. The concentric spirals of stars that we observe lie on concentric [[W:3-sphere|3-sphere]]s (4-dimensional spheres), not on concentric 2-ellipsoids (3-dimensional elliptical spirals). Our sun and solar system lies on one of those concentric 3-spheres. More generally, orbits are circular in 4-space, and elliptical in the 3-space of their elliptic hyperplane. ...rotating illustration of the 4-ball galaxy showimg its spirals of star clouds on the surface of concentric 3-spheres...obtained by reverse sterographic projection from 3D images of the galaxy... The galaxy as a whole, or more properly its orbital center point, is translating through 4-space at velocity <math>c</math>, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity <math>c</math> in the same direction, but on spiral trajectories relative to the galaxy's linear trajectory, as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the cylinder; their trajectories are screw-displacements, the compound of a simple rotation and a linear translation. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis. It is the tremendous inertial force of stars in motion at velocity <math>c</math> that holds the cylinder of motion together. The observed universe as a whole appears to be a 3-sphere expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. For all observers, the conjectured origin point of the universe corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space (the same point in the same Euclidean 4-space for all observers). The big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured separately, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. The enclosing surface of a spherical region of 4-space is itself a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. We live within such a 3-space, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our particular 3-sphere is one of the galaxy's concentric 3-spheres of spiral star-clouds. The solar system occupies a tiny patch of this filmy 4-dimensional soap-bubble of galactic size, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. Our entire 3-sphere manifold, as a 3-spherical shell within the moving 4-ball galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy, in a distinct direction that is orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of rotating objects through Euclidean space by screw translation. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold that is such an evolving surface boundary is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a three-dimensional smear of atoms no thicker than one atom in its fourth dimension, which is the direction of its linear translation through 4-space at velocity <math>c</math>. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to 4-dimensional lumps of matter as plasma, and have little experimental knowledge of their geometry or internal structure. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know nothing about its interior 4-ball. Every such moving 3-dimensional surface boundary of matter in the observed universe is evolving in four dimensions at velocity <math>c</math>. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from atoms to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, two orthogonal inertial motions each at the constant universal rate of transformation <math>c</math>. Every object moves relative to universal 4-coordinate space on its own distinct geodesic spiral, a screw translation trajectory that is the compound of its two orthogonal inertial motions. Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in motion at velocity <math>c</math> in all four dimensions concurrently, so they move diagonally through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c^\prime = 2c</math>, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold moving at velocity <math>c</math>. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math> in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper space. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <small><math>\mathrm{Q}</math></small> denote a rotation, <small><math>\mathrm{R}</math></small> a reflection, <small><math>\mathrm{T}</math></small> a translation, and let <small><math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math></small> denote a product of several such transformations, all commutative with one another. Then <small><math>\mathrm{RT}</math></small> is a glide-reflection (in two or three dimensions), <small><math>\mathrm{QR}</math></small> is a rotary-reflection, <small><math>\mathrm{QT}</math></small> is a screw-displacement, and <small><math>\mathrm{Q^2}</math></small> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<small><math>\mathrm{Q}^q \mathrm{R}^r</math></small><br> where <small><math>(2^q + r \le n)</math></small>, the number of dimensions.<br> Transformations involving a translation are expressible as:<br> :<small><math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math></small><br> where <small><math>(2^q + r + 1 \le n)</math></small>.<br> For <small><math>(n = 4)</math></small> in particular, every displacement is either a double rotation <small><math>\mathrm{Q}^2</math></small>, or a screw-displacement <small><math>\mathrm{QT}</math></small> [where the rotation component <small><math>\mathrm{Q}</math></small> is a simple rotation, but the <small><math>\mathrm{QT}</math></small> is chiral like a <small><math>\mathrm{Q^2}</math></small>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <small><math>\mathrm{QRT}</math></small>.</blockquote> If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a <small><math>\mathrm{Q^2}</math></small> or a <small><math>\mathrm{QT}</math></small>, because we can view any <small><math>\mathrm{QT}</math></small> as a <small><math>\mathrm{Q^2}</math></small> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <small><math>\mathrm{Q^2}</math></small>. By the same principle, we can view any <small><math>\mathrm{QT}</math></small> or <small><math>\mathrm{Q^2}</math></small> as an isoclinic (equi-angled) <small><math>\mathrm{Q^2}</math></small> by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (<small><math>\mathrm{T}</math></small>) for ''one'' of the two rotations (<small><math>\mathrm{Q}</math></small>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<small><math>\mathrm{Q}</math></small>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<small><math>\mathrm{T}</math></small>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <small><math>SO(4)</math></small> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <small><math>SO(4)</math></small> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, stationary atoms exhibit the <small><math>SO(4)</math></small> symmetries of the discrete isoclinic (equi-angled) double rotations (<small><math>\mathrm{Q^2}</math></small>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.<ref>{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional "space" because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was first described by Einstein himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction. Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity ''c'', in some 4-space direction. This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always ''c'', as measured by all observers in any inertial reference frame. This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity ''c''. In physics as it has been universally understood, observers are not supposed to be able to move at velocity ''c''. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four. In the Euclidean relativity alternative view, however, every observer is always moving at velocity ''c'' through the universe, which is real Euclidean 4-dimensional space <small><math>\mathbb{R^4}</math></small>. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity ''c''. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity ''c'', in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your coordinate systems; you are both embedded in the same Euclidean 4-dimensional space <small><math>\mathbb{R^4}</math></small>.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity ''c'' relative to universal 4-coordinate space, so the maximum relative velocity between two observers is 2''c'' when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to ''c'', it is the same measurement in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}} So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity ''c'' in Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" ''c'', although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity ''c'', but in at least slightly different directions. In Einstein's relativity, the invariant ''c'' is the speed of light through 3-space. In Euclidean relativity, the invariant ''c'' is the speed of matter through 4-space! The speed of light through 3-space is also perceived as ''c'' by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity ''c''. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space. ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity <math>c</math> through space and proper time ...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords. Euclidean relativity is not even a fringe theory; no physicists have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at ''c'' (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than ''c''. Euclidean relativity is a revolutionary theory indeed, in which ''c'' cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for it to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c^\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the diagonal of the unit 4-hypercube is the natural number <small><math>\sqrt{4}</math></small>. == An object's motion in space is the product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a small number of discrete self-reflections. Any action of a geometric object that transforms its position and orientation in space may be measured as a distinct group of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete set of local self-reflections. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which touch (intersect each other). When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, leaving itself right-side-out again. Coxeter's laws of motion are a geometric counterpart to Newton's laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as simple geometric pictures, by anyone baffled by algebraic formulas. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. For example, they give us simple geometric pictures of all the possible motions of objects in four dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<small><math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math></small><br> where <small><math>(2^q + r + t \le 4)</math></small>. Every displacement is either a double rotation <small><math>\mathrm{Q}^2</math></small>, or a screw-displacement <small><math>\mathrm{QT}</math></small> [where the rotation component <small><math>\mathrm{Q}</math></small> is a simple rotation, but the <small><math>\mathrm{QT}</math></small> is chiral like a <small><math>\mathrm{Q^2}</math></small>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <small><math>\mathrm{QRT}</math></small>.</blockquote> While this description should be understood as simple geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (<small><math>\mathrm{Q}</math></small>), reflection (<small><math>\mathrm{R}</math></small>) and translation (<small><math>\mathrm{T}</math></small>) are just what they are in three-dimensional space, but double rotation (<small><math>\mathrm{Q}^2</math></small>) is something new and unprecedented in our physical experience, because double rotations cannot occur until you have four or more dimensions of space to rotate in. ...to readers who have not studied Coxeter (almost all readers including TAC), the blockquote above is "just math", not visualizable geometry...but I could describe Coxeter's congruent transformations in 4-space here geometrically: I could say clearly what they mean in spatial terms, in language anyone can understand, because they don't require any math to be understood; the "math" here is really just simple pictures (reflections and rotations); even double rotations can be visualized by dimensional analogy, as compounds of simple rotations...since even most physicists are unacquainted with Coxeter geometry, it really is important that I do this here... == Light propagates through 4-space at twice its apparent velocity ''c''== Coxeter's geometric laws of motion apply to all objects with mass in 4-dimensional Euclidean space, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <small><math>\mathrm{R}^4</math></small>, which may be termed a double translation <small><math>\mathrm{T}^2</math></small>, a pure translation via two pairs of parallel reflections, without any rotation component <small><math>\mathrm{Q}</math></small>. Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by <small><math>\mathrm{QT}</math></small>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <small><math>\mathrm{Q^2}</math></small>, an isoclinically rotating object such as an atom. A simple rotation <small><math>\mathrm{Q}</math></small> or simple translation <small><math>\mathrm{T}</math></small> is a double reflection <small><math>\mathrm{R^2}</math></small>, so a <small><math>\mathrm{QT}</math></small> or <small><math>\mathrm{Q^2}</math></small> is also an <small><math>\mathrm{R^4}</math></small>, but not with the same group of reflection angles as a light signal <small><math>\mathrm{R^4}</math></small>. A translation <small><math>\mathrm{T = R^2}</math></small> is a double reflection in two parallel planes, and a rotation <small><math>\mathrm{Q = R^2}</math></small> is a double reflection in two intersecting planes, as in a <small><math>\mathrm{QT = R^4}</math></small> which is both at once. A double translation <small><math>\mathrm{T^2 = R^4}</math></small> is two double reflections in pairs of parallel planes at once, a reflection in four or more non-intersecting parallel planes; it is all translation and no rotation. In a <small><math>\mathrm{T^2}</math></small> all the motion goes to translation, so the translation goes twice as far as the simple translation <small><math>\mathrm{T}</math></small> in a <small><math>\mathrm{QT}</math></small>. A double translation <small><math>\mathrm{T^2 = R^4}</math></small> is the opposite of a double rotation <small><math>\mathrm{Q^2 = R^4}</math></small>, which is stationary but rotates twice as fast as the simple rotation <small><math>\mathrm{Q}</math></small> in a <small><math>\mathrm{QT}</math></small>. The product of the two translations in a <small><math>\mathrm{T^2}</math></small> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <small><math>\mathrm{T}</math></small> over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A <small><math>\mathrm{T^2}</math></small> cannot reposition a 4-polytope the way a <small><math>\mathrm{QT}</math></small> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter. ...lensing of double translations <small><math>\mathrm{T^2 = R^4}</math></small> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet... == The Kepler problem is framed in Euclidean 4-space == The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], arguably the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the "inverse Kepler problem": the orbit characteristics require the force to depend on the inverse square of the distance.<ref>{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton & Company|isbn=978-0393039184}}</ref> The inverse square law behind the Kepler problem is the [[W:Central force|central force]] law which governs not only [[W:Newtonian gravity|Newtonian gravity]] and celestial orbits, but also the motion of two charged particles in [[W:Coulomb’s law|Coulomb’s law]] of [[W:Electrostatics|electrostatics]]; it applies to attractive or repulsive forces. Problems in which two bodies interact by a central force that varies as the [[W:Inverse square law|inverse square]] of the distance between them are called Kepler problems. Thus the [[W:Hydrogen atom|hydrogen atom]] is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law, another inverse-square central force. Using classical mechanics, the solution to a Kepler problem can be expressed as a [[W:Kepler orbit|Kepler orbit]] using six kinematical variables or [[W:Orbital elements|orbital elements]]. The solution conserves an orbital element called the [[W:Laplace–Runge–Lenz vector|Laplace–Runge–Lenz (LRL) vector]], a [[W:Constant of motion|constant of motion]], meaning that it is the same no matter where it is calculated on the orbit. The LRL vector was essential in the first quantum mechanical derivation of the [[W:Atomic emission spectrum|spectrum]] of the hydrogen atom, but this approach has rarely been used since the development of the [[W:Schrödinger equation|Schrödinger equation]]. The conservation of the LRL vector corresponds to the <small><math>SO(4)</math></small> symmetry, by Nother's theorem. The LRL vector lies orthogonal to both the orbital plane and the angular momentum vector of the Kepler orbit; we observe that it lies in a fourth orthogonal dimension. Fock in 1935<ref>V. Fock, Zur Theorie des Wasserstoffatoms, Zeitschrift für Physik. 98 (3-4) (1935), 145–154.</ref> and Moser in 1970<ref>J. Moser, Regularization of Kepler’s problem and the averaging method on a manifold, Commun. Pure Appl. 23 (1970), 609–636</ref> observed that the Kepler problem is mathematically equivalent to non-affine geodesic motion (a particle moving freely) on the surface of a 3-sphere, so that the whole problem is symmetric under certain rotations of the four-dimensional space. This higher-dimensional symmetry results in two well-known properties of the Kepler problem: the momentum vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points. ... Relativity establishes that an orbit in space is viewed in a different way in each distinct inertial reference frame. Depending on the choice of reference frame, the same Kepler system may be seen to be performing any one of a sequence of relativistically equivalent rotations in 4-space, on a continuum from an isoclinic rotation (Q²) in the orbit's proper reference frame, to a screw transfer (QT) with a simple rotation component (Q) and a translation component (T) at velocity <math>c</math>, in the universal reference frame of 4-coordinate space wherein every object is seen to be translating at velocity <math>c</math>. In reference frames between these two limit cases, the orbit is seen to be performing a double rotation (Q²) at two unequal, completely orthogonal angular rates of rotation: an elliptical double rotation. These include the reference frames of most typical observers, who are moving slowly relative to the observed orbital system's reference frame (their relative motion is a small fraction of the speed of light). ...this is probably misplaced here and should not interrupt the discussion at this point: ...These typical observations agree closely with the predictions of special relativity, because the non-isoclinic elliptical (Q²) resembles a (QT), since one of its two completely orthogonal rotations (Q) has such a long period that it is almost indistinguishable from a straight translation (T). All orbits in 4-space are isoclinic in their own reference frame. Orbiting objects in their own proper Kepler systems follow circular geodesic isoclines through 4-space. Orbits in 4-space are perfectly circular in their own reference frame, as Copernicus assumed the orbits of planets to be. It is the orbit's path through the 3-space of its elliptic hyperplane that is an ellipse, as Kepler found it to be. ...cite Jesper Goransson's very concise paper The geodesic circle that an orbiting object follows through 4-space in the proper reference frame of its own Kepler system is not a simple great circle which turns in two orthogonal dimensions. It is a helical great circle that turns in four orthogonal dimensions at once.{{Efn|Geodesic orbits in 4-space are not simple 2-dimensional great circles; they are helical 4-dimensional great circles that curve in all four dimensions at once. Their circular trajectories are helixes which we call ''isoclines'', since they are the paths taken by points on a rigid object undergoing isoclinic rotation.}} Such circles lie outside our physical experience, since our local space has only three orthogonal dimensions. Nonetheless we can visualize them in imagination, because their helical, circular shape is perfectly well defined by the kinematical variables of the Kepler orbit. The real physical correlates of abstract orthogonal planes and rotation angles are already familiar to us viscerally in our body-language of physical experience, since we are endowed biologically with highly evolved visual signal processing engines. These enable us to see and understand spatial relations and motions, including rotations, without even thinking about angles and orthogonal planes. This physical endowment is an inborn capacity for dimensional analogy which our biologic evolution has provided. All our instinctive spatial reasoning is by dimensional analogy from flat 2-dimensional retinal images to 3-dimensional scenes, using our powerful inborn visualization capacities of reverse stereographic projection and pattern recognition. We humans are thus very well equipped with everything we need to see in four-dimensional space, except experience. ... Recently Anco and Moghadam found that through Noether’s theorem in reverse, the LRL vector gives rise to a corresponding infinitesimal dynamical symmetry on the kinematical variables, which they show to be the semi-direct product of <small><math>SO(3)</math></small> and <small><math>\mathbb{R^3}</math></small>, in contrast to the <small><math>SO(4)</math></small> symmetry group generated by the LRL symmetries and the rotations.{{Sfn|Anco|Moghadam|2026|ps=; The physically relevant part of the LRL vector is its direction ... since its magnitude is just a function of energy and angular momentum.}} This remarkable symmetry breaking is expressive of the ''dimensional relativity'' between ordinary 3-space <small><math>\mathbb{R^3}</math></small>, spherical space <small><math>S^3</math></small> and Euclidean space <small><math>\mathbb{R^4}</math></small>. Consider a hydrogen atom in a Kepler orbit: for example, a hydrogen atom moving freely in space in an orbit around the sun. It is a ''double'' Kepler problem: an electrostatic Kepler problem within itself, and a gravitational Kepler problem in its environment. The ''single'' electrostatic Kepler problem of a hydrogen atom moving freely in space beyond any gravitational influence is a problem in special relativity. In our Euclidean 4-space model, this atom viewed as stationary in its own proper reference frame exhibits an <small><math>SO(4)</math></small> rotation symmetry corresponding to an isoclinic double rotation (<small><math>\mathrm{Q^2}</math></small>). The fourth dimension in this reference frame is the atom's proper time vector; it has constant velocity <math>c</math> and constant direction. From the point of view of our universal 4-coordinate space (which cannot be the proper inertial reference frame of any physical observer, all of whom are moving relative to it at velocity ''c''), the entire Kepler system (the atom) is translating through 4-space via a screw translation (<small><math>\mathrm{QT}</math></small>) at constant velocity <math>c</math>. From this viewpoint the atom has only a simple <small><math>SO(3)</math></small> rotation component (<small><math>\mathrm{Q}</math></small>), breaking its stationary <small><math>SO(4)</math></small> isoclinic rotation symmetry (<small><math>\mathrm{Q^2}</math></small>). Because each discrete part of the rotating atom moves along a helical trajectory through 4-space, the atom is in orbit around a barycentric axis (like a star in a galaxy), but only in a tiny orbit within its own radius, which is its inertial domain of rotation. The straight 4-dimensional cylinder it progresses along at velocity <math>c</math> is very narrow: only the diameter of the rotating atom itself. The gravitational Kepler problem of a hydrogen atom in a Kepler orbit around the sun is a problem in general relativity. In our 4-space model, this atom viewed in its own proper reference frame exhibits the same <small><math>SO(4)</math></small> rotation symmetry as it did in the electrostatic Kepler problem where the atom was translating linearly through space. The Kepler system in this case is not just the atom; it is the entire solar system. The LRL vector of this Kepler system is the proper time vector of the atom's inertial reference frame; once again it has constant velocity ''and constant direction''. Although the momentum vector moves in a perfect circle as the atom orbits the sun, the 4-space LRL vector does not move at all: it is a constant of motion, of linear motion (<small><math>\mathrm{T}</math></small>) of the Kepler system (the entire solar system in this case) in a constant 4-space direction, the proper time direction of the system. The direction of the system's proper time vector would vary under some kinds of acceleration of the atom, but it is constant under this kind of orbital acceleration. It continues to point in the same direction, like a 4-space compass needle, as the atom winds its way along its spiral path around the axis of the sun's straight-line translation through 4-space at velocity <math>c</math>. This compass needle always points in the direction the sun is moving, not the direction the atom is moving at any instant. ...Its Kepler orbit around the sun is its <small><math>SO(3)</math></small> rotation component (<small><math>\mathrm{Q}</math></small>). Although the atom is moving on a geodesic circle in the second problem, by the [[equivalence principle]] the difference in the state of the atomic systems in these two problems cannot be observed by examining the atoms alone. Even from another inertial reference frame, where the atom in the second problem is seen to be translating through 4-space via a wide screw translation (<small><math>\mathrm{QT}</math></small>) around the sun's axis of motion, there is still no difference between the two problems which can be detected by examining only the atoms within their own proper reference frames (even over time), because the LRL vector (<small><math>\mathrm{T}</math></small>) is a constant of motion of the entire system in both cases. ...Anco and Maghadam found that <small><math>SO(4)</math></small>) breaks to ... <small><math>S^3</math></small>)... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... <small><math>H^3</math></small>) ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). ... Finally we consider a third problem in which a hydrogen atom enters the solar system as a comet, loops around the sun and exits the solar system again. This atom... ... As Hamilton found when he discovered the quaternions, we see that it is necessary to admit a fourth dimension to the system in order to properly model the problem: in Hamilton's case the general problem of ..., and in our case the Kepler problem. These are instances of the same problem in 4-dimensional Euclidean geometry, and indeed a solution to the Kepler problem in quaternions (the four Cartesian coordinates of Euclidean 4-space) is a solution to it in our model of the 4-coordinate Euclidean cosmos. == Distribution of stars in our galaxy == The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by its red shift, and by assuming that they are distributed in three dimensions of space, we have plotted their locations in 3-space. If we abandon the last of those three assumptions, we can just as easily reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits as we saw them in 3-space? That would be an expected consequence of the special rotational symmetry group of 4-space <small><math>SO(4)</math></small>, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits. ...have to perform this experiment somehow, at least as a conclusive thought experiment, before I publish this paper... == Rotations == The [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotations]] of the convex [[W:regular 4-polytope|regular 4-polytope]]s are usually described as discrete rotations of a rigid object. For example, the rigid [[24-cell]] can rotate in a [[24-cell#Great hexagons|hexagonal]] (6-vertex) central [[24-cell#Planes of rotation|plane of rotation]]. A 4-dimensional [[24-cell#Isoclinic rotations|''isoclinic'' rotation]] (as distinct from a [[24-cell#Simple rotations|''simple'' rotation]] like the ones that occur in 3-dimensional space) is a ''diagonal'' rotation in multiple [[W:Clifford parallel|Clifford parallel]] [[24-cell#Geodesics|central planes]] of rotation at once. It is diagonal because it is a [[W:SO(4)#Double rotations|double rotation]]: in addition to rotating in parallel (like wheels), the multiple planes of rotation also tilt sideways in the completely orthogonal plane of rotation (like coins flipping) into each other's planes. Consequently, the path taken by each vertex is a [[24-cell#Helical hexagrams and their isoclines|twisted helical circle]], rather than the ordinary flat great circle a vertex follows in a simple rotation. In a rigid 4-polytope rotating isoclinically, ''all'' the vertices lie in one of the parallel planes of rotation, so all the vertices move in parallel along Clifford parallel twisting circular paths. [[24-cell#Clifford parallel polytopes|Clifford parallel planes]] are not parallel in the normal sense of parallel planes in three dimensions; the vertices are all moving in different directions around the [[W:3-sphere|3-sphere]]. In one complete 360° isoclinic revolution, a rigid 4-polytope turns itself inside out. This is sufficiently different from the simple rotations of rigid bodies in our 3-dimensional experience that a [[24-cell#Rotations|detailed description]] enabling the reader to properly visualize its counter-intuitive consequences runs to many pages and illustrations, with many accompanying pages of explanatory notes on surprising phenomena that arise in 4-dimensional space: [[24-cell#Great squares|completely orthogonal planes]], [[24-cell#Clifford parallel polytopes|Clifford parallelism]]{{Efn|name=Clifford parallels}} and [[W:Hopf fibration|Hopf fiber bundles]], [[24-cell#Isoclinic rotations|isoclinic geodesic paths]], and [[24-cell#Double rotations|chiral (mirror image) pairs of rotations]], among other complexities. Moreover, the characteristic rotations of the various regular 4-polytopes are all different; each is a unique surprise. [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|The 6 regular convex 4-polytopes]] have different numbers of vertices (5, 8, 16, 24, 120 and 600 respectively) and those with fewer vertices occur inscribed in those with more vertices (with one exception), with the result that the more complex 4-polytopes subsume the kinds of rotations characteristic of their less complex predecessors, as well as each having a characteristic kind of rotation not found in their predecessors. None of these symmetries is to be found in 3-dimensional space, although their simpler 3-dimensional analogues are all present there. [[W:Euclidean geometry#Higher dimensions|Four dimensional Euclidean space]] is more complicated (and more interesting) than three dimensional space because there is more room in it, in which unprecedented things can happen. It subsumes 3-dimensional space, with all of the symmetries we are accustomed to, and adds astonishing new surprises. These are hard for us to visualize, because the only way we can experience them is in our imagination; we have no body of sensory experience in 4-dimensional space to draw upon, other than our evolution in time. For that reason (our difficulty in visualizing them), descriptions of isoclinic rotations usually begin and end with rigid rotations: [[24-cell#Isoclinic rotations|for example]], all 24 vertices of a single rigid 24-cell rotating in unison, with 6 vertices evenly spaced around each of 4 Clifford parallel twisted circles.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} But that is only the simplest case, which is easiest for us to understand. Compound and [[W:Kinematics|kinematic]] 24-cells (with moving parts) are even more interesting (and more complicated) than the rotation of a single rigid 24-cell. To begin with, when we examine the individual parts of a single rigid 24-cell that are moving in an isoclinic rotation, such as the orbits of individual vertices, we can imagine a case where fewer than 24 point-objects are orbiting on those twisted circular paths at once. [[24-cell#Reflections|For example]], if we imagine just 8 point-objects, evenly spaced around the 24-cell at [[24-cell#Reciprocal constructions from 8-cell and 16-cell|the 8 vertices that lie on the 4 coordinate axes]], and rotate them isoclinically along exactly the same orbits they would take in the above-mentioned rotation of a rigid 24-cell, then in the course of a single 360° rotation the 8 point-objects will trace out the whole 24-cell, with just one point-object reaching each of the 24 vertex positions just once, and no point-object colliding with (or even crossing the path of) any other at any time. This is an example of a discrete Hopf fibration. But it is still an example of a rigid object in a discrete isoclinic rotation: a rigid 8-vertex object (called the 4-[[W:orthoplex|orthoplex]] or [[16-cell]]) performing one half of the characteristic rotation of the 24-cell. We can also imagine ''combining'' distinct isoclinic rotations. What happens when multiple point-objects are orbiting at once, but do ''not'' all follow the Clifford parallel paths characteristic of the ''same'' distinct rigid rotation? What happens when we combine orbits from distinct rotations characteristic of different 4-polytopes, for example when different rigid 4-polytopes are concentric and rotating simultaneously in their characteristic ways? What kinds of such hybrid rotations are possible in the same 3-sphere shell without collisions? In adjacent concentric shells without asymmetric imbalance? What sort of [[Kinematics of the cuboctahedron|kinematic polytopes]] do they trace out, and how do their [[24-cell#Clifford parallel polytopes|component parts]] relate to each other as they move? Is there (sometimes) some kind of mutual stability amid their lack of combined rigidity? Visualizing isoclinic rotations (rigid and otherwise) allows us to explore such questions of [[W:kinematics|kinematics]], and where dynamic stabilities arise, of [[wikipedia:kinetics (physics)|kinetics]]. In four dimensions, we discover that space has more room in it than we have experienced, which permits previously unimagined motions. Even 3-space is more commodious than we thought; when it is curved and lies embedded in a higher-dimensional space, it permits previously impossible symmetric packings. Sadoc studied double-twisted 3-dimensional molecules, and imagined them embedded in 4-dimensional space as the Hopf fibrations of regular 4-polytopes. He found that these molecules would close-pack on the 3-sphere perfectly without exhibiting any torsion, although their packing in ordinary flat 3-space is imperfect, "frustrated" by their twisted geometry. <blockquote>The frustration, which arises when the molecular orientation is transported along the two [spiral] AB paths of figure 1 [double twist helix], is imposed by the very topological nature of the Euclidean space R³. It would not occur if the molecules were embedded in the non-Euclidean space of the [[W:3-sphere|3-sphere]] S³, or hypersphere. This space with a homogeneous positive curvature can indeed be described by equidistant and uniformly twisted fibers, along which the molecules can be aligned without any conflict between compactness and [[W:torsion of a curve|torsion]].... The fibres of this [[W:Hopf fibration|Hopf fibration]] are great circles of S³, the whole family of which is also called the [[W:Clifford parallel|Clifford parallel]]s.{{Efn|name=Clifford parallels}} Two of these fibers are C_∞ symmetry axes for the whole fibration; each fibre makes one turn around each axis and regularly rotates when moving from one axis to another.{{Efn|name=helical geodesic}} These fibers build a double twist configuration while staying parallel, i.e. without any frustration, in the whole volume of S³.{{Efn|name=Petrie polygon of a honeycomb}} They can therefore be used as models to study the condensation of long molecules in the presence of a double twist constraint.{{Sfn|Sadoc & Charvolin|2009|loc=§1.2 The curved space approach|ps=; studies the helical orientation of molecules in crystal structures and their imperfect packings ("frustrations") in 3-dimensional space.}}</blockquote> Of course we do not find molecules condensing to close-pack the 3-sphere in our experience, and Sadoc does not say that we do. We find 3-spheres in the atomic realm (if atoms are 4-polytopes), and in the cosmic realm (as the surface boundaries of stars, and the concentric surfaces of galaxies). But in between, in the realm of ordinary experience which includes the molecular realm, ourselves and all the objects we can materially handle or observe up close including the planets, we are confined together by gravity as inertia within a curved 3-dimensional space that is no more than one atom thick in the fourth spatial dimension. That is why in the molecular realm we find only objects that occupy 3-spaces which, though infinitesimally curved in the fourth dimension, are tiny patches on whole 3-spheres of galactic size. So Sadoc's exercise is a thought experiment, like Einstein's gedankenexperiments about railroad embankments and trains moving at nearly the speed of light. It is no less illuminating, despite the symmetry it reveals not having a realization as an actual 3-sphere of actual molecules. And might not something very like it have an actual realization in the atomic realm? We know that atoms have their own complex internal structure, which we are unable to model geometrically in ordinary 3-dimensional space. Suppose such a model is impossible because an atom is actually a 4-polytope occupying a tiny spherical region of 4-dimensional space, and so we only find its constituent particles in close-packed helical orbits on the 3-sphere, in the manner of Sadoc's imaginary twisted molecules, but as real 4-dimensional helices of atomic scale. We would expect to find the atomic orbit of a fundamental particle in some discrete Hopf fibration characteristic of a symmetry group, that is, on the maximally symmetric isoclines of a discrete isoclinic rotation characteristic of some regular 4-polytope and the particle. == A theory of the Euclidean atom == <blockquote>Because quantum physics could be tested without being understood, it allowed humans to see how the universe worked without knowing why.<ref>Sebastian Junger, In My Time of Dying</ref></blockquote> ... == Light and Mass are Reflection and Rotation == The phenomena of light and mass are expressions of reflection symmetries and rotation symmetries, respectively. ... Atoms are 4-polytopes, elementary objects with SO(4) rotational symmetry. Light is .... Motion in space is the propagation of the elementary objects of light and matter in Coxeter congruent transformations by kaleidoscopic self-reflections, like the motion of self-reproducing cellular automata in [[Conway's Game of Life|Conway's game of life]]. ... Light is discrete reflections. Mass is discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics. === Atoms are 4-polytopes === ... == Relativity in real space of four or more orthogonal dimensions == Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions. General relativity is Galilean relativity in a general space of four or more orthogonal dimensions, e.g. in Euclidean 4-space <math>R^4</math>, spherical 4-space <math>S^4</math>, and any orthogonal 4-manifold. Light is a consequence of symmetry group reflections at quantum scale. Gravity and the other fundamental forces are consequences of rotations, which are consequences of quantum reflections. Light is discrete reflections. Gravity and all forces are discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics. Every observer may properly see themself as stationary and the universe as an ''n''-sphere with themself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and can be measured by the observer as the speed of light. === Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions === ...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here... Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.) None of these Lorentz effects are beyond geometric explanation or paradoxical. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, successive Euclidean spaces are dimensionally analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries: that is Nother's great discovery. === General relativity is Galilean relativity in a general space of four orthogonal dimensions === ... == Dimensional relativity == Coxeter's kinetic law of <math>n</math>-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean <math>n</math>-spaces of increasing <math>n</math>, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating <math>n</math>-spaces. By the degree of dimensional analogy of which they are capable, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: "The universe is a sphere whose center is wherever there is intelligence."{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}} == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity ''c'', with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may, or may not, all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed us that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a sense related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system. When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw displacements), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw displacement has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal of that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. Anco and Maghadam found that <small><math>SO(4)</math></small> breaks to ... <small><math>S^3</math></small>... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... <small><math>H^3</math></small> ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). Because the planets orbit on ellipses in our 3-space, Euclidean 4-space is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the truer model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position of centrality as our exclusive conception of our place in space. ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others. The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.<ref>{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}</ref>}} == Boundaries == <blockquote>Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? ''What is the nature of the boundary which confines us to just three dimensions?'' We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined motions and objects. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. A boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three .... In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it. We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed "Dimensional Analogy."<ref>{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's and Coxeter's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover in imagination and then explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :We shall not cease from exploration :And the end of all our exploring :Will be to arrive where we started :And know the place for the first time. :Through the unknown, remembered gate :When the last of earth left to discover :Is that which was the beginning; :At the source of the longest river :The voice of the hidden waterfall :And the children in the apple-tree :Not known, because not looked for :But heard, half-heard, in the stillness :Between two waves of the sea. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. This project is forever beginning anew. Coxeter showed us that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether showed us all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions, in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::BEECH :Where my imaginary line :Bends square in woods, an iron spine :And pile of real rocks have been founded. :And off this corner in the wild, :Where these are driven in and piled, :One tree, by being deeply wounded, :Has been impressed as Witness Tree :And made commit to memory :My proof of being not unbounded. :Thus truth's established and borne out, :Though circumstanced with dark and doubt— :Though by a world of doubt surrounded. :::::::—''The Moodie Forester''<ref>{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|The long radius (center to vertex) of the 24-cell is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including the four-dimensional 24-cell and [[W:Tesseract#Radial equilateral symmetry|tesseract]], the three-dimensional [[W:Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[W:Hexagon#Regular hexagon|hexagon]]. (The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron.) '''Radially equilateral''' polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.|name=radially equilateral|group=}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|5}}(–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>){{spaces|3}}(–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} {{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}} {{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}} {{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles]."}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}} {{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}} {{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}} {{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}} {{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} {{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} {{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} {{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}} {{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} {{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90^o bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60^o bend, or as a straight line with one 60^o bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} {{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)"}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} {{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}} {{Efn|Departing from any vertex V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V₀ and V₂ are ''two'' {{radic|3}} chords apart{{Efn|V₀ and V₂ are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V₀ and V₂ are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹₄.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} b0qwz5c7u09vpapcqxcud3827bji6bu 2818505 2818504 2026-07-19T02:56:09Z Dc.samizdat 2856930 2818505 wikitext text/x-wiki = Real Euclidean four-dimensional space R⁴ = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - July 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as a Euclidean space of four orthogonal spatial dimensions. Space itself has a fourth orthogonal dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.</blockquote> == Summary == We observe that physical space has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is translating through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions. == A theory of the Euclidean cosmos == The physical universe is properly visualized as a [[w:Four-dimensional_space|Euclidean space of four orthogonal spatial dimensions]]. Space itself has a fourth orthogonal dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move through Euclidean 4-space at velocity <math>c</math> as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical spiral galaxy such as ours is a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our galaxy appear from our viewpoint to be distributed in a cloud of elliptical spirals occupying a flattened ellipsoid region of 3-dimensional space, but they are not so confined: they are distributed within a spherical region of 4-dimensional space. The galaxy's actual shape is spherical, not a flattened ellipsoid, but it is rounder than round can be in our ordinary experience: it occupies a hyperspherical region of space. The concentric spirals of stars that we observe lie on concentric [[W:3-sphere|3-sphere]]s (4-dimensional spheres), not on concentric 2-ellipsoids (3-dimensional elliptical spirals). Our sun and solar system lies on one of those concentric 3-spheres. More generally, orbits are circular in 4-space, and elliptical in the 3-space of their elliptic hyperplane. ...rotating illustration of the 4-ball galaxy showimg its spirals of star clouds on the surface of concentric 3-spheres...obtained by reverse sterographic projection from 3D images of the galaxy... The galaxy as a whole, or more properly its orbital center point, is translating through 4-space at velocity <math>c</math>, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity <math>c</math> in the same direction, but on spiral trajectories relative to the galaxy's linear trajectory, as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the cylinder; their trajectories are screw-displacements, the compound of a simple rotation and a linear translation. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis. It is the tremendous inertial force of stars in motion at velocity <math>c</math> that holds the cylinder of motion together. The observed universe as a whole appears to be a 3-sphere expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. For all observers, the conjectured origin point of the universe corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space (the same point in the same Euclidean 4-space for all observers). The big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured separately, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. The enclosing surface of a spherical region of 4-space is itself a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. We live within such a 3-space, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our particular 3-sphere is one of the galaxy's concentric 3-spheres of spiral star-clouds. The solar system occupies a tiny patch of this filmy 4-dimensional soap-bubble of galactic size, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. Our entire 3-sphere manifold, as a 3-spherical shell within the moving 4-ball galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy, in a distinct direction that is orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of rotating objects through Euclidean space by screw translation. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold that is such an evolving surface boundary is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a three-dimensional smear of atoms no thicker than one atom in its fourth dimension, which is the direction of its linear translation through 4-space at velocity <math>c</math>. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to 4-dimensional lumps of matter as plasma, and have little experimental knowledge of their geometry or internal structure. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know nothing about its interior 4-ball. Every such moving 3-dimensional surface boundary of matter in the observed universe is evolving in four dimensions at velocity <math>c</math>. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from atoms to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, two orthogonal inertial motions each at the constant universal rate of transformation <math>c</math>. Every object moves relative to universal 4-coordinate space on its own distinct geodesic spiral, a screw translation trajectory that is the compound of its two orthogonal inertial motions. Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in motion at velocity <math>c</math> in all four dimensions concurrently, so they move diagonally through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c^\prime = 2c</math>, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold moving at velocity <math>c</math>. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math> in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper space. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <small><math>\mathrm{Q}</math></small> denote a rotation, <small><math>\mathrm{R}</math></small> a reflection, <small><math>\mathrm{T}</math></small> a translation, and let <small><math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math></small> denote a product of several such transformations, all commutative with one another. Then <small><math>\mathrm{RT}</math></small> is a glide-reflection (in two or three dimensions), <small><math>\mathrm{QR}</math></small> is a rotary-reflection, <small><math>\mathrm{QT}</math></small> is a screw-displacement, and <small><math>\mathrm{Q^2}</math></small> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<small><math>\mathrm{Q}^q \mathrm{R}^r</math></small><br> where <small><math>(2^q + r \le n)</math></small>, the number of dimensions.<br> Transformations involving a translation are expressible as:<br> :<small><math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math></small><br> where <small><math>(2^q + r + 1 \le n)</math></small>.<br> For <small><math>(n = 4)</math></small> in particular, every displacement is either a double rotation <small><math>\mathrm{Q}^2</math></small>, or a screw-displacement <small><math>\mathrm{QT}</math></small> [where the rotation component <small><math>\mathrm{Q}</math></small> is a simple rotation, but the <small><math>\mathrm{QT}</math></small> is chiral like a <small><math>\mathrm{Q^2}</math></small>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <small><math>\mathrm{QRT}</math></small>.</blockquote> If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a <small><math>\mathrm{Q^2}</math></small> or a <small><math>\mathrm{QT}</math></small>, because we can view any <small><math>\mathrm{QT}</math></small> as a <small><math>\mathrm{Q^2}</math></small> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <small><math>\mathrm{Q^2}</math></small>. By the same principle, we can view any <small><math>\mathrm{QT}</math></small> or <small><math>\mathrm{Q^2}</math></small> as an isoclinic (equi-angled) <small><math>\mathrm{Q^2}</math></small> by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (<small><math>\mathrm{T}</math></small>) for ''one'' of the two rotations (<small><math>\mathrm{Q}</math></small>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<small><math>\mathrm{Q}</math></small>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<small><math>\mathrm{T}</math></small>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <small><math>SO(4)</math></small> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <small><math>SO(4)</math></small> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, stationary atoms exhibit the <small><math>SO(4)</math></small> symmetries of the discrete isoclinic (equi-angled) double rotations (<small><math>\mathrm{Q^2}</math></small>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.<ref>{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional "space" because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was first described by Einstein himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction. Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity ''c'', in some 4-space direction. This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always ''c'', as measured by all observers in any inertial reference frame. This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity ''c''. In physics as it has been universally understood, observers are not supposed to be able to move at velocity ''c''. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four. In the Euclidean relativity alternative view, however, every observer is always moving at velocity ''c'' through the universe, which is real Euclidean 4-dimensional space <small><math>\mathbb{R^4}</math></small>. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity ''c''. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity ''c'', in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your coordinate systems; you are both embedded in the same Euclidean 4-dimensional space <small><math>\mathbb{R^4}</math></small>.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity ''c'' relative to universal 4-coordinate space, so the maximum relative velocity between two observers is 2''c'' when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to ''c'', it is the same measurement in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}} So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity ''c'' in Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" ''c'', although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity ''c'', but in at least slightly different directions. In Einstein's relativity, the invariant ''c'' is the speed of light through 3-space. In Euclidean relativity, the invariant ''c'' is the speed of matter through 4-space! The speed of light through 3-space is also perceived as ''c'' by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity ''c''. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space. ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity <math>c</math> through space and proper time ...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords. Euclidean relativity is not even a fringe theory; no physicists have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at ''c'' (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than ''c''. Euclidean relativity is a revolutionary theory indeed, in which ''c'' cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for it to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c^\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the diagonal of the unit 4-hypercube is the natural number <small><math>\sqrt{4}</math></small>. == An object's motion in space is the product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a small number of discrete self-reflections. Any action of a geometric object that transforms its position and orientation in space may be measured as a distinct group of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete set of local self-reflections. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which touch (intersect each other). When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, leaving itself right-side-out again. Coxeter's laws of motion are a geometric counterpart to Newton's laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as simple geometric pictures, by anyone baffled by algebraic formulas. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. For example, they give us simple geometric pictures of all the possible motions of objects in four dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<small><math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math></small><br> where <small><math>(2^q + r + t \le 4)</math></small>. Every displacement is either a double rotation <small><math>\mathrm{Q}^2</math></small>, or a screw-displacement <small><math>\mathrm{QT}</math></small> [where the rotation component <small><math>\mathrm{Q}</math></small> is a simple rotation, but the <small><math>\mathrm{QT}</math></small> is chiral like a <small><math>\mathrm{Q^2}</math></small>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <small><math>\mathrm{QRT}</math></small>.</blockquote> While this description should be understood as simple geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (<small><math>\mathrm{Q}</math></small>), reflection (<small><math>\mathrm{R}</math></small>) and translation (<small><math>\mathrm{T}</math></small>) are just what they are in three-dimensional space, but double rotation (<small><math>\mathrm{Q}^2</math></small>) is something new and unprecedented in our physical experience, because double rotations cannot occur until you have four or more dimensions of space to rotate in. ...to readers who have not studied Coxeter (almost all readers including TAC), the blockquote above is "just math", not visualizable geometry...but I could describe Coxeter's congruent transformations in 4-space here geometrically: I could say clearly what they mean in spatial terms, in language anyone can understand, because they don't require any math to be understood; the "math" here is really just simple pictures (reflections and rotations); even double rotations can be visualized by dimensional analogy, as compounds of simple rotations...since even most physicists are unacquainted with Coxeter geometry, it really is important that I do this here... == Light propagates through 4-space at twice its apparent velocity ''c''== Coxeter's geometric laws of motion apply to all objects with mass in 4-dimensional Euclidean space, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <small><math>\mathrm{R}^4</math></small>, which may be termed a double translation <small><math>\mathrm{T}^2</math></small>, a pure translation via two pairs of parallel reflections, without any rotation component <small><math>\mathrm{Q}</math></small>. Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by <small><math>\mathrm{QT}</math></small>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <small><math>\mathrm{Q^2}</math></small>, an isoclinically rotating object such as an atom. A simple rotation <small><math>\mathrm{Q}</math></small> or simple translation <small><math>\mathrm{T}</math></small> is a double reflection <small><math>\mathrm{R^2}</math></small>, so a <small><math>\mathrm{QT}</math></small> or <small><math>\mathrm{Q^2}</math></small> is also an <small><math>\mathrm{R^4}</math></small>, but not with the same group of reflection angles as a light signal <small><math>\mathrm{R^4}</math></small>. A translation <small><math>\mathrm{T = R^2}</math></small> is a double reflection in two parallel planes, and a rotation <small><math>\mathrm{Q = R^2}</math></small> is a double reflection in two intersecting planes, as in a <small><math>\mathrm{QT = R^4}</math></small> which is both at once. A double translation <small><math>\mathrm{T^2 = R^4}</math></small> is two double reflections in pairs of parallel planes at once, a reflection in four or more non-intersecting parallel planes; it is all translation and no rotation. In a <small><math>\mathrm{T^2}</math></small> all the motion goes to translation, so the translation goes twice as far as the simple translation <small><math>\mathrm{T}</math></small> in a <small><math>\mathrm{QT}</math></small>. A double translation <small><math>\mathrm{T^2 = R^4}</math></small> is the opposite of a double rotation <small><math>\mathrm{Q^2 = R^4}</math></small>, which is stationary but rotates twice as fast as the simple rotation <small><math>\mathrm{Q}</math></small> in a <small><math>\mathrm{QT}</math></small>. The product of the two translations in a <small><math>\mathrm{T^2}</math></small> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <small><math>\mathrm{T}</math></small> over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A <small><math>\mathrm{T^2}</math></small> cannot reposition a 4-polytope the way a <small><math>\mathrm{QT}</math></small> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter. ...lensing of double translations <small><math>\mathrm{T^2 = R^4}</math></small> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet... == The Kepler problem is framed in Euclidean 4-space == The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], arguably the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the "inverse Kepler problem": the orbit characteristics require the force to depend on the inverse square of the distance.<ref>{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton & Company|isbn=978-0393039184}}</ref> The inverse square law behind the Kepler problem is the [[W:Central force|central force]] law which governs not only [[W:Newtonian gravity|Newtonian gravity]] and celestial orbits, but also the motion of two charged particles in [[W:Coulomb’s law|Coulomb’s law]] of [[W:Electrostatics|electrostatics]]; it applies to attractive or repulsive forces. Problems in which two bodies interact by a central force that varies as the [[W:Inverse square law|inverse square]] of the distance between them are called Kepler problems. Thus the [[W:Hydrogen atom|hydrogen atom]] is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law, another inverse-square central force. Using classical mechanics, the solution to a Kepler problem can be expressed as a [[W:Kepler orbit|Kepler orbit]] using six kinematical variables or [[W:Orbital elements|orbital elements]]. The solution conserves an orbital element called the [[W:Laplace–Runge–Lenz vector|Laplace–Runge–Lenz (LRL) vector]], a [[W:Constant of motion|constant of motion]], meaning that it is the same no matter where it is calculated on the orbit. The LRL vector was essential in the first quantum mechanical derivation of the [[W:Atomic emission spectrum|spectrum]] of the hydrogen atom, but this approach has rarely been used since the development of the [[W:Schrödinger equation|Schrödinger equation]]. The conservation of the LRL vector corresponds to the <small><math>SO(4)</math></small> symmetry, by Nother's theorem. The LRL vector lies orthogonal to both the orbital plane and the angular momentum vector of the Kepler orbit; we observe that it lies in a fourth orthogonal dimension. Fock in 1935<ref>V. Fock, Zur Theorie des Wasserstoffatoms, Zeitschrift für Physik. 98 (3-4) (1935), 145–154.</ref> and Moser in 1970<ref>J. Moser, Regularization of Kepler’s problem and the averaging method on a manifold, Commun. Pure Appl. 23 (1970), 609–636</ref> observed that the Kepler problem is mathematically equivalent to non-affine geodesic motion (a particle moving freely) on the surface of a 3-sphere, so that the whole problem is symmetric under certain rotations of the four-dimensional space. This higher-dimensional symmetry results in two well-known properties of the Kepler problem: the momentum vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points. ... Relativity establishes that an orbit in space is viewed in a different way in each distinct inertial reference frame. Depending on the choice of reference frame, the same Kepler system may be seen to be performing any one of a sequence of relativistically equivalent rotations in 4-space, on a continuum from an isoclinic rotation (Q²) in the orbit's proper reference frame, to a screw transfer (QT) with a simple rotation component (Q) and a translation component (T) at velocity <math>c</math>, in the universal reference frame of 4-coordinate space wherein every object is seen to be translating at velocity <math>c</math>. In reference frames between these two limit cases, the orbit is seen to be performing a double rotation (Q²) at two unequal, completely orthogonal angular rates of rotation: an elliptical double rotation. These include the reference frames of most typical observers, who are moving slowly relative to the observed orbital system's reference frame (their relative motion is a small fraction of the speed of light). ...this is probably misplaced here and should not interrupt the discussion at this point: ...These typical observations agree closely with the predictions of special relativity, because the non-isoclinic elliptical (Q²) resembles a (QT), since one of its two completely orthogonal rotations (Q) has such a long period that it is almost indistinguishable from a straight translation (T). All orbits in 4-space are isoclinic in their own reference frame. Orbiting objects in their own proper Kepler systems follow circular geodesic isoclines through 4-space. Orbits in 4-space are perfectly circular in their own reference frame, as Copernicus assumed the orbits of planets to be. It is the orbit's path through the 3-space of its elliptic hyperplane that is an ellipse, as Kepler found it to be. ...cite Jesper Goransson's very concise paper The geodesic circle that an orbiting object follows through 4-space in the proper reference frame of its own Kepler system is not a simple great circle which turns in two orthogonal dimensions. It is a helical great circle that turns in four orthogonal dimensions at once.{{Efn|Geodesic orbits in 4-space are not simple 2-dimensional great circles; they are helical 4-dimensional great circles that curve in all four dimensions at once. Their circular trajectories are helixes which we call ''isoclines'', since they are the paths taken by points on a rigid object undergoing isoclinic rotation.}} Such circles lie outside our physical experience, since our local space has only three orthogonal dimensions. Nonetheless we can visualize them in imagination, because their helical, circular shape is perfectly well defined by the kinematical variables of the Kepler orbit. The real physical correlates of abstract orthogonal planes and rotation angles are already familiar to us viscerally in our body-language of physical experience, since we are endowed biologically with highly evolved visual signal processing engines. These enable us to see and understand spatial relations and motions, including rotations, without even thinking about angles and orthogonal planes. This physical endowment is an inborn capacity for dimensional analogy which our biologic evolution has provided. All our instinctive spatial reasoning is by dimensional analogy from flat 2-dimensional retinal images to 3-dimensional scenes, using our powerful inborn visualization capacities of reverse stereographic projection and pattern recognition. We humans are thus very well equipped with everything we need to see in four-dimensional space, except experience. ... Recently Anco and Moghadam found that through Noether’s theorem in reverse, the LRL vector gives rise to a corresponding infinitesimal dynamical symmetry on the kinematical variables, which they show to be the semi-direct product of <small><math>SO(3)</math></small> and <small><math>\mathbb{R^3}</math></small>, in contrast to the <small><math>SO(4)</math></small> symmetry group generated by the LRL symmetries and the rotations.{{Sfn|Anco|Moghadam|2026|ps=; The physically relevant part of the LRL vector is its direction ... since its magnitude is just a function of energy and angular momentum.}} This remarkable symmetry breaking is expressive of the ''dimensional relativity'' between ordinary 3-space <small><math>\mathbb{R^3}</math></small>, spherical space <small><math>S^3</math></small> and Euclidean space <small><math>\mathbb{R^4}</math></small>. Consider a hydrogen atom in a Kepler orbit: for example, a hydrogen atom moving freely in space in an orbit around the sun. It is a ''double'' Kepler problem: an electrostatic Kepler problem within itself, and a gravitational Kepler problem in its environment. The ''single'' electrostatic Kepler problem of a hydrogen atom moving freely in space beyond any gravitational influence is a problem in special relativity. In our Euclidean 4-space model, this atom viewed as stationary in its own proper reference frame exhibits an <small><math>SO(4)</math></small> rotation symmetry corresponding to an isoclinic double rotation (<small><math>\mathrm{Q^2}</math></small>). The fourth dimension in this reference frame is the atom's proper time vector; it has constant velocity <math>c</math> and constant direction. From the point of view of our universal 4-coordinate space (which cannot be the proper inertial reference frame of any physical observer, all of whom are moving relative to it at velocity ''c''), the entire Kepler system (the atom) is translating through 4-space via a screw translation (<small><math>\mathrm{QT}</math></small>) at constant velocity <math>c</math>. From this viewpoint the atom has only a simple <small><math>SO(3)</math></small> rotation component (<small><math>\mathrm{Q}</math></small>), breaking its stationary <small><math>SO(4)</math></small> isoclinic rotation symmetry (<small><math>\mathrm{Q^2}</math></small>). Because each discrete part of the rotating atom moves along a helical trajectory through 4-space, the atom is in orbit around a barycentric axis (like a star in a galaxy), but only in a tiny orbit within its own radius, which is its inertial domain of rotation. The straight 4-dimensional cylinder it progresses along at velocity <math>c</math> is very narrow: only the diameter of the rotating atom itself. The gravitational Kepler problem of a hydrogen atom in a Kepler orbit around the sun is a problem in general relativity. In our 4-space model, this atom viewed in its own proper reference frame exhibits the same <small><math>SO(4)</math></small> rotation symmetry as it did in the electrostatic Kepler problem where the atom was translating linearly through space. The Kepler system in this case is not just the atom; it is the entire solar system. The LRL vector of this Kepler system is the proper time vector of the atom's inertial reference frame; once again it has constant velocity ''and constant direction''. Although the momentum vector moves in a perfect circle as the atom orbits the sun, the 4-space LRL vector does not move at all: it is a constant of motion, of linear motion (<small><math>\mathrm{T}</math></small>) of the Kepler system (the entire solar system in this case) in a constant 4-space direction, the proper time direction of the system. The direction of the system's proper time vector would vary under some kinds of acceleration of the atom, but it is constant under this kind of orbital acceleration. It continues to point in the same direction, like a 4-space compass needle, as the atom winds its way along its spiral path around the axis of the sun's straight-line translation through 4-space at velocity <math>c</math>. This compass needle always points in the direction the sun is moving, not the direction the atom is moving at any instant. ...Its Kepler orbit around the sun is its <small><math>SO(3)</math></small> rotation component (<small><math>\mathrm{Q}</math></small>). Although the atom is moving on a geodesic circle in the second problem, by the [[equivalence principle]] the difference in the state of the atomic systems in these two problems cannot be observed by examining the atoms alone. Even from another inertial reference frame, where the atom in the second problem is seen to be translating through 4-space via a wide screw translation (<small><math>\mathrm{QT}</math></small>) around the sun's axis of motion, there is still no difference between the two problems which can be detected by examining only the atoms within their own proper reference frames (even over time), because the LRL vector (<small><math>\mathrm{T}</math></small>) is a constant of motion of the entire system in both cases. ...Anco and Maghadam found that <small><math>SO(4)</math></small>) breaks to ... <small><math>S^3</math></small>)... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... <small><math>H^3</math></small>) ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). ... Finally we consider a third problem in which a hydrogen atom enters the solar system as a comet, loops around the sun and exits the solar system again. This atom... ... As Hamilton found when he discovered the quaternions, we see that it is necessary to admit a fourth dimension to the system in order to properly model the problem: in Hamilton's case the general problem of ..., and in our case the Kepler problem. These are instances of the same problem in 4-dimensional Euclidean geometry, and indeed a solution to the Kepler problem in quaternions (the four Cartesian coordinates of Euclidean 4-space) is a solution to it in our model of the 4-coordinate Euclidean cosmos. == Distribution of stars in our galaxy == The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by its red shift, and by assuming that they are distributed in three dimensions of space, we have plotted their locations in 3-space. If we abandon the last of those three assumptions, we can just as easily reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits as we saw them in 3-space? That would be an expected consequence of the special rotational symmetry group of 4-space <small><math>SO(4)</math></small>, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits. ...have to perform this experiment somehow, at least as a conclusive thought experiment, before I publish this paper... == Rotations == The [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotations]] of the convex [[W:regular 4-polytope|regular 4-polytope]]s are usually described as discrete rotations of a rigid object. For example, the rigid [[24-cell]] can rotate in a [[24-cell#Great hexagons|hexagonal]] (6-vertex) central [[24-cell#Planes of rotation|plane of rotation]]. A 4-dimensional [[24-cell#Isoclinic rotations|''isoclinic'' rotation]] (as distinct from a [[24-cell#Simple rotations|''simple'' rotation]] like the ones that occur in 3-dimensional space) is a ''diagonal'' rotation in multiple [[W:Clifford parallel|Clifford parallel]] [[24-cell#Geodesics|central planes]] of rotation at once. It is diagonal because it is a [[W:SO(4)#Double rotations|double rotation]]: in addition to rotating in parallel (like wheels), the multiple planes of rotation also tilt sideways in the completely orthogonal plane of rotation (like coins flipping) into each other's planes. Consequently, the path taken by each vertex is a [[24-cell#Helical hexagrams and their isoclines|twisted helical circle]], rather than the ordinary flat great circle a vertex follows in a simple rotation. In a rigid 4-polytope rotating isoclinically, ''all'' the vertices lie in one of the parallel planes of rotation, so all the vertices move in parallel along Clifford parallel twisting circular paths. [[24-cell#Clifford parallel polytopes|Clifford parallel planes]] are not parallel in the normal sense of parallel planes in three dimensions; the vertices are all moving in different directions around the [[W:3-sphere|3-sphere]]. In one complete 360° isoclinic revolution, a rigid 4-polytope turns itself inside out. This is sufficiently different from the simple rotations of rigid bodies in our 3-dimensional experience that a [[24-cell#Rotations|detailed description]] enabling the reader to properly visualize its counter-intuitive consequences runs to many pages and illustrations, with many accompanying pages of explanatory notes on surprising phenomena that arise in 4-dimensional space: [[24-cell#Great squares|completely orthogonal planes]], [[24-cell#Clifford parallel polytopes|Clifford parallelism]]{{Efn|name=Clifford parallels}} and [[W:Hopf fibration|Hopf fiber bundles]], [[24-cell#Isoclinic rotations|isoclinic geodesic paths]], and [[24-cell#Double rotations|chiral (mirror image) pairs of rotations]], among other complexities. Moreover, the characteristic rotations of the various regular 4-polytopes are all different; each is a unique surprise. [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|The 6 regular convex 4-polytopes]] have different numbers of vertices (5, 8, 16, 24, 120 and 600 respectively) and those with fewer vertices occur inscribed in those with more vertices (with one exception), with the result that the more complex 4-polytopes subsume the kinds of rotations characteristic of their less complex predecessors, as well as each having a characteristic kind of rotation not found in their predecessors. None of these symmetries is to be found in 3-dimensional space, although their simpler 3-dimensional analogues are all present there. [[W:Euclidean geometry#Higher dimensions|Four dimensional Euclidean space]] is more complicated (and more interesting) than three dimensional space because there is more room in it, in which unprecedented things can happen. It subsumes 3-dimensional space, with all of the symmetries we are accustomed to, and adds astonishing new surprises. These are hard for us to visualize, because the only way we can experience them is in our imagination; we have no body of sensory experience in 4-dimensional space to draw upon, other than our evolution in time. For that reason (our difficulty in visualizing them), descriptions of isoclinic rotations usually begin and end with rigid rotations: [[24-cell#Isoclinic rotations|for example]], all 24 vertices of a single rigid 24-cell rotating in unison, with 6 vertices evenly spaced around each of 4 Clifford parallel twisted circles.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} But that is only the simplest case, which is easiest for us to understand. Compound and [[W:Kinematics|kinematic]] 24-cells (with moving parts) are even more interesting (and more complicated) than the rotation of a single rigid 24-cell. To begin with, when we examine the individual parts of a single rigid 24-cell that are moving in an isoclinic rotation, such as the orbits of individual vertices, we can imagine a case where fewer than 24 point-objects are orbiting on those twisted circular paths at once. [[24-cell#Reflections|For example]], if we imagine just 8 point-objects, evenly spaced around the 24-cell at [[24-cell#Reciprocal constructions from 8-cell and 16-cell|the 8 vertices that lie on the 4 coordinate axes]], and rotate them isoclinically along exactly the same orbits they would take in the above-mentioned rotation of a rigid 24-cell, then in the course of a single 360° rotation the 8 point-objects will trace out the whole 24-cell, with just one point-object reaching each of the 24 vertex positions just once, and no point-object colliding with (or even crossing the path of) any other at any time. This is an example of a discrete Hopf fibration. But it is still an example of a rigid object in a discrete isoclinic rotation: a rigid 8-vertex object (called the 4-[[W:orthoplex|orthoplex]] or [[16-cell]]) performing one half of the characteristic rotation of the 24-cell. We can also imagine ''combining'' distinct isoclinic rotations. What happens when multiple point-objects are orbiting at once, but do ''not'' all follow the Clifford parallel paths characteristic of the ''same'' distinct rigid rotation? What happens when we combine orbits from distinct rotations characteristic of different 4-polytopes, for example when different rigid 4-polytopes are concentric and rotating simultaneously in their characteristic ways? What kinds of such hybrid rotations are possible in the same 3-sphere shell without collisions? In adjacent concentric shells without asymmetric imbalance? What sort of [[Kinematics of the cuboctahedron|kinematic polytopes]] do they trace out, and how do their [[24-cell#Clifford parallel polytopes|component parts]] relate to each other as they move? Is there (sometimes) some kind of mutual stability amid their lack of combined rigidity? Visualizing isoclinic rotations (rigid and otherwise) allows us to explore such questions of [[W:kinematics|kinematics]], and where dynamic stabilities arise, of [[wikipedia:kinetics (physics)|kinetics]]. In four dimensions, we discover that space has more room in it than we have experienced, which permits previously unimagined motions. Even 3-space is more commodious than we thought; when it is curved and lies embedded in a higher-dimensional space, it permits previously impossible symmetric packings. Sadoc studied double-twisted 3-dimensional molecules, and imagined them embedded in 4-dimensional space as the Hopf fibrations of regular 4-polytopes. He found that these molecules would close-pack on the 3-sphere perfectly without exhibiting any torsion, although their packing in ordinary flat 3-space is imperfect, "frustrated" by their twisted geometry. <blockquote>The frustration, which arises when the molecular orientation is transported along the two [spiral] AB paths of figure 1 [double twist helix], is imposed by the very topological nature of the Euclidean space R³. It would not occur if the molecules were embedded in the non-Euclidean space of the [[W:3-sphere|3-sphere]] S³, or hypersphere. This space with a homogeneous positive curvature can indeed be described by equidistant and uniformly twisted fibers, along which the molecules can be aligned without any conflict between compactness and [[W:torsion of a curve|torsion]].... The fibres of this [[W:Hopf fibration|Hopf fibration]] are great circles of S³, the whole family of which is also called the [[W:Clifford parallel|Clifford parallel]]s.{{Efn|name=Clifford parallels}} Two of these fibers are C_∞ symmetry axes for the whole fibration; each fibre makes one turn around each axis and regularly rotates when moving from one axis to another.{{Efn|name=helical geodesic}} These fibers build a double twist configuration while staying parallel, i.e. without any frustration, in the whole volume of S³.{{Efn|name=Petrie polygon of a honeycomb}} They can therefore be used as models to study the condensation of long molecules in the presence of a double twist constraint.{{Sfn|Sadoc & Charvolin|2009|loc=§1.2 The curved space approach|ps=; studies the helical orientation of molecules in crystal structures and their imperfect packings ("frustrations") in 3-dimensional space.}}</blockquote> Of course we do not find molecules condensing to close-pack the 3-sphere in our experience, and Sadoc does not say that we do. We find 3-spheres in the atomic realm (if atoms are 4-polytopes), and in the cosmic realm (as the surface boundaries of stars, and the concentric surfaces of galaxies). But in between, in the realm of ordinary experience which includes the molecular realm, ourselves and all the objects we can materially handle or observe up close including the planets, we are confined together by gravity as inertia within a curved 3-dimensional space that is no more than one atom thick in the fourth spatial dimension. That is why in the molecular realm we find only objects that occupy 3-spaces which, though infinitesimally curved in the fourth dimension, are tiny patches on whole 3-spheres of galactic size. So Sadoc's exercise is a thought experiment, like Einstein's gedankenexperiments about railroad embankments and trains moving at nearly the speed of light. It is no less illuminating, despite the symmetry it reveals not having a realization as an actual 3-sphere of actual molecules. And might not something very like it have an actual realization in the atomic realm? We know that atoms have their own complex internal structure, which we are unable to model geometrically in ordinary 3-dimensional space. Suppose such a model is impossible because an atom is actually a 4-polytope occupying a tiny spherical region of 4-dimensional space, and so we only find its constituent particles in close-packed helical orbits on the 3-sphere, in the manner of Sadoc's imaginary twisted molecules, but as real 4-dimensional helices of atomic scale. We would expect to find the atomic orbit of a fundamental particle in some discrete Hopf fibration characteristic of a symmetry group, that is, on the maximally symmetric isoclines of a discrete isoclinic rotation characteristic of some regular 4-polytope and the particle. == A theory of the Euclidean atom == <blockquote>Because quantum physics could be tested without being understood, it allowed humans to see how the universe worked without knowing why.<ref>Sebastian Junger, In My Time of Dying</ref></blockquote> ... == Light and Mass are Reflection and Rotation == The phenomena of light and mass are expressions of reflection symmetries and rotation symmetries, respectively. ... Atoms are 4-polytopes, elementary objects with SO(4) rotational symmetry. Light is .... Motion in space is the propagation of the elementary objects of light and matter in Coxeter congruent transformations by kaleidoscopic self-reflections, like the motion of self-reproducing cellular automata in [[Conway's Game of Life|Conway's game of life]]. ... Light is discrete reflections. Mass is discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics. === Atoms are 4-polytopes === ... == Relativity in real space of four or more orthogonal dimensions == Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions. General relativity is Galilean relativity in a general space of four or more orthogonal dimensions, e.g. in Euclidean 4-space <math>R^4</math>, spherical 4-space <math>S^4</math>, and any orthogonal 4-manifold. Light is a consequence of symmetry group reflections at quantum scale. Gravity and the other fundamental forces are consequences of rotations, which are consequences of quantum reflections. Light is discrete reflections. Gravity and all forces are discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics. Every observer may properly see themself as stationary and the universe as an ''n''-sphere with themself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and can be measured by the observer as the speed of light. === Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions === ...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here... Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.) None of these Lorentz effects are beyond geometric explanation or paradoxical. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, successive Euclidean spaces are dimensionally analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries: that is Nother's great discovery. === General relativity is Galilean relativity in a general space of four orthogonal dimensions === ... == Dimensional relativity == Coxeter's kinetic law of <math>n</math>-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean <math>n</math>-spaces of increasing <math>n</math>, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating <math>n</math>-spaces. By the degree of dimensional analogy of which they are capable, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: "The universe is a sphere whose center is wherever there is intelligence."{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}} == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity ''c'', with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may, or may not, all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed us that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a sense related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system. When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw displacements), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw displacement has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal of that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. Anco and Maghadam found that <small><math>SO(4)</math></small> breaks to ... <small><math>S^3</math></small>... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... <small><math>H^3</math></small> ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). Because the planets orbit on ellipses in our 3-space, Euclidean 4-space is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the truer model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position of centrality as our exclusive conception of our place in space. ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others. The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.<ref>{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}</ref>}} == Boundaries == <blockquote>Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? ''What is the nature of the boundary which confines us to just three dimensions?'' We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined motions and objects. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. A boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three .... In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it. We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed "Dimensional Analogy."<ref>{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's and Coxeter's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover in imagination and then explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :We shall not cease from exploration :And the end of all our exploring :Will be to arrive where we started :And know the place for the first time. :Through the unknown, remembered gate :When the last of earth left to discover :Is that which was the beginning; :At the source of the longest river :The voice of the hidden waterfall :And the children in the apple-tree :Not known, because not looked for :But heard, half-heard, in the stillness :Between two waves of the sea. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. This project is forever beginning anew. Coxeter showed us that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether showed us all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions, in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::BEECH :Where my imaginary line :Bends square in woods, an iron spine :And pile of real rocks have been founded. :And off this corner in the wild, :Where these are driven in and piled, :One tree, by being deeply wounded, :Has been impressed as Witness Tree :And made commit to memory :My proof of being not unbounded. :Thus truth's established and borne out, :Though circumstanced with dark and doubt— :Though by a world of doubt surrounded. :::::::—''The Moodie Forester''<ref>{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|The long radius (center to vertex) of the 24-cell is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including the four-dimensional 24-cell and [[W:Tesseract#Radial equilateral symmetry|tesseract]], the three-dimensional [[W:Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[W:Hexagon#Regular hexagon|hexagon]]. (The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron.) '''Radially equilateral''' polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.|name=radially equilateral|group=}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|5}}(–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>){{spaces|3}}(–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} {{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}} {{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}} {{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles]."}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}} {{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}} {{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}} {{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}} {{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} {{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} {{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} {{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}} {{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} {{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90^o bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60^o bend, or as a straight line with one 60^o bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} {{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)"}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} {{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}} {{Efn|Departing from any vertex V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V₀ and V₂ are ''two'' {{radic|3}} chords apart{{Efn|V₀ and V₂ are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V₀ and V₂ are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹₄.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} o3o7923k6qpindm74mjjl6p8nq5iciu 2818506 2818505 2026-07-19T03:04:47Z Dc.samizdat 2856930 /* Light propagates through 4-space at twice its apparent velocity c */ 2818506 wikitext text/x-wiki = Real Euclidean four-dimensional space R⁴ = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - July 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as a Euclidean space of four orthogonal spatial dimensions. Space itself has a fourth orthogonal dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.</blockquote> == Summary == We observe that physical space has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is translating through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions. == A theory of the Euclidean cosmos == The physical universe is properly visualized as a [[w:Four-dimensional_space|Euclidean space of four orthogonal spatial dimensions]]. Space itself has a fourth orthogonal dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move through Euclidean 4-space at velocity <math>c</math> as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical spiral galaxy such as ours is a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our galaxy appear from our viewpoint to be distributed in a cloud of elliptical spirals occupying a flattened ellipsoid region of 3-dimensional space, but they are not so confined: they are distributed within a spherical region of 4-dimensional space. The galaxy's actual shape is spherical, not a flattened ellipsoid, but it is rounder than round can be in our ordinary experience: it occupies a hyperspherical region of space. The concentric spirals of stars that we observe lie on concentric [[W:3-sphere|3-sphere]]s (4-dimensional spheres), not on concentric 2-ellipsoids (3-dimensional elliptical spirals). Our sun and solar system lies on one of those concentric 3-spheres. More generally, orbits are circular in 4-space, and elliptical in the 3-space of their elliptic hyperplane. ...rotating illustration of the 4-ball galaxy showimg its spirals of star clouds on the surface of concentric 3-spheres...obtained by reverse sterographic projection from 3D images of the galaxy... The galaxy as a whole, or more properly its orbital center point, is translating through 4-space at velocity <math>c</math>, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity <math>c</math> in the same direction, but on spiral trajectories relative to the galaxy's linear trajectory, as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the cylinder; their trajectories are screw-displacements, the compound of a simple rotation and a linear translation. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis. It is the tremendous inertial force of stars in motion at velocity <math>c</math> that holds the cylinder of motion together. The observed universe as a whole appears to be a 3-sphere expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. For all observers, the conjectured origin point of the universe corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space (the same point in the same Euclidean 4-space for all observers). The big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured separately, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. The enclosing surface of a spherical region of 4-space is itself a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. We live within such a 3-space, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our particular 3-sphere is one of the galaxy's concentric 3-spheres of spiral star-clouds. The solar system occupies a tiny patch of this filmy 4-dimensional soap-bubble of galactic size, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. Our entire 3-sphere manifold, as a 3-spherical shell within the moving 4-ball galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy, in a distinct direction that is orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of rotating objects through Euclidean space by screw translation. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold that is such an evolving surface boundary is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a three-dimensional smear of atoms no thicker than one atom in its fourth dimension, which is the direction of its linear translation through 4-space at velocity <math>c</math>. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to 4-dimensional lumps of matter as plasma, and have little experimental knowledge of their geometry or internal structure. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know nothing about its interior 4-ball. Every such moving 3-dimensional surface boundary of matter in the observed universe is evolving in four dimensions at velocity <math>c</math>. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from atoms to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, two orthogonal inertial motions each at the constant universal rate of transformation <math>c</math>. Every object moves relative to universal 4-coordinate space on its own distinct geodesic spiral, a screw translation trajectory that is the compound of its two orthogonal inertial motions. Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in motion at velocity <math>c</math> in all four dimensions concurrently, so they move diagonally through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c^\prime = 2c</math>, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold moving at velocity <math>c</math>. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math> in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper space. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <small><math>\mathrm{Q}</math></small> denote a rotation, <small><math>\mathrm{R}</math></small> a reflection, <small><math>\mathrm{T}</math></small> a translation, and let <small><math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math></small> denote a product of several such transformations, all commutative with one another. Then <small><math>\mathrm{RT}</math></small> is a glide-reflection (in two or three dimensions), <small><math>\mathrm{QR}</math></small> is a rotary-reflection, <small><math>\mathrm{QT}</math></small> is a screw-displacement, and <small><math>\mathrm{Q^2}</math></small> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<small><math>\mathrm{Q}^q \mathrm{R}^r</math></small><br> where <small><math>(2^q + r \le n)</math></small>, the number of dimensions.<br> Transformations involving a translation are expressible as:<br> :<small><math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math></small><br> where <small><math>(2^q + r + 1 \le n)</math></small>.<br> For <small><math>(n = 4)</math></small> in particular, every displacement is either a double rotation <small><math>\mathrm{Q}^2</math></small>, or a screw-displacement <small><math>\mathrm{QT}</math></small> [where the rotation component <small><math>\mathrm{Q}</math></small> is a simple rotation, but the <small><math>\mathrm{QT}</math></small> is chiral like a <small><math>\mathrm{Q^2}</math></small>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <small><math>\mathrm{QRT}</math></small>.</blockquote> If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a <small><math>\mathrm{Q^2}</math></small> or a <small><math>\mathrm{QT}</math></small>, because we can view any <small><math>\mathrm{QT}</math></small> as a <small><math>\mathrm{Q^2}</math></small> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <small><math>\mathrm{Q^2}</math></small>. By the same principle, we can view any <small><math>\mathrm{QT}</math></small> or <small><math>\mathrm{Q^2}</math></small> as an isoclinic (equi-angled) <small><math>\mathrm{Q^2}</math></small> by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (<small><math>\mathrm{T}</math></small>) for ''one'' of the two rotations (<small><math>\mathrm{Q}</math></small>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<small><math>\mathrm{Q}</math></small>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<small><math>\mathrm{T}</math></small>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <small><math>SO(4)</math></small> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <small><math>SO(4)</math></small> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, stationary atoms exhibit the <small><math>SO(4)</math></small> symmetries of the discrete isoclinic (equi-angled) double rotations (<small><math>\mathrm{Q^2}</math></small>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.<ref>{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional "space" because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was first described by Einstein himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction. Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity ''c'', in some 4-space direction. This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always ''c'', as measured by all observers in any inertial reference frame. This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity ''c''. In physics as it has been universally understood, observers are not supposed to be able to move at velocity ''c''. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four. In the Euclidean relativity alternative view, however, every observer is always moving at velocity ''c'' through the universe, which is real Euclidean 4-dimensional space <small><math>\mathbb{R^4}</math></small>. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity ''c''. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity ''c'', in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your coordinate systems; you are both embedded in the same Euclidean 4-dimensional space <small><math>\mathbb{R^4}</math></small>.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity ''c'' relative to universal 4-coordinate space, so the maximum relative velocity between two observers is 2''c'' when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to ''c'', it is the same measurement in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}} So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity ''c'' in Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" ''c'', although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity ''c'', but in at least slightly different directions. In Einstein's relativity, the invariant ''c'' is the speed of light through 3-space. In Euclidean relativity, the invariant ''c'' is the speed of matter through 4-space! The speed of light through 3-space is also perceived as ''c'' by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity ''c''. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space. ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity <math>c</math> through space and proper time ...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords. Euclidean relativity is not even a fringe theory; no physicists have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at ''c'' (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than ''c''. Euclidean relativity is a revolutionary theory indeed, in which ''c'' cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for it to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c^\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the diagonal of the unit 4-hypercube is the natural number <small><math>\sqrt{4}</math></small>. == An object's motion in space is the product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a small number of discrete self-reflections. Any action of a geometric object that transforms its position and orientation in space may be measured as a distinct group of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete set of local self-reflections. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which touch (intersect each other). When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, leaving itself right-side-out again. Coxeter's laws of motion are a geometric counterpart to Newton's laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as simple geometric pictures, by anyone baffled by algebraic formulas. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. For example, they give us simple geometric pictures of all the possible motions of objects in four dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<small><math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math></small><br> where <small><math>(2^q + r + t \le 4)</math></small>. Every displacement is either a double rotation <small><math>\mathrm{Q}^2</math></small>, or a screw-displacement <small><math>\mathrm{QT}</math></small> [where the rotation component <small><math>\mathrm{Q}</math></small> is a simple rotation, but the <small><math>\mathrm{QT}</math></small> is chiral like a <small><math>\mathrm{Q^2}</math></small>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <small><math>\mathrm{QRT}</math></small>.</blockquote> While this description should be understood as simple geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (<small><math>\mathrm{Q}</math></small>), reflection (<small><math>\mathrm{R}</math></small>) and translation (<small><math>\mathrm{T}</math></small>) are just what they are in three-dimensional space, but double rotation (<small><math>\mathrm{Q}^2</math></small>) is something new and unprecedented in our physical experience, because double rotations cannot occur until you have four or more dimensions of space to rotate in. ...to readers who have not studied Coxeter (almost all readers including TAC), the blockquote above is "just math", not visualizable geometry...but I could describe Coxeter's congruent transformations in 4-space here geometrically: I could say clearly what they mean in spatial terms, in language anyone can understand, because they don't require any math to be understood; the "math" here is really just simple pictures (reflections and rotations); even double rotations can be visualized by dimensional analogy, as compounds of simple rotations...since even most physicists are unacquainted with Coxeter geometry, it could be important that I do this here... == Light propagates through 4-space at twice its apparent velocity ''c''== Coxeter's geometric laws of motion apply to all objects with mass in 4-dimensional Euclidean space, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <small><math>\mathrm{R}^4</math></small>, which may be termed a double translation <small><math>\mathrm{T}^2</math></small>, a pure translation via two pairs of parallel reflections, without any rotation component <small><math>\mathrm{Q}</math></small>. Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by <small><math>\mathrm{QT}</math></small>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <small><math>\mathrm{Q^2}</math></small>, an isoclinically rotating object such as an atom. A simple rotation <small><math>\mathrm{Q}</math></small> or simple translation <small><math>\mathrm{T}</math></small> is a double reflection <small><math>\mathrm{R^2}</math></small>, so a <small><math>\mathrm{QT}</math></small> or <small><math>\mathrm{Q^2}</math></small> is also an <small><math>\mathrm{R^4}</math></small>, but not with the same group of reflection angles as a light signal <small><math>\mathrm{R^4}</math></small>. A translation <small><math>\mathrm{T = R^2}</math></small> is a double reflection in two parallel planes, and a rotation <small><math>\mathrm{Q = R^2}</math></small> is a double reflection in two intersecting planes, as in a <small><math>\mathrm{QT = R^4}</math></small> which is both at once. A double translation <small><math>\mathrm{T^2 = R^4}</math></small> is two double reflections in pairs of parallel planes at once, a reflection in four or more non-intersecting parallel planes; it is all translation and no rotation. In a <small><math>\mathrm{T^2}</math></small> all the motion goes to translation, so the translation goes twice as far as the simple translation <small><math>\mathrm{T}</math></small> in a <small><math>\mathrm{QT}</math></small>. A double translation <small><math>\mathrm{T^2 = R^4}</math></small> is the opposite of a double rotation <small><math>\mathrm{Q^2 = R^4}</math></small>, which is stationary but rotates twice as fast as the simple rotation <small><math>\mathrm{Q}</math></small> in a <small><math>\mathrm{QT}</math></small>. The product of the two translations in a <small><math>\mathrm{T^2}</math></small> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <small><math>\mathrm{T}</math></small> over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A <small><math>\mathrm{T^2}</math></small> cannot reposition a 4-polytope the way a <small><math>\mathrm{QT}</math></small> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter. ...lensing of double translations <small><math>\mathrm{T^2 = R^4}</math></small> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet... == The Kepler problem is framed in Euclidean 4-space == The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], arguably the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the "inverse Kepler problem": the orbit characteristics require the force to depend on the inverse square of the distance.<ref>{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton & Company|isbn=978-0393039184}}</ref> The inverse square law behind the Kepler problem is the [[W:Central force|central force]] law which governs not only [[W:Newtonian gravity|Newtonian gravity]] and celestial orbits, but also the motion of two charged particles in [[W:Coulomb’s law|Coulomb’s law]] of [[W:Electrostatics|electrostatics]]; it applies to attractive or repulsive forces. Problems in which two bodies interact by a central force that varies as the [[W:Inverse square law|inverse square]] of the distance between them are called Kepler problems. Thus the [[W:Hydrogen atom|hydrogen atom]] is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law, another inverse-square central force. Using classical mechanics, the solution to a Kepler problem can be expressed as a [[W:Kepler orbit|Kepler orbit]] using six kinematical variables or [[W:Orbital elements|orbital elements]]. The solution conserves an orbital element called the [[W:Laplace–Runge–Lenz vector|Laplace–Runge–Lenz (LRL) vector]], a [[W:Constant of motion|constant of motion]], meaning that it is the same no matter where it is calculated on the orbit. The LRL vector was essential in the first quantum mechanical derivation of the [[W:Atomic emission spectrum|spectrum]] of the hydrogen atom, but this approach has rarely been used since the development of the [[W:Schrödinger equation|Schrödinger equation]]. The conservation of the LRL vector corresponds to the <small><math>SO(4)</math></small> symmetry, by Nother's theorem. The LRL vector lies orthogonal to both the orbital plane and the angular momentum vector of the Kepler orbit; we observe that it lies in a fourth orthogonal dimension. Fock in 1935<ref>V. Fock, Zur Theorie des Wasserstoffatoms, Zeitschrift für Physik. 98 (3-4) (1935), 145–154.</ref> and Moser in 1970<ref>J. Moser, Regularization of Kepler’s problem and the averaging method on a manifold, Commun. Pure Appl. 23 (1970), 609–636</ref> observed that the Kepler problem is mathematically equivalent to non-affine geodesic motion (a particle moving freely) on the surface of a 3-sphere, so that the whole problem is symmetric under certain rotations of the four-dimensional space. This higher-dimensional symmetry results in two well-known properties of the Kepler problem: the momentum vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points. ... Relativity establishes that an orbit in space is viewed in a different way in each distinct inertial reference frame. Depending on the choice of reference frame, the same Kepler system may be seen to be performing any one of a sequence of relativistically equivalent rotations in 4-space, on a continuum from an isoclinic rotation (Q²) in the orbit's proper reference frame, to a screw transfer (QT) with a simple rotation component (Q) and a translation component (T) at velocity <math>c</math>, in the universal reference frame of 4-coordinate space wherein every object is seen to be translating at velocity <math>c</math>. In reference frames between these two limit cases, the orbit is seen to be performing a double rotation (Q²) at two unequal, completely orthogonal angular rates of rotation: an elliptical double rotation. These include the reference frames of most typical observers, who are moving slowly relative to the observed orbital system's reference frame (their relative motion is a small fraction of the speed of light). ...this is probably misplaced here and should not interrupt the discussion at this point: ...These typical observations agree closely with the predictions of special relativity, because the non-isoclinic elliptical (Q²) resembles a (QT), since one of its two completely orthogonal rotations (Q) has such a long period that it is almost indistinguishable from a straight translation (T). All orbits in 4-space are isoclinic in their own reference frame. Orbiting objects in their own proper Kepler systems follow circular geodesic isoclines through 4-space. Orbits in 4-space are perfectly circular in their own reference frame, as Copernicus assumed the orbits of planets to be. It is the orbit's path through the 3-space of its elliptic hyperplane that is an ellipse, as Kepler found it to be. ...cite Jesper Goransson's very concise paper The geodesic circle that an orbiting object follows through 4-space in the proper reference frame of its own Kepler system is not a simple great circle which turns in two orthogonal dimensions. It is a helical great circle that turns in four orthogonal dimensions at once.{{Efn|Geodesic orbits in 4-space are not simple 2-dimensional great circles; they are helical 4-dimensional great circles that curve in all four dimensions at once. Their circular trajectories are helixes which we call ''isoclines'', since they are the paths taken by points on a rigid object undergoing isoclinic rotation.}} Such circles lie outside our physical experience, since our local space has only three orthogonal dimensions. Nonetheless we can visualize them in imagination, because their helical, circular shape is perfectly well defined by the kinematical variables of the Kepler orbit. The real physical correlates of abstract orthogonal planes and rotation angles are already familiar to us viscerally in our body-language of physical experience, since we are endowed biologically with highly evolved visual signal processing engines. These enable us to see and understand spatial relations and motions, including rotations, without even thinking about angles and orthogonal planes. This physical endowment is an inborn capacity for dimensional analogy which our biologic evolution has provided. All our instinctive spatial reasoning is by dimensional analogy from flat 2-dimensional retinal images to 3-dimensional scenes, using our powerful inborn visualization capacities of reverse stereographic projection and pattern recognition. We humans are thus very well equipped with everything we need to see in four-dimensional space, except experience. ... Recently Anco and Moghadam found that through Noether’s theorem in reverse, the LRL vector gives rise to a corresponding infinitesimal dynamical symmetry on the kinematical variables, which they show to be the semi-direct product of <small><math>SO(3)</math></small> and <small><math>\mathbb{R^3}</math></small>, in contrast to the <small><math>SO(4)</math></small> symmetry group generated by the LRL symmetries and the rotations.{{Sfn|Anco|Moghadam|2026|ps=; The physically relevant part of the LRL vector is its direction ... since its magnitude is just a function of energy and angular momentum.}} This remarkable symmetry breaking is expressive of the ''dimensional relativity'' between ordinary 3-space <small><math>\mathbb{R^3}</math></small>, spherical space <small><math>S^3</math></small> and Euclidean space <small><math>\mathbb{R^4}</math></small>. Consider a hydrogen atom in a Kepler orbit: for example, a hydrogen atom moving freely in space in an orbit around the sun. It is a ''double'' Kepler problem: an electrostatic Kepler problem within itself, and a gravitational Kepler problem in its environment. The ''single'' electrostatic Kepler problem of a hydrogen atom moving freely in space beyond any gravitational influence is a problem in special relativity. In our Euclidean 4-space model, this atom viewed as stationary in its own proper reference frame exhibits an <small><math>SO(4)</math></small> rotation symmetry corresponding to an isoclinic double rotation (<small><math>\mathrm{Q^2}</math></small>). The fourth dimension in this reference frame is the atom's proper time vector; it has constant velocity <math>c</math> and constant direction. From the point of view of our universal 4-coordinate space (which cannot be the proper inertial reference frame of any physical observer, all of whom are moving relative to it at velocity ''c''), the entire Kepler system (the atom) is translating through 4-space via a screw translation (<small><math>\mathrm{QT}</math></small>) at constant velocity <math>c</math>. From this viewpoint the atom has only a simple <small><math>SO(3)</math></small> rotation component (<small><math>\mathrm{Q}</math></small>), breaking its stationary <small><math>SO(4)</math></small> isoclinic rotation symmetry (<small><math>\mathrm{Q^2}</math></small>). Because each discrete part of the rotating atom moves along a helical trajectory through 4-space, the atom is in orbit around a barycentric axis (like a star in a galaxy), but only in a tiny orbit within its own radius, which is its inertial domain of rotation. The straight 4-dimensional cylinder it progresses along at velocity <math>c</math> is very narrow: only the diameter of the rotating atom itself. The gravitational Kepler problem of a hydrogen atom in a Kepler orbit around the sun is a problem in general relativity. In our 4-space model, this atom viewed in its own proper reference frame exhibits the same <small><math>SO(4)</math></small> rotation symmetry as it did in the electrostatic Kepler problem where the atom was translating linearly through space. The Kepler system in this case is not just the atom; it is the entire solar system. The LRL vector of this Kepler system is the proper time vector of the atom's inertial reference frame; once again it has constant velocity ''and constant direction''. Although the momentum vector moves in a perfect circle as the atom orbits the sun, the 4-space LRL vector does not move at all: it is a constant of motion, of linear motion (<small><math>\mathrm{T}</math></small>) of the Kepler system (the entire solar system in this case) in a constant 4-space direction, the proper time direction of the system. The direction of the system's proper time vector would vary under some kinds of acceleration of the atom, but it is constant under this kind of orbital acceleration. It continues to point in the same direction, like a 4-space compass needle, as the atom winds its way along its spiral path around the axis of the sun's straight-line translation through 4-space at velocity <math>c</math>. This compass needle always points in the direction the sun is moving, not the direction the atom is moving at any instant. ...Its Kepler orbit around the sun is its <small><math>SO(3)</math></small> rotation component (<small><math>\mathrm{Q}</math></small>). Although the atom is moving on a geodesic circle in the second problem, by the [[equivalence principle]] the difference in the state of the atomic systems in these two problems cannot be observed by examining the atoms alone. Even from another inertial reference frame, where the atom in the second problem is seen to be translating through 4-space via a wide screw translation (<small><math>\mathrm{QT}</math></small>) around the sun's axis of motion, there is still no difference between the two problems which can be detected by examining only the atoms within their own proper reference frames (even over time), because the LRL vector (<small><math>\mathrm{T}</math></small>) is a constant of motion of the entire system in both cases. ...Anco and Maghadam found that <small><math>SO(4)</math></small>) breaks to ... <small><math>S^3</math></small>)... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... <small><math>H^3</math></small>) ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). ... Finally we consider a third problem in which a hydrogen atom enters the solar system as a comet, loops around the sun and exits the solar system again. This atom... ... As Hamilton found when he discovered the quaternions, we see that it is necessary to admit a fourth dimension to the system in order to properly model the problem: in Hamilton's case the general problem of ..., and in our case the Kepler problem. These are instances of the same problem in 4-dimensional Euclidean geometry, and indeed a solution to the Kepler problem in quaternions (the four Cartesian coordinates of Euclidean 4-space) is a solution to it in our model of the 4-coordinate Euclidean cosmos. == Distribution of stars in our galaxy == The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by its red shift, and by assuming that they are distributed in three dimensions of space, we have plotted their locations in 3-space. If we abandon the last of those three assumptions, we can just as easily reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits as we saw them in 3-space? That would be an expected consequence of the special rotational symmetry group of 4-space <small><math>SO(4)</math></small>, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits. ...have to perform this experiment somehow, at least as a conclusive thought experiment, before I publish this paper... == Rotations == The [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotations]] of the convex [[W:regular 4-polytope|regular 4-polytope]]s are usually described as discrete rotations of a rigid object. For example, the rigid [[24-cell]] can rotate in a [[24-cell#Great hexagons|hexagonal]] (6-vertex) central [[24-cell#Planes of rotation|plane of rotation]]. A 4-dimensional [[24-cell#Isoclinic rotations|''isoclinic'' rotation]] (as distinct from a [[24-cell#Simple rotations|''simple'' rotation]] like the ones that occur in 3-dimensional space) is a ''diagonal'' rotation in multiple [[W:Clifford parallel|Clifford parallel]] [[24-cell#Geodesics|central planes]] of rotation at once. It is diagonal because it is a [[W:SO(4)#Double rotations|double rotation]]: in addition to rotating in parallel (like wheels), the multiple planes of rotation also tilt sideways in the completely orthogonal plane of rotation (like coins flipping) into each other's planes. Consequently, the path taken by each vertex is a [[24-cell#Helical hexagrams and their isoclines|twisted helical circle]], rather than the ordinary flat great circle a vertex follows in a simple rotation. In a rigid 4-polytope rotating isoclinically, ''all'' the vertices lie in one of the parallel planes of rotation, so all the vertices move in parallel along Clifford parallel twisting circular paths. [[24-cell#Clifford parallel polytopes|Clifford parallel planes]] are not parallel in the normal sense of parallel planes in three dimensions; the vertices are all moving in different directions around the [[W:3-sphere|3-sphere]]. In one complete 360° isoclinic revolution, a rigid 4-polytope turns itself inside out. This is sufficiently different from the simple rotations of rigid bodies in our 3-dimensional experience that a [[24-cell#Rotations|detailed description]] enabling the reader to properly visualize its counter-intuitive consequences runs to many pages and illustrations, with many accompanying pages of explanatory notes on surprising phenomena that arise in 4-dimensional space: [[24-cell#Great squares|completely orthogonal planes]], [[24-cell#Clifford parallel polytopes|Clifford parallelism]]{{Efn|name=Clifford parallels}} and [[W:Hopf fibration|Hopf fiber bundles]], [[24-cell#Isoclinic rotations|isoclinic geodesic paths]], and [[24-cell#Double rotations|chiral (mirror image) pairs of rotations]], among other complexities. Moreover, the characteristic rotations of the various regular 4-polytopes are all different; each is a unique surprise. [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|The 6 regular convex 4-polytopes]] have different numbers of vertices (5, 8, 16, 24, 120 and 600 respectively) and those with fewer vertices occur inscribed in those with more vertices (with one exception), with the result that the more complex 4-polytopes subsume the kinds of rotations characteristic of their less complex predecessors, as well as each having a characteristic kind of rotation not found in their predecessors. None of these symmetries is to be found in 3-dimensional space, although their simpler 3-dimensional analogues are all present there. [[W:Euclidean geometry#Higher dimensions|Four dimensional Euclidean space]] is more complicated (and more interesting) than three dimensional space because there is more room in it, in which unprecedented things can happen. It subsumes 3-dimensional space, with all of the symmetries we are accustomed to, and adds astonishing new surprises. These are hard for us to visualize, because the only way we can experience them is in our imagination; we have no body of sensory experience in 4-dimensional space to draw upon, other than our evolution in time. For that reason (our difficulty in visualizing them), descriptions of isoclinic rotations usually begin and end with rigid rotations: [[24-cell#Isoclinic rotations|for example]], all 24 vertices of a single rigid 24-cell rotating in unison, with 6 vertices evenly spaced around each of 4 Clifford parallel twisted circles.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} But that is only the simplest case, which is easiest for us to understand. Compound and [[W:Kinematics|kinematic]] 24-cells (with moving parts) are even more interesting (and more complicated) than the rotation of a single rigid 24-cell. To begin with, when we examine the individual parts of a single rigid 24-cell that are moving in an isoclinic rotation, such as the orbits of individual vertices, we can imagine a case where fewer than 24 point-objects are orbiting on those twisted circular paths at once. [[24-cell#Reflections|For example]], if we imagine just 8 point-objects, evenly spaced around the 24-cell at [[24-cell#Reciprocal constructions from 8-cell and 16-cell|the 8 vertices that lie on the 4 coordinate axes]], and rotate them isoclinically along exactly the same orbits they would take in the above-mentioned rotation of a rigid 24-cell, then in the course of a single 360° rotation the 8 point-objects will trace out the whole 24-cell, with just one point-object reaching each of the 24 vertex positions just once, and no point-object colliding with (or even crossing the path of) any other at any time. This is an example of a discrete Hopf fibration. But it is still an example of a rigid object in a discrete isoclinic rotation: a rigid 8-vertex object (called the 4-[[W:orthoplex|orthoplex]] or [[16-cell]]) performing one half of the characteristic rotation of the 24-cell. We can also imagine ''combining'' distinct isoclinic rotations. What happens when multiple point-objects are orbiting at once, but do ''not'' all follow the Clifford parallel paths characteristic of the ''same'' distinct rigid rotation? What happens when we combine orbits from distinct rotations characteristic of different 4-polytopes, for example when different rigid 4-polytopes are concentric and rotating simultaneously in their characteristic ways? What kinds of such hybrid rotations are possible in the same 3-sphere shell without collisions? In adjacent concentric shells without asymmetric imbalance? What sort of [[Kinematics of the cuboctahedron|kinematic polytopes]] do they trace out, and how do their [[24-cell#Clifford parallel polytopes|component parts]] relate to each other as they move? Is there (sometimes) some kind of mutual stability amid their lack of combined rigidity? Visualizing isoclinic rotations (rigid and otherwise) allows us to explore such questions of [[W:kinematics|kinematics]], and where dynamic stabilities arise, of [[wikipedia:kinetics (physics)|kinetics]]. In four dimensions, we discover that space has more room in it than we have experienced, which permits previously unimagined motions. Even 3-space is more commodious than we thought; when it is curved and lies embedded in a higher-dimensional space, it permits previously impossible symmetric packings. Sadoc studied double-twisted 3-dimensional molecules, and imagined them embedded in 4-dimensional space as the Hopf fibrations of regular 4-polytopes. He found that these molecules would close-pack on the 3-sphere perfectly without exhibiting any torsion, although their packing in ordinary flat 3-space is imperfect, "frustrated" by their twisted geometry. <blockquote>The frustration, which arises when the molecular orientation is transported along the two [spiral] AB paths of figure 1 [double twist helix], is imposed by the very topological nature of the Euclidean space R³. It would not occur if the molecules were embedded in the non-Euclidean space of the [[W:3-sphere|3-sphere]] S³, or hypersphere. This space with a homogeneous positive curvature can indeed be described by equidistant and uniformly twisted fibers, along which the molecules can be aligned without any conflict between compactness and [[W:torsion of a curve|torsion]].... The fibres of this [[W:Hopf fibration|Hopf fibration]] are great circles of S³, the whole family of which is also called the [[W:Clifford parallel|Clifford parallel]]s.{{Efn|name=Clifford parallels}} Two of these fibers are C_∞ symmetry axes for the whole fibration; each fibre makes one turn around each axis and regularly rotates when moving from one axis to another.{{Efn|name=helical geodesic}} These fibers build a double twist configuration while staying parallel, i.e. without any frustration, in the whole volume of S³.{{Efn|name=Petrie polygon of a honeycomb}} They can therefore be used as models to study the condensation of long molecules in the presence of a double twist constraint.{{Sfn|Sadoc & Charvolin|2009|loc=§1.2 The curved space approach|ps=; studies the helical orientation of molecules in crystal structures and their imperfect packings ("frustrations") in 3-dimensional space.}}</blockquote> Of course we do not find molecules condensing to close-pack the 3-sphere in our experience, and Sadoc does not say that we do. We find 3-spheres in the atomic realm (if atoms are 4-polytopes), and in the cosmic realm (as the surface boundaries of stars, and the concentric surfaces of galaxies). But in between, in the realm of ordinary experience which includes the molecular realm, ourselves and all the objects we can materially handle or observe up close including the planets, we are confined together by gravity as inertia within a curved 3-dimensional space that is no more than one atom thick in the fourth spatial dimension. That is why in the molecular realm we find only objects that occupy 3-spaces which, though infinitesimally curved in the fourth dimension, are tiny patches on whole 3-spheres of galactic size. So Sadoc's exercise is a thought experiment, like Einstein's gedankenexperiments about railroad embankments and trains moving at nearly the speed of light. It is no less illuminating, despite the symmetry it reveals not having a realization as an actual 3-sphere of actual molecules. And might not something very like it have an actual realization in the atomic realm? We know that atoms have their own complex internal structure, which we are unable to model geometrically in ordinary 3-dimensional space. Suppose such a model is impossible because an atom is actually a 4-polytope occupying a tiny spherical region of 4-dimensional space, and so we only find its constituent particles in close-packed helical orbits on the 3-sphere, in the manner of Sadoc's imaginary twisted molecules, but as real 4-dimensional helices of atomic scale. We would expect to find the atomic orbit of a fundamental particle in some discrete Hopf fibration characteristic of a symmetry group, that is, on the maximally symmetric isoclines of a discrete isoclinic rotation characteristic of some regular 4-polytope and the particle. == A theory of the Euclidean atom == <blockquote>Because quantum physics could be tested without being understood, it allowed humans to see how the universe worked without knowing why.<ref>Sebastian Junger, In My Time of Dying</ref></blockquote> ... == Light and Mass are Reflection and Rotation == The phenomena of light and mass are expressions of reflection symmetries and rotation symmetries, respectively. ... Atoms are 4-polytopes, elementary objects with SO(4) rotational symmetry. Light is .... Motion in space is the propagation of the elementary objects of light and matter in Coxeter congruent transformations by kaleidoscopic self-reflections, like the motion of self-reproducing cellular automata in [[Conway's Game of Life|Conway's game of life]]. ... Light is discrete reflections. Mass is discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics. === Atoms are 4-polytopes === ... == Relativity in real space of four or more orthogonal dimensions == Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions. General relativity is Galilean relativity in a general space of four or more orthogonal dimensions, e.g. in Euclidean 4-space <math>R^4</math>, spherical 4-space <math>S^4</math>, and any orthogonal 4-manifold. Light is a consequence of symmetry group reflections at quantum scale. Gravity and the other fundamental forces are consequences of rotations, which are consequences of quantum reflections. Light is discrete reflections. Gravity and all forces are discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics. Every observer may properly see themself as stationary and the universe as an ''n''-sphere with themself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and can be measured by the observer as the speed of light. === Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions === ...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here... Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.) None of these Lorentz effects are beyond geometric explanation or paradoxical. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, successive Euclidean spaces are dimensionally analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries: that is Nother's great discovery. === General relativity is Galilean relativity in a general space of four orthogonal dimensions === ... == Dimensional relativity == Coxeter's kinetic law of <math>n</math>-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean <math>n</math>-spaces of increasing <math>n</math>, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating <math>n</math>-spaces. By the degree of dimensional analogy of which they are capable, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: "The universe is a sphere whose center is wherever there is intelligence."{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}} == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity ''c'', with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may, or may not, all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed us that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a sense related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system. When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw displacements), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw displacement has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal of that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. Anco and Maghadam found that <small><math>SO(4)</math></small> breaks to ... <small><math>S^3</math></small>... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... <small><math>H^3</math></small> ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). Because the planets orbit on ellipses in our 3-space, Euclidean 4-space is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the truer model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position of centrality as our exclusive conception of our place in space. ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others. The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.<ref>{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}</ref>}} == Boundaries == <blockquote>Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? ''What is the nature of the boundary which confines us to just three dimensions?'' We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined motions and objects. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. A boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three .... In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it. We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed "Dimensional Analogy."<ref>{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's and Coxeter's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover in imagination and then explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :We shall not cease from exploration :And the end of all our exploring :Will be to arrive where we started :And know the place for the first time. :Through the unknown, remembered gate :When the last of earth left to discover :Is that which was the beginning; :At the source of the longest river :The voice of the hidden waterfall :And the children in the apple-tree :Not known, because not looked for :But heard, half-heard, in the stillness :Between two waves of the sea. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. This project is forever beginning anew. Coxeter showed us that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether showed us all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions, in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::BEECH :Where my imaginary line :Bends square in woods, an iron spine :And pile of real rocks have been founded. :And off this corner in the wild, :Where these are driven in and piled, :One tree, by being deeply wounded, :Has been impressed as Witness Tree :And made commit to memory :My proof of being not unbounded. :Thus truth's established and borne out, :Though circumstanced with dark and doubt— :Though by a world of doubt surrounded. :::::::—''The Moodie Forester''<ref>{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|The long radius (center to vertex) of the 24-cell is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including the four-dimensional 24-cell and [[W:Tesseract#Radial equilateral symmetry|tesseract]], the three-dimensional [[W:Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[W:Hexagon#Regular hexagon|hexagon]]. (The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron.) '''Radially equilateral''' polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.|name=radially equilateral|group=}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|5}}(–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>){{spaces|3}}(–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} {{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}} {{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}} {{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles]."}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}} {{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}} {{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}} {{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}} {{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} {{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} {{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} {{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}} {{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} {{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90^o bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60^o bend, or as a straight line with one 60^o bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} {{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)"}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} {{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}} {{Efn|Departing from any vertex V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V₀ and V₂ are ''two'' {{radic|3}} chords apart{{Efn|V₀ and V₂ are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V₀ and V₂ are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹₄.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} qeo0uojoc6ctchsbz2piync3gbnu87u 2818507 2818506 2026-07-19T03:05:40Z Dc.samizdat 2856930 /* Light propagates through 4-space at twice its apparent velocity c */ 2818507 wikitext text/x-wiki = Real Euclidean four-dimensional space R⁴ = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - July 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as a Euclidean space of four orthogonal spatial dimensions. Space itself has a fourth orthogonal dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.</blockquote> == Summary == We observe that physical space has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is translating through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions. == A theory of the Euclidean cosmos == The physical universe is properly visualized as a [[w:Four-dimensional_space|Euclidean space of four orthogonal spatial dimensions]]. Space itself has a fourth orthogonal dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move through Euclidean 4-space at velocity <math>c</math> as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical spiral galaxy such as ours is a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our galaxy appear from our viewpoint to be distributed in a cloud of elliptical spirals occupying a flattened ellipsoid region of 3-dimensional space, but they are not so confined: they are distributed within a spherical region of 4-dimensional space. The galaxy's actual shape is spherical, not a flattened ellipsoid, but it is rounder than round can be in our ordinary experience: it occupies a hyperspherical region of space. The concentric spirals of stars that we observe lie on concentric [[W:3-sphere|3-sphere]]s (4-dimensional spheres), not on concentric 2-ellipsoids (3-dimensional elliptical spirals). Our sun and solar system lies on one of those concentric 3-spheres. More generally, orbits are circular in 4-space, and elliptical in the 3-space of their elliptic hyperplane. ...rotating illustration of the 4-ball galaxy showimg its spirals of star clouds on the surface of concentric 3-spheres...obtained by reverse sterographic projection from 3D images of the galaxy... The galaxy as a whole, or more properly its orbital center point, is translating through 4-space at velocity <math>c</math>, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity <math>c</math> in the same direction, but on spiral trajectories relative to the galaxy's linear trajectory, as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the cylinder; their trajectories are screw-displacements, the compound of a simple rotation and a linear translation. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis. It is the tremendous inertial force of stars in motion at velocity <math>c</math> that holds the cylinder of motion together. The observed universe as a whole appears to be a 3-sphere expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. For all observers, the conjectured origin point of the universe corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space (the same point in the same Euclidean 4-space for all observers). The big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured separately, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. The enclosing surface of a spherical region of 4-space is itself a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. We live within such a 3-space, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our particular 3-sphere is one of the galaxy's concentric 3-spheres of spiral star-clouds. The solar system occupies a tiny patch of this filmy 4-dimensional soap-bubble of galactic size, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. Our entire 3-sphere manifold, as a 3-spherical shell within the moving 4-ball galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy, in a distinct direction that is orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of rotating objects through Euclidean space by screw translation. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold that is such an evolving surface boundary is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a three-dimensional smear of atoms no thicker than one atom in its fourth dimension, which is the direction of its linear translation through 4-space at velocity <math>c</math>. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to 4-dimensional lumps of matter as plasma, and have little experimental knowledge of their geometry or internal structure. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know nothing about its interior 4-ball. Every such moving 3-dimensional surface boundary of matter in the observed universe is evolving in four dimensions at velocity <math>c</math>. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from atoms to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, two orthogonal inertial motions each at the constant universal rate of transformation <math>c</math>. Every object moves relative to universal 4-coordinate space on its own distinct geodesic spiral, a screw translation trajectory that is the compound of its two orthogonal inertial motions. Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in motion at velocity <math>c</math> in all four dimensions concurrently, so they move diagonally through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c^\prime = 2c</math>, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold moving at velocity <math>c</math>. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math> in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper space. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <small><math>\mathrm{Q}</math></small> denote a rotation, <small><math>\mathrm{R}</math></small> a reflection, <small><math>\mathrm{T}</math></small> a translation, and let <small><math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math></small> denote a product of several such transformations, all commutative with one another. Then <small><math>\mathrm{RT}</math></small> is a glide-reflection (in two or three dimensions), <small><math>\mathrm{QR}</math></small> is a rotary-reflection, <small><math>\mathrm{QT}</math></small> is a screw-displacement, and <small><math>\mathrm{Q^2}</math></small> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<small><math>\mathrm{Q}^q \mathrm{R}^r</math></small><br> where <small><math>(2^q + r \le n)</math></small>, the number of dimensions.<br> Transformations involving a translation are expressible as:<br> :<small><math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math></small><br> where <small><math>(2^q + r + 1 \le n)</math></small>.<br> For <small><math>(n = 4)</math></small> in particular, every displacement is either a double rotation <small><math>\mathrm{Q}^2</math></small>, or a screw-displacement <small><math>\mathrm{QT}</math></small> [where the rotation component <small><math>\mathrm{Q}</math></small> is a simple rotation, but the <small><math>\mathrm{QT}</math></small> is chiral like a <small><math>\mathrm{Q^2}</math></small>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <small><math>\mathrm{QRT}</math></small>.</blockquote> If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a <small><math>\mathrm{Q^2}</math></small> or a <small><math>\mathrm{QT}</math></small>, because we can view any <small><math>\mathrm{QT}</math></small> as a <small><math>\mathrm{Q^2}</math></small> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <small><math>\mathrm{Q^2}</math></small>. By the same principle, we can view any <small><math>\mathrm{QT}</math></small> or <small><math>\mathrm{Q^2}</math></small> as an isoclinic (equi-angled) <small><math>\mathrm{Q^2}</math></small> by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (<small><math>\mathrm{T}</math></small>) for ''one'' of the two rotations (<small><math>\mathrm{Q}</math></small>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<small><math>\mathrm{Q}</math></small>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<small><math>\mathrm{T}</math></small>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <small><math>SO(4)</math></small> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <small><math>SO(4)</math></small> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, stationary atoms exhibit the <small><math>SO(4)</math></small> symmetries of the discrete isoclinic (equi-angled) double rotations (<small><math>\mathrm{Q^2}</math></small>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.<ref>{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional "space" because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was first described by Einstein himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction. Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity ''c'', in some 4-space direction. This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always ''c'', as measured by all observers in any inertial reference frame. This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity ''c''. In physics as it has been universally understood, observers are not supposed to be able to move at velocity ''c''. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four. In the Euclidean relativity alternative view, however, every observer is always moving at velocity ''c'' through the universe, which is real Euclidean 4-dimensional space <small><math>\mathbb{R^4}</math></small>. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity ''c''. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity ''c'', in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your coordinate systems; you are both embedded in the same Euclidean 4-dimensional space <small><math>\mathbb{R^4}</math></small>.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity ''c'' relative to universal 4-coordinate space, so the maximum relative velocity between two observers is 2''c'' when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to ''c'', it is the same measurement in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}} So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity ''c'' in Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" ''c'', although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity ''c'', but in at least slightly different directions. In Einstein's relativity, the invariant ''c'' is the speed of light through 3-space. In Euclidean relativity, the invariant ''c'' is the speed of matter through 4-space! The speed of light through 3-space is also perceived as ''c'' by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity ''c''. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space. ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity <math>c</math> through space and proper time ...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords. Euclidean relativity is not even a fringe theory; no physicists have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at ''c'' (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than ''c''. Euclidean relativity is a revolutionary theory indeed, in which ''c'' cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for it to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c^\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the diagonal of the unit 4-hypercube is the natural number <small><math>\sqrt{4}</math></small>. == An object's motion in space is the product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a small number of discrete self-reflections. Any action of a geometric object that transforms its position and orientation in space may be measured as a distinct group of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete set of local self-reflections. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which touch (intersect each other). When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, leaving itself right-side-out again. Coxeter's laws of motion are a geometric counterpart to Newton's laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as simple geometric pictures, by anyone baffled by algebraic formulas. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. For example, they give us simple geometric pictures of all the possible motions of objects in four dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<small><math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math></small><br> where <small><math>(2^q + r + t \le 4)</math></small>. Every displacement is either a double rotation <small><math>\mathrm{Q}^2</math></small>, or a screw-displacement <small><math>\mathrm{QT}</math></small> [where the rotation component <small><math>\mathrm{Q}</math></small> is a simple rotation, but the <small><math>\mathrm{QT}</math></small> is chiral like a <small><math>\mathrm{Q^2}</math></small>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <small><math>\mathrm{QRT}</math></small>.</blockquote> While this description should be understood as simple geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (<small><math>\mathrm{Q}</math></small>), reflection (<small><math>\mathrm{R}</math></small>) and translation (<small><math>\mathrm{T}</math></small>) are just what they are in three-dimensional space, but double rotation (<small><math>\mathrm{Q}^2</math></small>) is something new and unprecedented in our physical experience, because double rotations cannot occur until you have four or more dimensions of space to rotate in. ...to readers who have not studied Coxeter (almost all readers including TAC), the blockquote above is "just math", not visualizable geometry...but I could describe Coxeter's congruent transformations in 4-space here geometrically: I could say clearly what they mean in spatial terms, in language anyone can understand, because they don't require any math to be understood; the "math" here is really just simple pictures (reflections and rotations); even double rotations can be visualized by dimensional analogy, as compounds of simple rotations...since even most physicists are unacquainted with Coxeter geometry, it might be useful to do this here... == Light propagates through 4-space at twice its apparent velocity ''c''== Coxeter's geometric laws of motion apply to all objects with mass in 4-dimensional Euclidean space, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <small><math>\mathrm{R}^4</math></small>, which may be termed a double translation <small><math>\mathrm{T}^2</math></small>, a pure translation via two pairs of parallel reflections, without any rotation component <small><math>\mathrm{Q}</math></small>. Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by <small><math>\mathrm{QT}</math></small>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <small><math>\mathrm{Q^2}</math></small>, an isoclinically rotating object such as an atom. A simple rotation <small><math>\mathrm{Q}</math></small> or simple translation <small><math>\mathrm{T}</math></small> is a double reflection <small><math>\mathrm{R^2}</math></small>, so a <small><math>\mathrm{QT}</math></small> or <small><math>\mathrm{Q^2}</math></small> is also an <small><math>\mathrm{R^4}</math></small>, but not with the same group of reflection angles as a light signal <small><math>\mathrm{R^4}</math></small>. A translation <small><math>\mathrm{T = R^2}</math></small> is a double reflection in two parallel planes, and a rotation <small><math>\mathrm{Q = R^2}</math></small> is a double reflection in two intersecting planes, as in a <small><math>\mathrm{QT = R^4}</math></small> which is both at once. A double translation <small><math>\mathrm{T^2 = R^4}</math></small> is two double reflections in pairs of parallel planes at once, a reflection in four or more non-intersecting parallel planes; it is all translation and no rotation. In a <small><math>\mathrm{T^2}</math></small> all the motion goes to translation, so the translation goes twice as far as the simple translation <small><math>\mathrm{T}</math></small> in a <small><math>\mathrm{QT}</math></small>. A double translation <small><math>\mathrm{T^2 = R^4}</math></small> is the opposite of a double rotation <small><math>\mathrm{Q^2 = R^4}</math></small>, which is stationary but rotates twice as fast as the simple rotation <small><math>\mathrm{Q}</math></small> in a <small><math>\mathrm{QT}</math></small>. The product of the two translations in a <small><math>\mathrm{T^2}</math></small> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <small><math>\mathrm{T}</math></small> over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A <small><math>\mathrm{T^2}</math></small> cannot reposition a 4-polytope the way a <small><math>\mathrm{QT}</math></small> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter. ...lensing of double translations <small><math>\mathrm{T^2 = R^4}</math></small> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet... == The Kepler problem is framed in Euclidean 4-space == The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], arguably the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the "inverse Kepler problem": the orbit characteristics require the force to depend on the inverse square of the distance.<ref>{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton & Company|isbn=978-0393039184}}</ref> The inverse square law behind the Kepler problem is the [[W:Central force|central force]] law which governs not only [[W:Newtonian gravity|Newtonian gravity]] and celestial orbits, but also the motion of two charged particles in [[W:Coulomb’s law|Coulomb’s law]] of [[W:Electrostatics|electrostatics]]; it applies to attractive or repulsive forces. Problems in which two bodies interact by a central force that varies as the [[W:Inverse square law|inverse square]] of the distance between them are called Kepler problems. Thus the [[W:Hydrogen atom|hydrogen atom]] is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law, another inverse-square central force. Using classical mechanics, the solution to a Kepler problem can be expressed as a [[W:Kepler orbit|Kepler orbit]] using six kinematical variables or [[W:Orbital elements|orbital elements]]. The solution conserves an orbital element called the [[W:Laplace–Runge–Lenz vector|Laplace–Runge–Lenz (LRL) vector]], a [[W:Constant of motion|constant of motion]], meaning that it is the same no matter where it is calculated on the orbit. The LRL vector was essential in the first quantum mechanical derivation of the [[W:Atomic emission spectrum|spectrum]] of the hydrogen atom, but this approach has rarely been used since the development of the [[W:Schrödinger equation|Schrödinger equation]]. The conservation of the LRL vector corresponds to the <small><math>SO(4)</math></small> symmetry, by Nother's theorem. The LRL vector lies orthogonal to both the orbital plane and the angular momentum vector of the Kepler orbit; we observe that it lies in a fourth orthogonal dimension. Fock in 1935<ref>V. Fock, Zur Theorie des Wasserstoffatoms, Zeitschrift für Physik. 98 (3-4) (1935), 145–154.</ref> and Moser in 1970<ref>J. Moser, Regularization of Kepler’s problem and the averaging method on a manifold, Commun. Pure Appl. 23 (1970), 609–636</ref> observed that the Kepler problem is mathematically equivalent to non-affine geodesic motion (a particle moving freely) on the surface of a 3-sphere, so that the whole problem is symmetric under certain rotations of the four-dimensional space. This higher-dimensional symmetry results in two well-known properties of the Kepler problem: the momentum vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points. ... Relativity establishes that an orbit in space is viewed in a different way in each distinct inertial reference frame. Depending on the choice of reference frame, the same Kepler system may be seen to be performing any one of a sequence of relativistically equivalent rotations in 4-space, on a continuum from an isoclinic rotation (Q²) in the orbit's proper reference frame, to a screw transfer (QT) with a simple rotation component (Q) and a translation component (T) at velocity <math>c</math>, in the universal reference frame of 4-coordinate space wherein every object is seen to be translating at velocity <math>c</math>. In reference frames between these two limit cases, the orbit is seen to be performing a double rotation (Q²) at two unequal, completely orthogonal angular rates of rotation: an elliptical double rotation. These include the reference frames of most typical observers, who are moving slowly relative to the observed orbital system's reference frame (their relative motion is a small fraction of the speed of light). ...this is probably misplaced here and should not interrupt the discussion at this point: ...These typical observations agree closely with the predictions of special relativity, because the non-isoclinic elliptical (Q²) resembles a (QT), since one of its two completely orthogonal rotations (Q) has such a long period that it is almost indistinguishable from a straight translation (T). All orbits in 4-space are isoclinic in their own reference frame. Orbiting objects in their own proper Kepler systems follow circular geodesic isoclines through 4-space. Orbits in 4-space are perfectly circular in their own reference frame, as Copernicus assumed the orbits of planets to be. It is the orbit's path through the 3-space of its elliptic hyperplane that is an ellipse, as Kepler found it to be. ...cite Jesper Goransson's very concise paper The geodesic circle that an orbiting object follows through 4-space in the proper reference frame of its own Kepler system is not a simple great circle which turns in two orthogonal dimensions. It is a helical great circle that turns in four orthogonal dimensions at once.{{Efn|Geodesic orbits in 4-space are not simple 2-dimensional great circles; they are helical 4-dimensional great circles that curve in all four dimensions at once. Their circular trajectories are helixes which we call ''isoclines'', since they are the paths taken by points on a rigid object undergoing isoclinic rotation.}} Such circles lie outside our physical experience, since our local space has only three orthogonal dimensions. Nonetheless we can visualize them in imagination, because their helical, circular shape is perfectly well defined by the kinematical variables of the Kepler orbit. The real physical correlates of abstract orthogonal planes and rotation angles are already familiar to us viscerally in our body-language of physical experience, since we are endowed biologically with highly evolved visual signal processing engines. These enable us to see and understand spatial relations and motions, including rotations, without even thinking about angles and orthogonal planes. This physical endowment is an inborn capacity for dimensional analogy which our biologic evolution has provided. All our instinctive spatial reasoning is by dimensional analogy from flat 2-dimensional retinal images to 3-dimensional scenes, using our powerful inborn visualization capacities of reverse stereographic projection and pattern recognition. We humans are thus very well equipped with everything we need to see in four-dimensional space, except experience. ... Recently Anco and Moghadam found that through Noether’s theorem in reverse, the LRL vector gives rise to a corresponding infinitesimal dynamical symmetry on the kinematical variables, which they show to be the semi-direct product of <small><math>SO(3)</math></small> and <small><math>\mathbb{R^3}</math></small>, in contrast to the <small><math>SO(4)</math></small> symmetry group generated by the LRL symmetries and the rotations.{{Sfn|Anco|Moghadam|2026|ps=; The physically relevant part of the LRL vector is its direction ... since its magnitude is just a function of energy and angular momentum.}} This remarkable symmetry breaking is expressive of the ''dimensional relativity'' between ordinary 3-space <small><math>\mathbb{R^3}</math></small>, spherical space <small><math>S^3</math></small> and Euclidean space <small><math>\mathbb{R^4}</math></small>. Consider a hydrogen atom in a Kepler orbit: for example, a hydrogen atom moving freely in space in an orbit around the sun. It is a ''double'' Kepler problem: an electrostatic Kepler problem within itself, and a gravitational Kepler problem in its environment. The ''single'' electrostatic Kepler problem of a hydrogen atom moving freely in space beyond any gravitational influence is a problem in special relativity. In our Euclidean 4-space model, this atom viewed as stationary in its own proper reference frame exhibits an <small><math>SO(4)</math></small> rotation symmetry corresponding to an isoclinic double rotation (<small><math>\mathrm{Q^2}</math></small>). The fourth dimension in this reference frame is the atom's proper time vector; it has constant velocity <math>c</math> and constant direction. From the point of view of our universal 4-coordinate space (which cannot be the proper inertial reference frame of any physical observer, all of whom are moving relative to it at velocity ''c''), the entire Kepler system (the atom) is translating through 4-space via a screw translation (<small><math>\mathrm{QT}</math></small>) at constant velocity <math>c</math>. From this viewpoint the atom has only a simple <small><math>SO(3)</math></small> rotation component (<small><math>\mathrm{Q}</math></small>), breaking its stationary <small><math>SO(4)</math></small> isoclinic rotation symmetry (<small><math>\mathrm{Q^2}</math></small>). Because each discrete part of the rotating atom moves along a helical trajectory through 4-space, the atom is in orbit around a barycentric axis (like a star in a galaxy), but only in a tiny orbit within its own radius, which is its inertial domain of rotation. The straight 4-dimensional cylinder it progresses along at velocity <math>c</math> is very narrow: only the diameter of the rotating atom itself. The gravitational Kepler problem of a hydrogen atom in a Kepler orbit around the sun is a problem in general relativity. In our 4-space model, this atom viewed in its own proper reference frame exhibits the same <small><math>SO(4)</math></small> rotation symmetry as it did in the electrostatic Kepler problem where the atom was translating linearly through space. The Kepler system in this case is not just the atom; it is the entire solar system. The LRL vector of this Kepler system is the proper time vector of the atom's inertial reference frame; once again it has constant velocity ''and constant direction''. Although the momentum vector moves in a perfect circle as the atom orbits the sun, the 4-space LRL vector does not move at all: it is a constant of motion, of linear motion (<small><math>\mathrm{T}</math></small>) of the Kepler system (the entire solar system in this case) in a constant 4-space direction, the proper time direction of the system. The direction of the system's proper time vector would vary under some kinds of acceleration of the atom, but it is constant under this kind of orbital acceleration. It continues to point in the same direction, like a 4-space compass needle, as the atom winds its way along its spiral path around the axis of the sun's straight-line translation through 4-space at velocity <math>c</math>. This compass needle always points in the direction the sun is moving, not the direction the atom is moving at any instant. ...Its Kepler orbit around the sun is its <small><math>SO(3)</math></small> rotation component (<small><math>\mathrm{Q}</math></small>). Although the atom is moving on a geodesic circle in the second problem, by the [[equivalence principle]] the difference in the state of the atomic systems in these two problems cannot be observed by examining the atoms alone. Even from another inertial reference frame, where the atom in the second problem is seen to be translating through 4-space via a wide screw translation (<small><math>\mathrm{QT}</math></small>) around the sun's axis of motion, there is still no difference between the two problems which can be detected by examining only the atoms within their own proper reference frames (even over time), because the LRL vector (<small><math>\mathrm{T}</math></small>) is a constant of motion of the entire system in both cases. ...Anco and Maghadam found that <small><math>SO(4)</math></small>) breaks to ... <small><math>S^3</math></small>)... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... <small><math>H^3</math></small>) ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). ... Finally we consider a third problem in which a hydrogen atom enters the solar system as a comet, loops around the sun and exits the solar system again. This atom... ... As Hamilton found when he discovered the quaternions, we see that it is necessary to admit a fourth dimension to the system in order to properly model the problem: in Hamilton's case the general problem of ..., and in our case the Kepler problem. These are instances of the same problem in 4-dimensional Euclidean geometry, and indeed a solution to the Kepler problem in quaternions (the four Cartesian coordinates of Euclidean 4-space) is a solution to it in our model of the 4-coordinate Euclidean cosmos. == Distribution of stars in our galaxy == The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by its red shift, and by assuming that they are distributed in three dimensions of space, we have plotted their locations in 3-space. If we abandon the last of those three assumptions, we can just as easily reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits as we saw them in 3-space? That would be an expected consequence of the special rotational symmetry group of 4-space <small><math>SO(4)</math></small>, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits. ...have to perform this experiment somehow, at least as a conclusive thought experiment, before I publish this paper... == Rotations == The [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotations]] of the convex [[W:regular 4-polytope|regular 4-polytope]]s are usually described as discrete rotations of a rigid object. For example, the rigid [[24-cell]] can rotate in a [[24-cell#Great hexagons|hexagonal]] (6-vertex) central [[24-cell#Planes of rotation|plane of rotation]]. A 4-dimensional [[24-cell#Isoclinic rotations|''isoclinic'' rotation]] (as distinct from a [[24-cell#Simple rotations|''simple'' rotation]] like the ones that occur in 3-dimensional space) is a ''diagonal'' rotation in multiple [[W:Clifford parallel|Clifford parallel]] [[24-cell#Geodesics|central planes]] of rotation at once. It is diagonal because it is a [[W:SO(4)#Double rotations|double rotation]]: in addition to rotating in parallel (like wheels), the multiple planes of rotation also tilt sideways in the completely orthogonal plane of rotation (like coins flipping) into each other's planes. Consequently, the path taken by each vertex is a [[24-cell#Helical hexagrams and their isoclines|twisted helical circle]], rather than the ordinary flat great circle a vertex follows in a simple rotation. In a rigid 4-polytope rotating isoclinically, ''all'' the vertices lie in one of the parallel planes of rotation, so all the vertices move in parallel along Clifford parallel twisting circular paths. [[24-cell#Clifford parallel polytopes|Clifford parallel planes]] are not parallel in the normal sense of parallel planes in three dimensions; the vertices are all moving in different directions around the [[W:3-sphere|3-sphere]]. In one complete 360° isoclinic revolution, a rigid 4-polytope turns itself inside out. This is sufficiently different from the simple rotations of rigid bodies in our 3-dimensional experience that a [[24-cell#Rotations|detailed description]] enabling the reader to properly visualize its counter-intuitive consequences runs to many pages and illustrations, with many accompanying pages of explanatory notes on surprising phenomena that arise in 4-dimensional space: [[24-cell#Great squares|completely orthogonal planes]], [[24-cell#Clifford parallel polytopes|Clifford parallelism]]{{Efn|name=Clifford parallels}} and [[W:Hopf fibration|Hopf fiber bundles]], [[24-cell#Isoclinic rotations|isoclinic geodesic paths]], and [[24-cell#Double rotations|chiral (mirror image) pairs of rotations]], among other complexities. Moreover, the characteristic rotations of the various regular 4-polytopes are all different; each is a unique surprise. [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|The 6 regular convex 4-polytopes]] have different numbers of vertices (5, 8, 16, 24, 120 and 600 respectively) and those with fewer vertices occur inscribed in those with more vertices (with one exception), with the result that the more complex 4-polytopes subsume the kinds of rotations characteristic of their less complex predecessors, as well as each having a characteristic kind of rotation not found in their predecessors. None of these symmetries is to be found in 3-dimensional space, although their simpler 3-dimensional analogues are all present there. [[W:Euclidean geometry#Higher dimensions|Four dimensional Euclidean space]] is more complicated (and more interesting) than three dimensional space because there is more room in it, in which unprecedented things can happen. It subsumes 3-dimensional space, with all of the symmetries we are accustomed to, and adds astonishing new surprises. These are hard for us to visualize, because the only way we can experience them is in our imagination; we have no body of sensory experience in 4-dimensional space to draw upon, other than our evolution in time. For that reason (our difficulty in visualizing them), descriptions of isoclinic rotations usually begin and end with rigid rotations: [[24-cell#Isoclinic rotations|for example]], all 24 vertices of a single rigid 24-cell rotating in unison, with 6 vertices evenly spaced around each of 4 Clifford parallel twisted circles.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} But that is only the simplest case, which is easiest for us to understand. Compound and [[W:Kinematics|kinematic]] 24-cells (with moving parts) are even more interesting (and more complicated) than the rotation of a single rigid 24-cell. To begin with, when we examine the individual parts of a single rigid 24-cell that are moving in an isoclinic rotation, such as the orbits of individual vertices, we can imagine a case where fewer than 24 point-objects are orbiting on those twisted circular paths at once. [[24-cell#Reflections|For example]], if we imagine just 8 point-objects, evenly spaced around the 24-cell at [[24-cell#Reciprocal constructions from 8-cell and 16-cell|the 8 vertices that lie on the 4 coordinate axes]], and rotate them isoclinically along exactly the same orbits they would take in the above-mentioned rotation of a rigid 24-cell, then in the course of a single 360° rotation the 8 point-objects will trace out the whole 24-cell, with just one point-object reaching each of the 24 vertex positions just once, and no point-object colliding with (or even crossing the path of) any other at any time. This is an example of a discrete Hopf fibration. But it is still an example of a rigid object in a discrete isoclinic rotation: a rigid 8-vertex object (called the 4-[[W:orthoplex|orthoplex]] or [[16-cell]]) performing one half of the characteristic rotation of the 24-cell. We can also imagine ''combining'' distinct isoclinic rotations. What happens when multiple point-objects are orbiting at once, but do ''not'' all follow the Clifford parallel paths characteristic of the ''same'' distinct rigid rotation? What happens when we combine orbits from distinct rotations characteristic of different 4-polytopes, for example when different rigid 4-polytopes are concentric and rotating simultaneously in their characteristic ways? What kinds of such hybrid rotations are possible in the same 3-sphere shell without collisions? In adjacent concentric shells without asymmetric imbalance? What sort of [[Kinematics of the cuboctahedron|kinematic polytopes]] do they trace out, and how do their [[24-cell#Clifford parallel polytopes|component parts]] relate to each other as they move? Is there (sometimes) some kind of mutual stability amid their lack of combined rigidity? Visualizing isoclinic rotations (rigid and otherwise) allows us to explore such questions of [[W:kinematics|kinematics]], and where dynamic stabilities arise, of [[wikipedia:kinetics (physics)|kinetics]]. In four dimensions, we discover that space has more room in it than we have experienced, which permits previously unimagined motions. Even 3-space is more commodious than we thought; when it is curved and lies embedded in a higher-dimensional space, it permits previously impossible symmetric packings. Sadoc studied double-twisted 3-dimensional molecules, and imagined them embedded in 4-dimensional space as the Hopf fibrations of regular 4-polytopes. He found that these molecules would close-pack on the 3-sphere perfectly without exhibiting any torsion, although their packing in ordinary flat 3-space is imperfect, "frustrated" by their twisted geometry. <blockquote>The frustration, which arises when the molecular orientation is transported along the two [spiral] AB paths of figure 1 [double twist helix], is imposed by the very topological nature of the Euclidean space R³. It would not occur if the molecules were embedded in the non-Euclidean space of the [[W:3-sphere|3-sphere]] S³, or hypersphere. This space with a homogeneous positive curvature can indeed be described by equidistant and uniformly twisted fibers, along which the molecules can be aligned without any conflict between compactness and [[W:torsion of a curve|torsion]].... The fibres of this [[W:Hopf fibration|Hopf fibration]] are great circles of S³, the whole family of which is also called the [[W:Clifford parallel|Clifford parallel]]s.{{Efn|name=Clifford parallels}} Two of these fibers are C_∞ symmetry axes for the whole fibration; each fibre makes one turn around each axis and regularly rotates when moving from one axis to another.{{Efn|name=helical geodesic}} These fibers build a double twist configuration while staying parallel, i.e. without any frustration, in the whole volume of S³.{{Efn|name=Petrie polygon of a honeycomb}} They can therefore be used as models to study the condensation of long molecules in the presence of a double twist constraint.{{Sfn|Sadoc & Charvolin|2009|loc=§1.2 The curved space approach|ps=; studies the helical orientation of molecules in crystal structures and their imperfect packings ("frustrations") in 3-dimensional space.}}</blockquote> Of course we do not find molecules condensing to close-pack the 3-sphere in our experience, and Sadoc does not say that we do. We find 3-spheres in the atomic realm (if atoms are 4-polytopes), and in the cosmic realm (as the surface boundaries of stars, and the concentric surfaces of galaxies). But in between, in the realm of ordinary experience which includes the molecular realm, ourselves and all the objects we can materially handle or observe up close including the planets, we are confined together by gravity as inertia within a curved 3-dimensional space that is no more than one atom thick in the fourth spatial dimension. That is why in the molecular realm we find only objects that occupy 3-spaces which, though infinitesimally curved in the fourth dimension, are tiny patches on whole 3-spheres of galactic size. So Sadoc's exercise is a thought experiment, like Einstein's gedankenexperiments about railroad embankments and trains moving at nearly the speed of light. It is no less illuminating, despite the symmetry it reveals not having a realization as an actual 3-sphere of actual molecules. And might not something very like it have an actual realization in the atomic realm? We know that atoms have their own complex internal structure, which we are unable to model geometrically in ordinary 3-dimensional space. Suppose such a model is impossible because an atom is actually a 4-polytope occupying a tiny spherical region of 4-dimensional space, and so we only find its constituent particles in close-packed helical orbits on the 3-sphere, in the manner of Sadoc's imaginary twisted molecules, but as real 4-dimensional helices of atomic scale. We would expect to find the atomic orbit of a fundamental particle in some discrete Hopf fibration characteristic of a symmetry group, that is, on the maximally symmetric isoclines of a discrete isoclinic rotation characteristic of some regular 4-polytope and the particle. == A theory of the Euclidean atom == <blockquote>Because quantum physics could be tested without being understood, it allowed humans to see how the universe worked without knowing why.<ref>Sebastian Junger, In My Time of Dying</ref></blockquote> ... == Light and Mass are Reflection and Rotation == The phenomena of light and mass are expressions of reflection symmetries and rotation symmetries, respectively. ... Atoms are 4-polytopes, elementary objects with SO(4) rotational symmetry. Light is .... Motion in space is the propagation of the elementary objects of light and matter in Coxeter congruent transformations by kaleidoscopic self-reflections, like the motion of self-reproducing cellular automata in [[Conway's Game of Life|Conway's game of life]]. ... Light is discrete reflections. Mass is discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics. === Atoms are 4-polytopes === ... == Relativity in real space of four or more orthogonal dimensions == Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions. General relativity is Galilean relativity in a general space of four or more orthogonal dimensions, e.g. in Euclidean 4-space <math>R^4</math>, spherical 4-space <math>S^4</math>, and any orthogonal 4-manifold. Light is a consequence of symmetry group reflections at quantum scale. Gravity and the other fundamental forces are consequences of rotations, which are consequences of quantum reflections. Light is discrete reflections. Gravity and all forces are discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics. Every observer may properly see themself as stationary and the universe as an ''n''-sphere with themself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and can be measured by the observer as the speed of light. === Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions === ...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here... Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.) None of these Lorentz effects are beyond geometric explanation or paradoxical. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, successive Euclidean spaces are dimensionally analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries: that is Nother's great discovery. === General relativity is Galilean relativity in a general space of four orthogonal dimensions === ... == Dimensional relativity == Coxeter's kinetic law of <math>n</math>-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean <math>n</math>-spaces of increasing <math>n</math>, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating <math>n</math>-spaces. By the degree of dimensional analogy of which they are capable, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: "The universe is a sphere whose center is wherever there is intelligence."{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}} == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity ''c'', with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may, or may not, all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed us that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a sense related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system. When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw displacements), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw displacement has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal of that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. Anco and Maghadam found that <small><math>SO(4)</math></small> breaks to ... <small><math>S^3</math></small>... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... <small><math>H^3</math></small> ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). Because the planets orbit on ellipses in our 3-space, Euclidean 4-space is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the truer model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position of centrality as our exclusive conception of our place in space. ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others. The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.<ref>{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}</ref>}} == Boundaries == <blockquote>Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? ''What is the nature of the boundary which confines us to just three dimensions?'' We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined motions and objects. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. A boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three .... In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it. We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed "Dimensional Analogy."<ref>{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's and Coxeter's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover in imagination and then explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :We shall not cease from exploration :And the end of all our exploring :Will be to arrive where we started :And know the place for the first time. :Through the unknown, remembered gate :When the last of earth left to discover :Is that which was the beginning; :At the source of the longest river :The voice of the hidden waterfall :And the children in the apple-tree :Not known, because not looked for :But heard, half-heard, in the stillness :Between two waves of the sea. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. This project is forever beginning anew. Coxeter showed us that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether showed us all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions, in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::BEECH :Where my imaginary line :Bends square in woods, an iron spine :And pile of real rocks have been founded. :And off this corner in the wild, :Where these are driven in and piled, :One tree, by being deeply wounded, :Has been impressed as Witness Tree :And made commit to memory :My proof of being not unbounded. :Thus truth's established and borne out, :Though circumstanced with dark and doubt— :Though by a world of doubt surrounded. :::::::—''The Moodie Forester''<ref>{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|The long radius (center to vertex) of the 24-cell is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including the four-dimensional 24-cell and [[W:Tesseract#Radial equilateral symmetry|tesseract]], the three-dimensional [[W:Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[W:Hexagon#Regular hexagon|hexagon]]. (The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron.) '''Radially equilateral''' polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.|name=radially equilateral|group=}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|5}}(–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>){{spaces|3}}(–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} {{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}} {{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}} {{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles]."}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}} {{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}} {{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}} {{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}} {{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} {{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} {{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} {{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}} {{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} {{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90^o bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60^o bend, or as a straight line with one 60^o bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} {{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)"}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} {{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}} {{Efn|Departing from any vertex V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V₀ and V₂ are ''two'' {{radic|3}} chords apart{{Efn|V₀ and V₂ are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V₀ and V₂ are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹₄.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} c0fa2u3xijcj98evuwmk58o8mt9em8d 2 296524 2818517 2777994 2026-07-19T11:31:35Z Solstag 64708 partial 2026 2818517 wikitext text/x-wiki === 2026 === [[User:Solstag/Educators and researchers, get on with Wikiversity!|Educators and researchers, get on with Wikiversity!]] [https://solstag.gitlab.io/presentations/cortext-lisis-2026-07/ Analyses socio-sémantiques de corpus] pour une [[:fr:Cortext/Formations/2026-07-08+09_LISIS|formation Cortext le 08 juillet au LISIS]] === 2025 === [https://solstag.gitlab.io/presentations/anf-istex-2025/ Ma présentation] @ Action Nationale de Formation « Corpus Istex » [https://solstag.gitlab.io/presentations/waysout2025/ Exploring the research concern of destabilisation beyond the domain of transitions studies] @ Ways-Out Project Conclusion Meeting [https://solstag.gitlab.io/presentations/migpsyhome2025/ My presentation] @ Launch event for the project Home is Where the Wi-Fi Connects: Migration, Digital Media and Psychological Home === 2024 === [https://solstag.gitlab.io/presentations/easst-4s-2024/ The Political Shaping of Argentine Pesticide Science] @ EASST-4S 2024 [https://sdrive.cnrs.fr/s/tp9MZH9F4s8ncb6 Transformando uma infraestrutura de pesquisa para ciências sociais madura em um projeto de software livre] @ Grupo de Sistemas de Software, IME-USP === 2023 === [https://solstag.gitlab.io/presentations/waysout2023/ Explorations into the existence of DDD as a research problem beyond the domain of transitions studies] @ Ways-Out Project Meeting [https://solstag.gitlab.io/presentations/netsci2023/ A mixed-methods approach to study socio-semantic networks] (poster) @ NetSci 2023 [https://solstag.gitlab.io/presentations/sunbelt2023/ SASHIMI and new frontiers in the study of socio-semantic networks with mixed-methods on the Cortext Platform] @ Sunbelt 2023 [https://md.cortext.net/p/0p14WRE0N#/ An empirical assessment of Open Hardware practices in the academic community and their policy implications] @ Eu-SPRI Annual Conference [https://md.cortext.net/p/BhezPIOW7#/ So many ways out! Explorations into the existence of destabilisation as a research problem beyond the domain of transitions studies] @ Eu-SPRI Annual Conference '''[fr]''' [https://md.cortext.net/p/0VC3BMtR9#/ Une approche de méthodes mixtes pour l'étude de réseaux socio-sémantiques] @ Frognet 2023 === 2022 === [https://md.cortext.net/p/6E1odeLTA#/ So many ways out! Explorations into the existence of destabilisation as a research problem beyond the domain of transitions studies] @ 13th International Sustainability Transitions Conference – IST 2022 '''[fr]''' [https://md.cortext.net/p/OAhx_MR7d#/ Étudier les modes d’existence de la problématique de la déstabilisation dans la recherche en sciences sociales par l’analyse d’un corpus bibliographique] @ Colloque FRACTEXT [https://md.cortext.net/p/p43FrnlAo#/ An Empirical Assessment of the Evolution of Open Hardware Documentation Practices in the Academic Community] @ 4S / ESOCITE 2022 [https://md.cortext.net/p/J43WsiUsq#/ Thoughts on Computational Environmental STS] @ CESTS Workshop '''[pt]''' [https://md.cortext.net/p/sZenPJNfd#/ Métodos mistos no estudo da regulação da inteligência artificial, da desestabilização de sistemas sociotécnicos, e de práticas de ciência aberta] @ Grupo TransObjeto, PUC-SP [https://md.cortext.net/p/eitEtnxV-#/ Mapping destabilisation as a research problem across scholarly domains] @ WAYS-OUT Project Seminar === 2021 === '''[fr]''' [https://solstag.gitlab.io/presentations/20211208-netconf/ Cartographie d’un domaine émergent dans une grande conférence en oncologie] @ Congrès scientifiques et réseaux savants - Journée d'étude NetConf [https://docs.google.com/presentation/d/1knOO-BgE9HEMyGbVrLf-LxyezoUsoNfwsfTcnPGnLNA/edit?usp=sharing Time to open the tools of research: Policy recommendations from the Gathering for Open Science Hardware] @ EU Open Source Policy Community Meeting 2021, OpenForum Europe [https://solstag.gitlab.io/presentations/lecsinvite5/ Makers & Open Science : common roots, field actions, new agencements] @ Le CS Invite #5, Réseaux Français des Fablabs [https://solstag.gitlab.io/presentations/foruminnovation2021/ Producing a policy brief on open science hardware to engage technology transfer offices: an account from the Gathering for Open Science Hardware] @ RNI Forum Innovation 2021 === 2019 === '''[fr]''' [[:fr:Recherche:Les_projets_Wikimédia_:_un_environnement_de_recherche_pour_amateurs_&_scientifiques#Solstag_-_Croisements_Wikiversit%C3%A9,_universit%C3%A9_et_autres_:_exp%C3%A9riences_en_France_et_au_Br%C3%A9sil|Croisements Wikiversité, université et autres : expériences en France et au Brésil]] @ WikiConvention francophone 2019 === 2017 === [[User:Solstag/Wikiversities: get over the content, serve the flow|Wikiversities: get over the content, serve the flow]] @ Wikimania 2017 (see [[wm2017:Submissions/Wikiversities:_get_over_the_content,_serve_the_flow|proposal]]) Capturing Oncology Dynamics from Textual Content of Conference Abstracts: Word Embedding and Stochastic Block Models @ [https://www.ixxi.fr/agenda/seminaires/understanding-the-dynamics-of-science-an-interdisciplinary-workshop Understanding the dynamics of science, an interdisciplinary workshop, IXXI ENS Lyon] ==== ''Past presentations not included'' ==== rvx4wbbmbbewi6mdksdbn7eitxid39p 2818518 2818517 2026-07-19T11:33:10Z Solstag 64708 2818518 wikitext text/x-wiki === 2026 === [[User:Solstag/Educators and researchers, get on with Wikiversity!|Educators and researchers, get on with Wikiversity!]] [https://solstag.gitlab.io/presentations/cortext-lisis-2026-07/ Analyses socio-sémantiques de corpus] for a [[:fr:Cortext/Formations/2026-07-08+09_LISIS|Cortext training session at LISIS]] [https://solstag.gitlab.io/presentations/cortext-usp-2026/ Análises do texto] for a [[:pt:Cortext/Treinamentos/2026-04-27_UFBA|Cortext training session at the University of São Paulo]] === 2025 === [https://solstag.gitlab.io/presentations/anf-istex-2025/ Ma présentation] @ Action Nationale de Formation « Corpus Istex » [https://solstag.gitlab.io/presentations/waysout2025/ Exploring the research concern of destabilisation beyond the domain of transitions studies] @ Ways-Out Project Conclusion Meeting [https://solstag.gitlab.io/presentations/migpsyhome2025/ My presentation] @ Launch event for the project Home is Where the Wi-Fi Connects: Migration, Digital Media and Psychological Home === 2024 === [https://solstag.gitlab.io/presentations/easst-4s-2024/ The Political Shaping of Argentine Pesticide Science] @ EASST-4S 2024 [https://sdrive.cnrs.fr/s/tp9MZH9F4s8ncb6 Transformando uma infraestrutura de pesquisa para ciências sociais madura em um projeto de software livre] @ Grupo de Sistemas de Software, IME-USP === 2023 === [https://solstag.gitlab.io/presentations/waysout2023/ Explorations into the existence of DDD as a research problem beyond the domain of transitions studies] @ Ways-Out Project Meeting [https://solstag.gitlab.io/presentations/netsci2023/ A mixed-methods approach to study socio-semantic networks] (poster) @ NetSci 2023 [https://solstag.gitlab.io/presentations/sunbelt2023/ SASHIMI and new frontiers in the study of socio-semantic networks with mixed-methods on the Cortext Platform] @ Sunbelt 2023 [https://md.cortext.net/p/0p14WRE0N#/ An empirical assessment of Open Hardware practices in the academic community and their policy implications] @ Eu-SPRI Annual Conference [https://md.cortext.net/p/BhezPIOW7#/ So many ways out! Explorations into the existence of destabilisation as a research problem beyond the domain of transitions studies] @ Eu-SPRI Annual Conference '''[fr]''' [https://md.cortext.net/p/0VC3BMtR9#/ Une approche de méthodes mixtes pour l'étude de réseaux socio-sémantiques] @ Frognet 2023 === 2022 === [https://md.cortext.net/p/6E1odeLTA#/ So many ways out! Explorations into the existence of destabilisation as a research problem beyond the domain of transitions studies] @ 13th International Sustainability Transitions Conference – IST 2022 '''[fr]''' [https://md.cortext.net/p/OAhx_MR7d#/ Étudier les modes d’existence de la problématique de la déstabilisation dans la recherche en sciences sociales par l’analyse d’un corpus bibliographique] @ Colloque FRACTEXT [https://md.cortext.net/p/p43FrnlAo#/ An Empirical Assessment of the Evolution of Open Hardware Documentation Practices in the Academic Community] @ 4S / ESOCITE 2022 [https://md.cortext.net/p/J43WsiUsq#/ Thoughts on Computational Environmental STS] @ CESTS Workshop '''[pt]''' [https://md.cortext.net/p/sZenPJNfd#/ Métodos mistos no estudo da regulação da inteligência artificial, da desestabilização de sistemas sociotécnicos, e de práticas de ciência aberta] @ Grupo TransObjeto, PUC-SP [https://md.cortext.net/p/eitEtnxV-#/ Mapping destabilisation as a research problem across scholarly domains] @ WAYS-OUT Project Seminar === 2021 === '''[fr]''' [https://solstag.gitlab.io/presentations/20211208-netconf/ Cartographie d’un domaine émergent dans une grande conférence en oncologie] @ Congrès scientifiques et réseaux savants - Journée d'étude NetConf [https://docs.google.com/presentation/d/1knOO-BgE9HEMyGbVrLf-LxyezoUsoNfwsfTcnPGnLNA/edit?usp=sharing Time to open the tools of research: Policy recommendations from the Gathering for Open Science Hardware] @ EU Open Source Policy Community Meeting 2021, OpenForum Europe [https://solstag.gitlab.io/presentations/lecsinvite5/ Makers & Open Science : common roots, field actions, new agencements] @ Le CS Invite #5, Réseaux Français des Fablabs [https://solstag.gitlab.io/presentations/foruminnovation2021/ Producing a policy brief on open science hardware to engage technology transfer offices: an account from the Gathering for Open Science Hardware] @ RNI Forum Innovation 2021 === 2019 === '''[fr]''' [[:fr:Recherche:Les_projets_Wikimédia_:_un_environnement_de_recherche_pour_amateurs_&_scientifiques#Solstag_-_Croisements_Wikiversit%C3%A9,_universit%C3%A9_et_autres_:_exp%C3%A9riences_en_France_et_au_Br%C3%A9sil|Croisements Wikiversité, université et autres : expériences en France et au Brésil]] @ WikiConvention francophone 2019 === 2017 === [[User:Solstag/Wikiversities: get over the content, serve the flow|Wikiversities: get over the content, serve the flow]] @ Wikimania 2017 (see [[wm2017:Submissions/Wikiversities:_get_over_the_content,_serve_the_flow|proposal]]) Capturing Oncology Dynamics from Textual Content of Conference Abstracts: Word Embedding and Stochastic Block Models @ [https://www.ixxi.fr/agenda/seminaires/understanding-the-dynamics-of-science-an-interdisciplinary-workshop Understanding the dynamics of science, an interdisciplinary workshop, IXXI ENS Lyon] ==== ''Past presentations not included'' ==== kd40mapsxhydckyhlo7ez2x9j86xye8 2818519 2818518 2026-07-19T11:34:00Z Solstag 64708 corrige link 2818519 wikitext text/x-wiki === 2026 === [[User:Solstag/Educators and researchers, get on with Wikiversity!|Educators and researchers, get on with Wikiversity!]] [https://solstag.gitlab.io/presentations/cortext-lisis-2026-07/ Analyses socio-sémantiques de corpus] for a [[:fr:Cortext/Formations/2026-07-08+09_LISIS|Cortext training session at LISIS]] [https://solstag.gitlab.io/presentations/cortext-usp-2026/ Análises do texto] for a [[:pt:Cortext/Treinamentos/2026-02-12_USP|Cortext training session at the University of São Paulo]] === 2025 === [https://solstag.gitlab.io/presentations/anf-istex-2025/ Ma présentation] @ Action Nationale de Formation « Corpus Istex » [https://solstag.gitlab.io/presentations/waysout2025/ Exploring the research concern of destabilisation beyond the domain of transitions studies] @ Ways-Out Project Conclusion Meeting [https://solstag.gitlab.io/presentations/migpsyhome2025/ My presentation] @ Launch event for the project Home is Where the Wi-Fi Connects: Migration, Digital Media and Psychological Home === 2024 === [https://solstag.gitlab.io/presentations/easst-4s-2024/ The Political Shaping of Argentine Pesticide Science] @ EASST-4S 2024 [https://sdrive.cnrs.fr/s/tp9MZH9F4s8ncb6 Transformando uma infraestrutura de pesquisa para ciências sociais madura em um projeto de software livre] @ Grupo de Sistemas de Software, IME-USP === 2023 === [https://solstag.gitlab.io/presentations/waysout2023/ Explorations into the existence of DDD as a research problem beyond the domain of transitions studies] @ Ways-Out Project Meeting [https://solstag.gitlab.io/presentations/netsci2023/ A mixed-methods approach to study socio-semantic networks] (poster) @ NetSci 2023 [https://solstag.gitlab.io/presentations/sunbelt2023/ SASHIMI and new frontiers in the study of socio-semantic networks with mixed-methods on the Cortext Platform] @ Sunbelt 2023 [https://md.cortext.net/p/0p14WRE0N#/ An empirical assessment of Open Hardware practices in the academic community and their policy implications] @ Eu-SPRI Annual Conference [https://md.cortext.net/p/BhezPIOW7#/ So many ways out! Explorations into the existence of destabilisation as a research problem beyond the domain of transitions studies] @ Eu-SPRI Annual Conference '''[fr]''' [https://md.cortext.net/p/0VC3BMtR9#/ Une approche de méthodes mixtes pour l'étude de réseaux socio-sémantiques] @ Frognet 2023 === 2022 === [https://md.cortext.net/p/6E1odeLTA#/ So many ways out! Explorations into the existence of destabilisation as a research problem beyond the domain of transitions studies] @ 13th International Sustainability Transitions Conference – IST 2022 '''[fr]''' [https://md.cortext.net/p/OAhx_MR7d#/ Étudier les modes d’existence de la problématique de la déstabilisation dans la recherche en sciences sociales par l’analyse d’un corpus bibliographique] @ Colloque FRACTEXT [https://md.cortext.net/p/p43FrnlAo#/ An Empirical Assessment of the Evolution of Open Hardware Documentation Practices in the Academic Community] @ 4S / ESOCITE 2022 [https://md.cortext.net/p/J43WsiUsq#/ Thoughts on Computational Environmental STS] @ CESTS Workshop '''[pt]''' [https://md.cortext.net/p/sZenPJNfd#/ Métodos mistos no estudo da regulação da inteligência artificial, da desestabilização de sistemas sociotécnicos, e de práticas de ciência aberta] @ Grupo TransObjeto, PUC-SP [https://md.cortext.net/p/eitEtnxV-#/ Mapping destabilisation as a research problem across scholarly domains] @ WAYS-OUT Project Seminar === 2021 === '''[fr]''' [https://solstag.gitlab.io/presentations/20211208-netconf/ Cartographie d’un domaine émergent dans une grande conférence en oncologie] @ Congrès scientifiques et réseaux savants - Journée d'étude NetConf [https://docs.google.com/presentation/d/1knOO-BgE9HEMyGbVrLf-LxyezoUsoNfwsfTcnPGnLNA/edit?usp=sharing Time to open the tools of research: Policy recommendations from the Gathering for Open Science Hardware] @ EU Open Source Policy Community Meeting 2021, OpenForum Europe [https://solstag.gitlab.io/presentations/lecsinvite5/ Makers & Open Science : common roots, field actions, new agencements] @ Le CS Invite #5, Réseaux Français des Fablabs [https://solstag.gitlab.io/presentations/foruminnovation2021/ Producing a policy brief on open science hardware to engage technology transfer offices: an account from the Gathering for Open Science Hardware] @ RNI Forum Innovation 2021 === 2019 === '''[fr]''' [[:fr:Recherche:Les_projets_Wikimédia_:_un_environnement_de_recherche_pour_amateurs_&_scientifiques#Solstag_-_Croisements_Wikiversit%C3%A9,_universit%C3%A9_et_autres_:_exp%C3%A9riences_en_France_et_au_Br%C3%A9sil|Croisements Wikiversité, université et autres : expériences en France et au Brésil]] @ WikiConvention francophone 2019 === 2017 === [[User:Solstag/Wikiversities: get over the content, serve the flow|Wikiversities: get over the content, serve the flow]] @ Wikimania 2017 (see [[wm2017:Submissions/Wikiversities:_get_over_the_content,_serve_the_flow|proposal]]) Capturing Oncology Dynamics from Textual Content of Conference Abstracts: Word Embedding and Stochastic Block Models @ [https://www.ixxi.fr/agenda/seminaires/understanding-the-dynamics-of-science-an-interdisciplinary-workshop Understanding the dynamics of science, an interdisciplinary workshop, IXXI ENS Lyon] ==== ''Past presentations not included'' ==== 7kw4tm1jrcrwe5fe53y00vo9n61ahkw 0 305362 2818494 2812204 2026-07-18T21:23:33Z Dc.samizdat 2856930 /* Chiral symmetry operations */ 2818494 wikitext text/x-wiki {{Short description|Regular object in four dimensional geometry}} {{Polyscheme|radius=an '''expanded version''' of|active=is the focus of active research}} {{Infobox 4-polytope | Name=24-cell | Image_File=Schlegel wireframe 24-cell.png | Image_Caption=[[W:Schlegel diagram|Schlegel diagram]]<br>(vertices and edges) | Type=[[W:Convex regular 4-polytope|Convex regular 4-polytope]] | Last=[[W:Omnitruncated tesseract|21]] | Index=22 | Next=[[W:Rectified 24-cell|23]] | Schläfli={3,4,3}<br>r{3,3,4} = <math>\left\{\begin{array}{l}3\\3,4\end{array}\right\}</math><br>{3^1,1,1} = <math>\left\{\begin{array}{l}3\\3\\3\end{array}\right\}</math> | CD={{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}<br>{{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}} or {{Coxeter–Dynkin diagram|node_1|split1|nodes|4a|nodea}}<br>{{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}} or {{Coxeter–Dynkin diagram|node_1|splitsplit1|branch3|node}} | Cell_List=24 [[W:Octahedron|{3,4}]] [[File:Octahedron.png|20px]] | Face_List=96 [[W:Triangle|{3}]] | Edge_Count=96 | Vertex_Count= 24 | Petrie_Polygon=[[W:Dodecagon|{12}]] | Coxeter_Group=[[W:F4 (mathematics)|F₄]], [3,4,3], order 1152<br>B₄, [4,3,3], order 384<br>D₄, [3^1,1,1], order 192 | Vertex_Figure=[[W:Cube|cube]] | Dual=[[W:Polytope#Self-dual polytopes|self-dual]] | Property_List=[[W:Convex polytope|convex]], [[W:Isogonal figure|isogonal]], [[W:Isotoxal figure|isotoxal]], [[W:Isohedral figure|isohedral]] }} [[File:24-cell net.png|thumb|right|[[W:Net (polyhedron)|Net]]]] In [[W:four-dimensional space|four-dimensional geometry]], the '''24-cell''' is the convex [[W:Regular 4-polytope|regular 4-polytope]]{{Sfn|Coxeter|1973|p=118|loc=Chapter VII: Ordinary Polytopes in Higher Space}} (four-dimensional analogue of a [[W:Platonic solid|Platonic solid]]]) with [[W:Schläfli symbol|Schläfli symbol]] {3,4,3}. It is also called '''C₂₄''', or the '''icositetrachoron''',{{Sfn|Johnson|2018|p=249|loc=11.5}} '''octaplex''' (short for "octahedral complex"), '''icosatetrahedroid''',{{sfn|Ghyka|1977|p=68}} '''[[W:Octacube (sculpture)|octacube]]''', '''hyper-diamond''' or '''polyoctahedron''', being constructed of [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]]. The boundary of the 24-cell is composed of 24 [[W:Octahedron|octahedral]] cells with six meeting at each vertex, and three at each edge. Together they have 96 triangular faces, 96 edges, and 24 vertices. The [[W:Vertex figure|vertex figure]] is a [[W:Cube|cube]]. The 24-cell is [[W:Self-dual polyhedron|self-dual]].{{Efn|The 24-cell is one of only three self-dual regular Euclidean polytopes which are neither a [[W:Polygon|polygon]] nor a [[W:Simplex|simplex]]. The other two are also 4-polytopes, but not convex: the [[W:Grand stellated 120-cell|grand stellated 120-cell]] and the [[W:Great 120-cell|great 120-cell]]. The 24-cell is nearly unique among self-dual regular convex polytopes in that it and the even polygons are the only such polytopes where a face is not opposite an edge.|name=|group=}} The 24-cell and the [[W:Tesseract|tesseract]] are the only convex regular 4-polytopes in which the edge length equals the radius.{{Efn||name=radially equilateral|group=}} The 24-cell does not have a regular analogue in [[W:Three dimensions|three dimensions]] or any other number of dimensions, either below or above.{{Sfn|Coxeter|1973|p=289|loc=Epilogue|ps=; "Another peculiarity of four-dimensional space is the occurrence of the 24-cell {3,4,3}, which stands quite alone, having no analogue above or below."}} It is the only one of the six convex regular 4-polytopes which is not the analogue of one of the five Platonic solids. However, it can be seen as the analogue of a pair of irregular solids: the [[W:Cuboctahedron|cuboctahedron]] and its dual the [[W:Rhombic dodecahedron|rhombic dodecahedron]].{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|p=25}} Translated copies of the 24-cell can [[W:Tesselate|tesselate]] four-dimensional space face-to-face, forming the [[W:24-cell honeycomb|24-cell honeycomb]]. As a polytope that can tile by translation, the 24-cell is an example of a [[W:Parallelohedron|parallelotope]], the simplest one that is not also a [[W:Zonotope|zonotope]].{{Sfn|Coxeter|1968|p=70|loc=§4.12 The Classification of Zonohedra}} ==Geometry== The 24-cell incorporates the geometries of every convex regular polytope in the first four dimensions, except the 5-cell, those with a 5 in their Schlӓfli symbol,{{Efn|The convex regular polytopes in the first four dimensions with a 5 in their Schlӓfli symbol are the [[W:Pentagon|pentagon]] {5}, the [[W:Icosahedron|icosahedron]] {3, 5}, the [[W:Dodecahedron|dodecahedron]] {5, 3}, the [[600-cell]] {3,3,5} and the [[120-cell]] {5,3,3}. The [[5-cell]] {3, 3, 3} is also pentagonal in the sense that its [[W:Petrie polygon|Petrie polygon]] is the pentagon.|name=pentagonal polytopes|group=}} and the regular polygons with 7 or more sides. In other words, the 24-cell contains ''all'' of the regular polytopes made of triangles and squares that exist in four dimensions except the regular 5-cell, but ''none'' of the pentagonal polytopes. It is especially useful to explore the 24-cell, because one can see the geometric relationships among all of these regular polytopes in a single 24-cell or [[W:24-cell honeycomb|its honeycomb]]. The 24-cell is the fourth in the sequence of six [[W:Convex regular 4-polytope|convex regular 4-polytope]]s (in order of size and complexity).{{Efn|name=4-polytopes ordered by size and complexity}}{{Sfn|Goucher|2020|loc=Subsumptions of regular polytopes}} It can be deconstructed into 3 overlapping instances of its predecessor the [[W:Tesseract|tesseract]] (8-cell), as the 8-cell can be deconstructed into 2 instances of its predecessor the [[16-cell]].{{Sfn|Coxeter|1973|p=302|pp=|loc=Table VI (ii): 𝐈𝐈 = {3,4,3}|ps=: see Result column}} The reverse procedure to construct each of these from an instance of its predecessor preserves the radius of the predecessor, but generally produces a successor with a smaller edge length.{{Efn|name=edge length of successor}} === Coordinates === The 24-cell has two natural systems of Cartesian coordinates, which reveal distinct structure. ==== Great squares ==== The 24-cell is the [[W:Convex hull|convex hull]] of its vertices which can be described as the 24 coordinate [[W:Permutation|permutation]]s of: <math display="block">(\pm1, \pm 1, 0, 0) \in \mathbb{R}^4 .</math> Those coordinates{{Sfn|Coxeter|1973|p=156|loc=§8.7. Cartesian Coordinates}} can be constructed as {{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}}, [[W:Rectification (geometry)|rectifying]] the [[16-cell]] {{Coxeter–Dynkin diagram|node_1|3|node|3|node|4|node}} with the 8 vertices that are permutations of (±2,0,0,0). The vertex figure of a 16-cell is the [[W:Octahedron|octahedron]]; thus, cutting the vertices of the 16-cell at the midpoint of its incident edges produces 8 octahedral cells. This process{{Sfn|Coxeter|1973|p=|pp=145-146|loc=§8.1 The simple truncations of the general regular polytope}} also rectifies the tetrahedral cells of the 16-cell which become 16 octahedra, giving the 24-cell 24 octahedral cells. In this frame of reference the 24-cell has edges of length {{sqrt|2}} and is inscribed in a [[W:3-sphere|3-sphere]] of radius {{sqrt|2}}. Remarkably, the edge length equals the circumradius, as in the [[W:Hexagon|hexagon]], or the [[W:Cuboctahedron|cuboctahedron]]. Such polytopes are ''radially equilateral''.{{Efn|name=radially equilateral|group=}} {{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}|instance=1}} The 24 vertices form 18 great squares{{Efn|The edges of six of the squares are aligned with the grid lines of the ''{{radic|2}} radius coordinate system''. For example: {{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}0, −1, −1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1, −1,{{spaces|2}}0)<br> is the square in the ''xy'' plane. The edges of the squares are not 24-cell edges, they are interior chords joining two vertices 90^o distant from each other; so the squares are merely invisible configurations of four of the 24-cell's vertices, not visible 24-cell features.|name=|group=}} (3 sets of 6 orthogonal{{Efn|Up to 6 planes can be mutually orthogonal in 4 dimensions. 3 dimensional space accommodates only 3 perpendicular axes and 3 perpendicular planes through a single point. In 4 dimensional space we may have 4 perpendicular axes and 6 perpendicular planes through a point (for the same reason that the tetrahedron has 6 edges, not 4): there are 6 ways to take 4 dimensions 2 at a time.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Three such perpendicular planes (pairs of axes) meet at each vertex of the 24-cell (for the same reason that three edges meet at each vertex of the tetrahedron). Each of the 6 planes is [[W:Completely orthogonal|completely orthogonal]] to just one of the other planes: the only one with which it does not share a line (for the same reason that each edge of the tetrahedron is orthogonal to just one of the other edges: the only one with which it does not share a point). Two completely orthogonal planes are perpendicular and opposite each other, as two edges of the tetrahedron are perpendicular and opposite.|name=six orthogonal planes tetrahedral symmetry}} central squares), 3 of which intersect at each vertex. By viewing just one square at each vertex, the 24-cell can be seen as the vertices of 3 pairs of [[W:Completely orthogonal|completely orthogonal]] great squares which intersect{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} if they are [[W:Completely orthogonal|completely orthogonal]].|name=how planes intersect}} at no vertices.{{Efn|name=three square fibrations}} ==== Great hexagons ==== The 24-cell is [[W:Self-dual|self-dual]], having the same number of vertices (24) as cells and the same number of edges (96) as faces. If the dual of the above 24-cell of edge length {{sqrt|2}} is taken by reciprocating it about its ''inscribed'' sphere, another 24-cell is found which has edge length and circumradius 1, and its coordinates reveal more structure. In this frame of reference the 24-cell lies vertex-up, and its vertices can be given as follows: 8 vertices obtained by permuting the ''integer'' coordinates: <math display="block">\left( \pm 1, 0, 0, 0 \right)</math> and 16 vertices with ''half-integer'' coordinates of the form: <math display="block">\left( \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2} \right)</math> all 24 of which lie at distance 1 from the origin. [[#Quaternionic interpretation|Viewed as quaternions]],{{Efn|name=quaternions}} these are the unit [[W:Hurwitz quaternions|Hurwitz quaternions]]. The 24-cell has unit radius and unit edge length{{Efn||name=radially equilateral}} in this coordinate system. We refer to the system as ''unit radius coordinates'' to distinguish it from others, such as the {{sqrt|2}} radius coordinates used [[#Great squares|above]].{{Efn|The edges of the orthogonal great squares are ''not'' aligned with the grid lines of the ''unit radius coordinate system''. Six of the squares do lie in the 6 orthogonal planes of this coordinate system, but their edges are the {{sqrt|2}} ''diagonals'' of unit edge length squares of the coordinate lattice. For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}0,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0,{{spaces|2}}0) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)<br> is the square in the ''xy'' plane. Notice that the 8 ''integer'' coordinates comprise the vertices of the 6 orthogonal squares.|name=orthogonal squares|group=}} {{Regular convex 4-polytopes|wiki=W:|radius=1}} The 24 vertices and 96 edges form 16 non-orthogonal great hexagons,{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|5}}(−<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}(−<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} four of which intersect{{Efn||name=how planes intersect}} at each vertex.{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:Cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:Cubic pyramid|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} By viewing just one hexagon at each vertex, the 24-cell can be seen as the 24 vertices of 4 non-intersecting hexagonal great circles which are [[W:Clifford parallel|Clifford parallel]] to each other.{{Efn|name=four hexagonal fibrations}} The 12 axes and 16 hexagons of the 24-cell constitute a [[W:Reye configuration|Reye configuration]], which in the language of [[W:Configuration (geometry)|configurations]] is written as 12₄16₃ to indicate that each axis belongs to 4 hexagons, and each hexagon contains 3 axes.{{Sfn|Waegell & Aravind|2009|loc=§3.4 The 24-cell: points, lines and Reye's configuration|pp=4-5|ps=; In the 24-cell Reye's "points" and "lines" are axes and hexagons, respectively.}} ==== Great triangles ==== The 24 vertices form 32 equilateral great triangles, of edge length {{radic|3}} in the unit-radius 24-cell,{{Efn|These triangles' edges of length {{sqrt|3}} are the diagonals{{Efn|name=missing the nearest vertices}} of cubical cells of unit edge length found within the 24-cell, but those cubical (tesseract){{Efn|name=three 8-cells}} cells are not cells of the unit radius coordinate lattice.|name=cube diagonals}} inscribed in the 16 great hexagons.{{Efn|These triangles lie in the same planes containing the hexagons;{{Efn|name=non-orthogonal hexagons}} two triangles of edge length {{sqrt|3}} are inscribed in each hexagon. For example, in unit radius coordinates: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|5}}(−<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}(−<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)<br> are two opposing central triangles on the ''y'' axis, with each triangle formed by the vertices in alternating rows. Unlike the hexagons, the {{sqrt|3}} triangles are not made of actual 24-cell edges, so they are invisible features of the 24-cell, like the {{sqrt|2}} squares.|name=central triangles|group=}} Each great triangle is a ring linking three completely disjoint{{Efn|name=completely disjoint}} great squares.{{Efn|The 18 great squares of the 24-cell occur as three sets of 6 orthogonal great squares,{{Efn|name=Six orthogonal planes of the Cartesian basis}} each forming a [[16-cell]].{{Efn|name=three isoclinic 16-cells}} The three 16-cells are completely disjoint (and [[#Clifford parallel polytopes|Clifford parallel]]): each has its own 8 vertices (on 4 orthogonal axes) and its own 24 edges (of length {{radic|2}}). The 18 square great circles are crossed by 16 hexagonal great circles; each hexagon has one axis (2 vertices) in each 16-cell.{{Efn|name=non-orthogonal hexagons}} The two great triangles inscribed in each great hexagon (occupying its alternate vertices, and with edges that are its {{radic|3}} chords) have one vertex in each 16-cell. Thus ''each great triangle is a ring linking the three completely disjoint 16-cells''. There are four different ways (four different ''fibrations'' of the 24-cell) in which the 8 vertices of the 16-cells correspond by being triangles of vertices {{radic|3}} apart: there are 32 distinct linking triangles. Each ''pair'' of 16-cells forms a tesseract (8-cell).{{Efn|name=three 16-cells form three tesseracts}} Each great triangle has one {{radic|3}} edge in each tesseract, so it is also a ring linking the three tesseracts.|name=great linking triangles}} ==== Hypercubic chords ==== [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral{{Efn||name=radially equilateral|group=}} 24-cell, showing its 3 great circle polygons and its 4 chord lengths.|alt=]] The 24 vertices of the 24-cell are distributed{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-Dimensional Polytopes in Parallel Solid Sections (§13.1); (i) Sections of {3,4,3} (edge 2) beginning with a vertex; see column ''a''|5=}} at four different [[W:Chord (geometry)|chord]] lengths from each other: {{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}} and {{sqrt|4}}. The {{sqrt|1}} chords (the 24-cell edges) are the edges of central hexagons, and the {{sqrt|3}} chords are the diagonals of central hexagons. The {{sqrt|2}} chords are the edges of central squares, and the {{sqrt|4}} chords are the diagonals of central squares. Each vertex is joined to 8 others{{Efn|The 8 nearest neighbor vertices surround the vertex (in the curved 3-dimensional space of the 24-cell's boundary surface) the way a cube's 8 corners surround its center. (The [[W:Vertex figure|vertex figure]] of the 24-cell is a cube.)|name=8 nearest vertices}} by an edge of length 1, spanning 60° = <small>{{sfrac|{{pi}}|3}}</small> of arc. Next nearest are 6 vertices{{Efn|The 6 second-nearest neighbor vertices surround the vertex in curved 3-dimensional space the way an octahedron's 6 corners surround its center.|name=6 second-nearest vertices}} located 90° = <small>{{sfrac|{{pi}}|2}}</small> away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = <small>{{sfrac|2{{pi}}|3}}</small> away, along an interior chord of length {{sqrt|3}}.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The opposite vertex is 180° = <small>{{pi}}</small> away along a diameter of length 2. Finally, as the 24-cell is radially equilateral, its center is 1 edge length away from all vertices. To visualize how the interior polytopes of the 24-cell fit together (as described [[#Constructions|below]]), keep in mind that the four chord lengths ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the long diameters of the [[W:Hypercube|hypercube]]s of dimensions 1 through 4: the long diameter of the square is {{sqrt|2}}; the long diameter of the cube is {{sqrt|3}}; and the long diameter of the tesseract is {{sqrt|4}}.{{Efn|Thus ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the vertex chord lengths of the tesseract as well as of the 24-cell. They are also the diameters of the tesseract (from short to long), though not of the 24-cell.}} Moreover, the long diameter of the octahedron is {{sqrt|2}} like the square; and the long diameter of the 24-cell itself is {{sqrt|4}} like the tesseract. ==== Geodesics ==== [[Image:stereographic polytope 24cell faces.png|thumb|[[W:Stereographic projection|Stereographic projection]] of the 24-cell's 16 central hexagons onto their great circles. Each great circle is divided into 6 arc-edges at the intersections where 4 great circles cross.]] The vertex chords of the 24-cell are arranged in [[W:Geodesic|geodesic]] [[W:great circle|great circle]] polygons.{{Efn|A geodesic great circle lies in a 2-dimensional plane which passes through the center of the polytope. Notice that in 4 dimensions this central plane does ''not'' bisect the polytope into two equal-sized parts, as it would in 3 dimensions, just as a diameter (a central line) bisects a circle but does not bisect a sphere. Another difference is that in 4 dimensions not all pairs of great circles intersect at two points, as they do in 3 dimensions; some pairs do, but some pairs of great circles are non-intersecting Clifford parallels.{{Efn|name=Clifford parallels}}}} The [[W:Geodesic distance|geodesic distance]] between two 24-cell vertices along a path of {{sqrt|1}} edges is always 1, 2, or 3, and it is 3 only for opposite vertices.{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90^o bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60^o bend, or as a straight line with one 60^o bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} The {{sqrt|1}} edges occur in 16 [[#Great hexagons|hexagonal great circles]] (in planes inclined at 60 degrees to each other), 4 of which cross{{Efn|name=cuboctahedral hexagons}} at each vertex.{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:Vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:Cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The cube is not radially equilateral in Euclidean 3-space <math>\mathbb{R}^3</math>, but a cubic pyramid is radially equilateral in the curved 3-space of the 24-cell's surface, the [[W:3-sphere|3-sphere]] <math>\mathbb{S}^3</math>. In 4-space the 8 edges radiating from its apex are not actually its radii: the apex of the [[W:Cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices. But in curved 3-space the edges radiating symmetrically from the apex ''are'' radii, so the cube is radially equilateral ''in that curved 3-space'' <math>\mathbb{S}^3</math>. In Euclidean 4-space <math>\mathbb{R}^4</math> 24 edges radiating symmetrically from a central point make the radially equilateral 24-cell,{{Efn|name=radially equilateral}} and a symmetrical subset of 16 of those edges make the [[W:Tesseract#Radial equilateral symmetry|radially equilateral tesseract]].}}|name=24-cell vertex figure}} The 96 distinct {{sqrt|1}} edges divide the surface into 96 triangular faces and 24 octahedral cells: a 24-cell. The 16 hexagonal great circles can be divided into 4 sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]] geodesics, such that only one hexagonal great circle in each set passes through each vertex, and the 4 hexagons in each set reach all 24 vertices.{{Efn|name=hexagonal fibrations}} {| class="wikitable floatright" |+ [[W:Orthographic projection|Orthogonal projection]]s of the 24-cell |- style="text-align:center;" ![[W:Coxeter plane|Coxeter plane]] !colspan=2|F₄ |- style="text-align:center;" !Graph |colspan=2|[[File:24-cell t0_F4.svg|100px]] |- style="text-align:center;" ![[W:Dihedral symmetry|Dihedral symmetry]] |colspan=2|[12] |- style="text-align:center;" !Coxeter plane !B₃ / A₂ (a) !B₃ / A₂ (b) |- style="text-align:center;" !Graph |[[File:24-cell t0_B3.svg|100px]] |[[File:24-cell t3_B3.svg|100px]] |- style="text-align:center;" !Dihedral symmetry |[6] |[6] |- style="text-align:center;" !Coxeter plane !B₄ !B₂ / A₃ |- style="text-align:center;" !Graph |[[File:24-cell t0_B4.svg|100px]] |[[File:24-cell t0_B2.svg|100px]] |- style="text-align:center;" !Dihedral symmetry |[8] |[4] |} The {{sqrt|2}} chords occur in 18 [[#Great squares|square great circles]] (3 sets of 6 orthogonal planes{{Efn|name=Six orthogonal planes of the Cartesian basis}}), 3 of which cross at each vertex.{{Efn|Six {{sqrt|2}} chords converge in 3-space from the face centers of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 3 straight lines which cross there perpendicularly. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell, and eight {{sqrt|1}} edges converge from there, but let us ignore them now, since 7 straight lines crossing at the center is confusing to visualize all at once. Each of the six {{sqrt|2}} chords runs from this cube's center (the vertex) through a face center to the center of an adjacent (face-bonded) cube, which is another vertex of the 24-cell: not a nearest vertex (at the cube corners), but one located 90° away in a second concentric shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices. The face-center through which the {{sqrt|2}} chord passes is the mid-point of the {{sqrt|2}} chord, so it lies inside the 24-cell.|name=|group=}} The 72 distinct {{sqrt|2}} chords do not run in the same planes as the hexagonal great circles; they do not follow the 24-cell's edges, they pass through its octagonal cell centers.{{Efn|One can cut the 24-cell through 6 vertices (in any hexagonal great circle plane), or through 4 vertices (in any square great circle plane). One can see this in the [[W:Cuboctahedron|cuboctahedron]] (the central [[W:hyperplane|hyperplane]] of the 24-cell), where there are four hexagonal great circles (along the edges) and six square great circles (across the square faces diagonally).}} The 72 {{sqrt|2}} chords are the 3 orthogonal axes of the 24 octahedral cells, joining vertices which are 2 {{radic|1}} edges apart. The 18 square great circles can be divided into 3 sets of 6 non-intersecting Clifford parallel geodesics,{{Efn|[[File:Hopf band wikipedia.png|thumb|Two [[W:Clifford parallel|Clifford parallel]] [[W:Great circle|great circle]]s on the [[W:3-sphere|3-sphere]] spanned by a twisted [[W:Annulus (mathematics)|annulus]]. They have a common center point in [[W:Rotations in 4-dimensional Euclidean space|4-dimensional Euclidean space]], and could lie in [[W:Completely orthogonal|completely orthogonal]] rotation planes.]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point.{{Sfn|Tyrrell & Semple|1971|loc=§3. Clifford's original definition of parallelism|pp=5-6}} A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the 2-sphere will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect; various sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. Perhaps the simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Each completely orthogonal pair is Clifford parallel. The two circles cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 3-sphere.{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} Because they are perpendicular and share a common center,{{Efn|In 4-space, two great circles can be perpendicular and share a common center ''which is their only point of intersection'', because there is more than one great [[W:2-sphere|2-sphere]] on the [[W:3-sphere|3-sphere]]. The dimensionally analogous structure to a [[W:Great circle|great circle]] (a great 1-sphere) is a great 2-sphere,{{Sfn|Stillwell|2001|p=24}} which is an ordinary sphere that constitutes an ''equator'' boundary dividing the 3-sphere into two equal halves, just as a great circle divides the 2-sphere. Although two Clifford parallel great circles{{Efn|name=Clifford parallels}} occupy the same 3-sphere, they lie on different great 2-spheres. The great 2-spheres are [[#Clifford parallel polytopes|Clifford parallel 3-dimensional objects]], displaced relative to each other by a fixed distance ''d'' in the fourth dimension. Their corresponding points (on their two surfaces) are ''d'' apart. The 2-spheres (by which we mean their surfaces) do not intersect at all, although they have a common center point in 4-space. The displacement ''d'' between a pair of their corresponding points is the [[#Geodesics|chord of a great circle]] which intersects both 2-spheres, so ''d'' can be represented equivalently as a linear chordal distance, or as an angular distance.|name=great 2-spheres}} the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]].|name=Clifford parallels}} such that only one square great circle in each set passes through each vertex, and the 6 squares in each set reach all 24 vertices.{{Efn|name=square fibrations}} The {{sqrt|3}} chords occur in 32 [[#Great triangles|triangular great circles]] in 16 planes, 4 of which cross at each vertex.{{Efn|Eight {{sqrt|3}} chords converge from the corners of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. Each of the eight {{sqrt|3}} chords runs from this cube's center to the center of a diagonally adjacent (vertex-bonded) cube,{{Efn|name=missing the nearest vertices}} which is another vertex of the 24-cell: one located 120° away in a third concentric shell of eight {{sqrt|3}}-distant vertices surrounding the second shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices.|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The 96 distinct {{sqrt|3}} chords{{Efn|name=cube diagonals}} run vertex-to-every-other-vertex in the same planes as the hexagonal great circles.{{Efn|name=central triangles}} They are the 3 edges of the 32 great triangles inscribed in the 16 great hexagons, joining vertices which are 2 {{sqrt|1}} edges apart on a great circle.{{Efn|name=three 8-cells}} The {{sqrt|4}} chords occur as 12 vertex-to-vertex diameters (3 sets of 4 orthogonal axes), the 24 radii around the 25th central vertex. The sum of the squared lengths{{Efn|The sum of 1・96 + 2・72 + 3・96 + 4・12 is 576.}} of all these distinct chords of the 24-cell is 576 = 24².{{Efn|The sum of the squared lengths of all the distinct chords of any regular convex n-polytope of unit radius is the square of the number of vertices.{{Sfn|Copher|2019|loc=§3.2 Theorem 3.4|p=6}}}} These are all the central polygons through vertices, but in 4-space there are geodesics on the 3-sphere which do not lie in central planes at all. There are geodesic shortest paths between two 24-cell vertices that are helical rather than simply circular; they correspond to diagonal [[#Isoclinic rotations|isoclinic rotations]] rather than [[#Simple rotations|simple rotations]].{{Efn|name=isoclinic geodesic}} The {{sqrt|1}} edges occur in 48 parallel pairs, {{sqrt|3}} apart. The {{sqrt|2}} chords occur in 36 parallel pairs, {{sqrt|2}} apart. The {{sqrt|3}} chords occur in 48 parallel pairs, {{sqrt|1}} apart.{{Efn|Each pair of parallel {{sqrt|1}} edges joins a pair of parallel {{sqrt|3}} chords to form one of 48 rectangles (inscribed in the 16 central hexagons), and each pair of parallel {{sqrt|2}} chords joins another pair of parallel {{sqrt|2}} chords to form one of the 18 central squares.|name=|group=}} The central planes of the 24-cell can be divided into 4 orthogonal central hyperplanes (3-spaces) each forming a [[W:Cuboctahedron|cuboctahedron]]. The great hexagons are 60 degrees apart; the great squares are 90 degrees or 60 degrees apart; a great square and a great hexagon are 90 degrees ''and'' 60 degrees apart.{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)".}} Since all planes in the same hyperplane{{Efn|name=hyperplanes}} are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles ([[W:Completely orthogonal|completely orthogonal]]) or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes ''may'' be isoclinic, but often they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} Each set of similar central polygons (squares or hexagons) can be divided into 4 sets of non-intersecting Clifford parallel polygons (of 6 squares or 4 hexagons).{{Efn|Each pair of Clifford parallel polygons lies in two different hyperplanes (cuboctahedrons). The 4 Clifford parallel hexagons lie in 4 different cuboctahedrons.}} Each set of Clifford parallel great circles is a parallel [[W:Hopf fibration|fiber bundle]] which visits all 24 vertices just once. Each great circle intersects{{Efn|name=how planes intersect}} with the other great circles to which it is not Clifford parallel at one {{sqrt|4}} diameter of the 24-cell.{{Efn|Two intersecting great squares or great hexagons share two opposing vertices, but squares or hexagons on Clifford parallel great circles share no vertices. Two intersecting great triangles share only one vertex, since they lack opposing vertices.|name=how great circle planes intersect|group=}} Great circles which are [[W:Completely orthogonal|completely orthogonal]] or otherwise Clifford parallel{{Efn|name=Clifford parallels}} do not intersect at all: they pass through disjoint sets of vertices.{{Efn|name=pairs of completely orthogonal planes}} === Constructions === [[File:24-cell-3CP.gif|thumb|The 24-point 24-cell contains three 8-point 16-cells (red, green, and blue), double-rotated by 60 degrees with respect to each other.{{Efn|name=three isoclinic 16-cells}} Each 8-point 16-cell is a coordinate system basis frame of four perpendicular (w,x,y,z) axes, just as a 6-point [[w:Octahedron|octahedron]] is a coordinate system basis frame of three perpendicular (x,y,z) axes.{{Efn|name=three basis 16-cells}} One octahedral cell of the 24 cells is emphasized. Each octahedral cell has two vertices of each color, delimiting an invisible perpendicular axis of the octahedron, which is a {{radic|2}} edge of the red, green, or blue 16-cell.{{Efn|name=octahedral diameters}}]] Triangles and squares come together uniquely in the 24-cell to generate, as interior features,{{Efn|Interior features are not considered elements of the polytope. For example, the center of a 24-cell is a noteworthy feature (as are its long radii), but these interior features do not count as elements in [[#As a configuration|its configuration matrix]], which counts only elementary features (which are not interior to any other feature including the polytope itself). Interior features are not rendered in most of the diagrams and illustrations in this article (they are normally invisible). In illustrations showing interior features, we always draw interior edges as dashed lines, to distinguish them from elementary edges.|name=interior features|group=}} all of the triangle-faced and square-faced regular convex polytopes in the first four dimensions (with caveats for the [[5-cell]] and the [[600-cell]]).{{Efn|The 600-cell is larger than the 24-cell, and contains the 24-cell as an interior feature.{{Sfn|Coxeter|1973|p=153|loc=8.5. Gosset's construction for {3,3,5}|ps=: "In fact, the vertices of {3,3,5}, each taken 5 times, are the vertices of 25 {3,4,3}'s."}} The regular 5-cell is not found in the interior of any convex regular 4-polytope except the [[120-cell]],{{Sfn|Coxeter|1973|p=304|loc=Table VI(iv) II={5,3,3}|ps=: Faceting {5,3,3}[120𝛼₄]{3,3,5} of the 120-cell reveals 120 regular 5-cells.}} though every convex 4-polytope can be [[#Characteristic orthoscheme|deconstructed into irregular 5-cells.]]|name=|group=}} Consequently, there are numerous ways to construct or deconstruct the 24-cell. ==== Reciprocal constructions from 8-cell and 16-cell ==== The 8 integer vertices (±1, 0, 0, 0) are the vertices of a regular [[16-cell]], and the 16 half-integer vertices (±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}) are the vertices of its dual, the [[W:Tesseract|tesseract]] (8-cell).{{Sfn|Egan|2021|loc=animation of a rotating 24-cell|ps=: {{color|red}} half-integer vertices (tesseract), {{Font color|fg=yellow|bg=black|text=yellow}} and {{color|black}} integer vertices (16-cell).}} The tesseract gives Gosset's construction{{Sfn|Coxeter|1973|p=150|loc=Gosset}} of the 24-cell, equivalent to cutting a tesseract into 8 [[W:Cubic pyramid|cubic pyramid]]s, and then attaching them to the facets of a second tesseract. The analogous construction in 3-space gives the [[W:Rhombic dodecahedron|rhombic dodecahedron]] which, however, is not regular.{{Efn|[[File:R1-cube.gif|thumb|150px|Construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube.]]This animation shows the construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube, by inverting the center-to-face pyramids of a cube. Gosset's construction of a 24-cell from a tesseract is the 4-dimensional analogue of this process, inverting the center-to-cell pyramids of an 8-cell (tesseract).{{Sfn|Coxeter|1973|p=150|loc=Gosset}}|name=rhombic dodecahedron from a cube}} The 16-cell gives the reciprocal construction of the 24-cell, Cesaro's construction,{{Sfn|Coxeter|1973|p=148|loc=§8.2. Cesaro's construction for {3, 4, 3}.}} equivalent to rectifying a 16-cell (truncating its corners at the mid-edges, as described [[#Great squares|above]]). The analogous construction in 3-space gives the [[W:Cuboctahedron|cuboctahedron]] (dual of the rhombic dodecahedron) which, however, is not regular. The tesseract and the 16-cell are the only regular 4-polytopes in the 24-cell.{{Sfn|Coxeter|1973|p=302|loc=Table VI(ii) II={3,4,3}, Result column}} We can further divide the 16 half-integer vertices into two groups: those whose coordinates contain an even number of minus (−) signs and those with an odd number. Each of these groups of 8 vertices also define a regular 16-cell. This shows that the vertices of the 24-cell can be grouped into three disjoint sets of eight with each set defining a regular 16-cell, and with the complement defining the dual tesseract.{{Sfn|Coxeter|1973|pp=149-150|loc=§8.22. see illustrations Fig. 8.2<small>A</small> and Fig 8.2<small>B</small>|p=|ps=}} This also shows that the symmetries of the 16-cell form a subgroup of index 3 of the symmetry group of the 24-cell.{{Efn|name=three 16-cells form three tesseracts}} ==== Diminishings ==== We can [[W:Faceting|facet]] the 24-cell by cutting{{Efn|We can cut a vertex off a polygon with a 0-dimensional cutting instrument (like the point of a knife, or the head of a zipper) by sweeping it along a 1-dimensional line, exposing a new edge. We can cut a vertex off a polyhedron with a 1-dimensional cutting edge (like a knife) by sweeping it through a 2-dimensional face plane, exposing a new face. We can cut a vertex off a polychoron (a 4-polytope) with a 2-dimensional cutting plane (like a snowplow), by sweeping it through a 3-dimensional cell volume, exposing a new cell. Notice that as within the new edge length of the polygon or the new face area of the polyhedron, every point within the new cell volume is now exposed on the surface of the polychoron.}} through interior cells bounded by vertex chords to remove vertices, exposing the [[W:Facet (geometry)|facets]] of interior 4-polytopes [[W:Inscribed figure|inscribed]] in the 24-cell. One can cut a 24-cell through any planar hexagon of 6 vertices, any planar rectangle of 4 vertices, or any triangle of 3 vertices. The great circle central planes ([[#Geodesics|above]]) are only some of those planes. Here we shall expose some of the others: the face planes{{Efn|Each cell face plane intersects with the other face planes of its kind to which it is not completely orthogonal or parallel at their characteristic vertex chord edge. Adjacent face planes of orthogonally-faced cells (such as cubes) intersect at an edge since they are not completely orthogonal.{{Efn|name=how planes intersect}} Although their dihedral angle is 90 degrees in the boundary 3-space, they lie in the same hyperplane{{Efn|name=hyperplanes}} (they are coincident rather than perpendicular in the fourth dimension); thus they intersect in a line, as non-parallel planes do in any 3-space.|name=how face planes intersect}} of interior polytopes.{{Efn|The only planes through exactly 6 vertices of the 24-cell (not counting the central vertex) are the '''16 hexagonal great circles'''. There are no planes through exactly 5 vertices. There are several kinds of planes through exactly 4 vertices: the 18 {{sqrt|2}} square great circles, the '''72 {{sqrt|1}} square (tesseract) faces''', and 144 {{sqrt|1}} by {{sqrt|2}} rectangles. The planes through exactly 3 vertices are the 96 {{sqrt|2}} equilateral triangle (16-cell) faces, and the '''96 {{sqrt|1}} equilateral triangle (24-cell) faces'''. There are an infinite number of central planes through exactly two vertices (great circle [[W:Digon|digon]]s); 16 are distinguished, as each is [[W:Completely orthogonal|completely orthogonal]] to one of the 16 hexagonal great circles. '''Only the polygons composed of 24-cell {{radic|1}} edges are visible''' in the projections and rotating animations illustrating this article; the others contain invisible interior chords.{{Efn|name=interior features}}|name=planes through vertices|group=}} ===== 8-cell ===== Starting with a complete 24-cell, remove the 8 orthogonal vertices of a 16-cell (4 opposite pairs on 4 perpendicular axes), and the 8 edges which radiate from each, by cutting through 8 cubic cells bounded by {{sqrt|1}} edges to remove 8 [[W:Cubic pyramid|cubic pyramid]]s whose [[W:Apex (geometry)|apexes]] are the vertices to be removed. This removes 4 edges from each hexagonal great circle (retaining just one opposite pair of edges), so no continuous hexagonal great circles remain. Now 3 perpendicular edges meet and form the corner of a cube at each of the 16 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to a tetrahedral vertex figure (see [[#Relationships among interior polytopes|Kepler's drawing]]). The vertex cube has vanished, and now there are only 4 corners of the vertex figure where before there were 8. Four tesseract edges converge from the tetrahedron vertices and meet at its center, where they do not cross (since the tetrahedron does not have opposing vertices).|name=|group=}} and the 32 remaining edges divide the surface into 24 square faces and 8 cubic cells: a [[W:Tesseract|tesseract]]. There are three ways you can do this (choose a set of 8 orthogonal vertices out of 24), so there are three such tesseracts inscribed in the 24-cell.{{Efn|name=three 8-cells}} They overlap with each other, but most of their element sets are disjoint: they share some vertex count, but no edge length, face area, or cell volume.{{Efn|name=vertex-bonded octahedra}} They do share 4-content, their common core.{{Efn||name=common core|group=}} ===== 16-cell ===== Starting with a complete 24-cell, remove the 16 vertices of a tesseract (retaining the 8 vertices you removed above), by cutting through 16 tetrahedral cells bounded by {{sqrt|2}} chords to remove 16 [[W:Tetrahedral pyramid|tetrahedral pyramid]]s whose apexes are the vertices to be removed. This removes 12 great squares (retaining just one orthogonal set of 6) and all the {{sqrt|1}} edges, exposing {{sqrt|2}} chords as the new edges. Now the remaining 6 great squares cross perpendicularly, 3 at each of 8 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to an octahedral vertex figure. The vertex cube has vanished, and now there are only 6 corners of the vertex figure where before there were 8. The 6 {{sqrt|2}} chords which formerly converged from cube face centers now converge from octahedron vertices; but just as before, they meet at the center where 3 straight lines cross perpendicularly. The octahedron vertices are located 90° away outside the vanished cube, at the new nearest vertices; before truncation those were 24-cell vertices in the second shell of surrounding vertices.|name=|group=}} and their 24 edges divide the surface into 32 triangular faces and 16 tetrahedral cells: a [[16-cell]]. There are three ways you can do this (remove 1 of 3 sets of tesseract vertices), so there are three such 16-cells inscribed in the 24-cell.{{Efn|name=three isoclinic 16-cells}} They overlap with each other, but all of their element sets are disjoint:{{Efn|name=completely disjoint}} they do not share any vertex count, edge length,{{Efn|name=root 2 chords}} or face area, but they do share cell volume. They also share 4-content, their common core.{{Efn||name=common core|group=}} ==== Tetrahedral constructions ==== The 24-cell can be constructed radially from 96 equilateral triangles of edge length {{sqrt|1}} which meet at the center of the polytope, each contributing two radii and an edge.{{Efn|name=radially equilateral|group=}} They form 96 {{sqrt|1}} tetrahedra (each contributing one 24-cell face), all sharing the 25th central apex vertex. These form 24 octahedral pyramids (half-16-cells) with their apexes at the center. The 24-cell can be constructed from 96 equilateral triangles of edge length {{sqrt|2}}, where the three vertices of each triangle are located 90° = <small>{{sfrac|{{pi}}|2}}</small> away from each other on the 3-sphere. They form 48 {{sqrt|2}}-edge tetrahedra (the cells of the [[#16-cell|three 16-cells]]), centered at the 24 mid-edge-radii of the 24-cell.{{Efn|Each of the 72 {{sqrt|2}} chords in the 24-cell is a face diagonal in two distinct cubical cells (of different 8-cells) and an edge of four tetrahedral cells (in just one 16-cell).|name=root 2 chords}} The 24-cell can be constructed directly from its [[#Characteristic orthoscheme|characteristic simplex]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, the [[5-cell#Irregular 5-cells|irregular 5-cell]] which is the [[W:Fundamental region|fundamental region]] of its [[W:Coxeter group|symmetry group]] [[W:F4 polytope|F₄]], by reflection of that 4-[[W:Orthoscheme|orthoscheme]] in its own cells (which are 3-orthoschemes).{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:Facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} ==== Cubic constructions ==== The 24-cell is not only the 24-octahedral-cell, it is also the 24-cubical-cell, although the cubes are cells of the three 8-cells, not cells of the 24-cell, in which they are not volumetrically disjoint. The 24-cell can be constructed from 24 cubes of its own edge length (three 8-cells).{{Efn|name=three 8-cells}} Each of the cubes is shared by 2 8-cells, each of the cubes' square faces is shared by 4 cubes (in 2 8-cells), each of the 96 edges is shared by 8 square faces (in 4 cubes in 2 8-cells), and each of the 96 vertices is shared by 16 edges (in 8 square faces in 4 cubes in 2 8-cells). ==== Relationships among interior polytopes ==== The 24-cell, three tesseracts, and three 16-cells are deeply entwined around their common center, and intersect in a common core.{{Efn|A simple way of stating this relationship is that the common core of the {{radic|2}}-radius 4-polytopes is the unit-radius 24-cell. The common core of the 24-cell and its inscribed 8-cells and 16-cells is the unit-radius 24-cell's insphere-inscribed dual 24-cell of edge length and radius {{radic|1/2}}.{{Sfn|Coxeter|1995|p=29|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|ps=; "The common content of the 4-cube and the 16-cell is a smaller {3,4,3} whose vertices are the permutations of [(±{{sfrac|1|2}}, ±{{sfrac|1|2}}, 0, 0)]".}} Rectifying any of the three 16-cells reveals this smaller 24-cell, which has a 4-content of only 1/2 (1/4 that of the unit-radius 24-cell). Its vertices lie at the centers of the 24-cell's octahedral cells, which are also the centers of the tesseracts' square faces, and are also the centers of the 16-cells' edges. {{Sfn|Coxeter|1973|p=147|loc=§8.1 The simple truncations of the general regular polytope|ps=; "At a point of contact, [elements of a regular polytope and elements of its dual in which it is inscribed in some manner] lie in [[W:completely orthogonal|completely orthogonal]] subspaces of the tangent hyperplane to the sphere [of reciprocation], so their only common point is the point of contact itself....{{Efn|name=how planes intersect}} In fact, the [various] radii ₀𝑹, ₁𝑹, ₂𝑹, ... determine the polytopes ... whose vertices are the centers of elements 𝐈𝐈₀, 𝐈𝐈₁, 𝐈𝐈₂, ... of the original polytope."}}|name=common core|group=}} The tesseracts and the 16-cells are rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other. This means that the corresponding vertices of two tesseracts or two 16-cells are {{radic|3}} (120°) apart.{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diameters). The 8-cells are not completely disjoint (they share vertices),{{Efn|name=completely disjoint}} but each {{radic|3}} chord occurs as a cube long diameter in just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell as cube long diameters.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}}|name=three 8-cells}} The tesseracts are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used twice, are the vertices of three 16-vertex tesseracts.|name=|group=}} such that their vertices and edges are exterior elements of the 24-cell, but their square faces and cubical cells lie inside the 24-cell (they are not elements of the 24-cell). The 16-cells are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used once, are the vertices of three 8-vertex 16-cells.{{Efn|name=three basis 16-cells}}|name=|group=}} such that only their vertices are exterior elements of the 24-cell: their edges, triangular faces, and tetrahedral cells lie inside the 24-cell. The interior{{Efn|The edges of the 16-cells are not shown in any of the renderings in this article; if we wanted to show interior edges, they could be drawn as dashed lines. The edges of the inscribed tesseracts are always visible, because they are also edges of the 24-cell.}} 16-cell edges have length {{sqrt|2}}.{{Efn|name=great linking triangles}}[[File:Kepler's tetrahedron in cube.png|thumb|Kepler's drawing of tetrahedra in the cube.{{Sfn|Kepler|1619|p=181}}]] The 16-cells are also inscribed in the tesseracts: their {{sqrt|2}} edges are the face diagonals of the tesseract, and their 8 vertices occupy every other vertex of the tesseract. Each tesseract has two 16-cells inscribed in it (occupying the opposite vertices and face diagonals), so each 16-cell is inscribed in two of the three 8-cells.{{Sfn|van Ittersum|2020|loc=§4.2|pp=73-79}}{{Efn|name=three 16-cells form three tesseracts}} This is reminiscent of the way, in 3 dimensions, two opposing regular tetrahedra can be inscribed in a cube, as discovered by Kepler.{{Sfn|Kepler|1619|p=181}} In fact it is the exact dimensional analogy (the [[W:Demihypercube|demihypercube]]s), and the 48 tetrahedral cells are inscribed in the 24 cubical cells in just that way.{{Sfn|Coxeter|1973|p=269|loc=§14.32|ps=. "For instance, in the case of <math>\gamma_4[2\beta_4]</math>...."}}{{Efn|name=root 2 chords}} The 24-cell encloses the three tesseracts within its envelope of octahedral facets, leaving 4-dimensional space in some places between its envelope and each tesseract's envelope of cubes. Each tesseract encloses two of the three 16-cells, leaving 4-dimensional space in some places between its envelope and each 16-cell's envelope of tetrahedra. Thus there are measurable{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {''p,q,r''} in four dimensions|ps=; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of the same radius.}} 4-dimensional interstices{{Efn|The 4-dimensional content of the unit edge length tesseract is 1 (by definition). The content of the unit edge length 24-cell is 2, so half its content is inside each tesseract, and half is between their envelopes. Each 16-cell (edge length {{sqrt|2}}) encloses a content of 2/3, leaving 1/3 of an enclosing tesseract between their envelopes.|name=|group=}} between the 24-cell, 8-cell and 16-cell envelopes. The shapes filling these gaps are [[W:Hyperpyramid|4-pyramids]], alluded to above.{{Efn|Between the 24-cell envelope and the 8-cell envelope, we have the 8 cubic pyramids of Gosset's construction. Between the 8-cell envelope and the 16-cell envelope, we have 16 right [[5-cell#Irregular 5-cell|tetrahedral pyramids]], with their apexes filling the corners of the tesseract.}} ==== Boundary cells ==== Despite the 4-dimensional interstices between 24-cell, 8-cell and 16-cell envelopes, their 3-dimensional volumes overlap. The different envelopes are separated in some places, and in contact in other places (where no 4-pyramid lies between them). Where they are in contact, they merge and share cell volume: they are the same 3-membrane in those places, not two separate but adjacent 3-dimensional layers.{{Efn|Because there are three overlapping tesseracts inscribed in the 24-cell,{{Efn|name=three 8-cells}} each octahedral cell lies ''on'' a cubic cell of one tesseract (in the cubic pyramid based on the cube, but not in the cube's volume), and ''in'' two cubic cells of each of the other two tesseracts (cubic cells which it spans, sharing their volume).{{Efn|name=octahedral diameters}}|name=octahedra both on and in cubes}} Because there are a total of 7 envelopes, there are places where several envelopes come together and merge volume, and also places where envelopes interpenetrate (cross from inside to outside each other). Some interior features lie within the 3-space of the (outer) boundary envelope of the 24-cell itself: each octahedral cell is bisected by three perpendicular squares (one from each of the tesseracts), and the diagonals of those squares (which cross each other perpendicularly at the center of the octahedron) are 16-cell edges (one from each 16-cell). Each square bisects an octahedron into two square pyramids, and also bonds two adjacent cubic cells of a tesseract together as their common face.{{Efn|Consider the three perpendicular {{sqrt|2}} long diameters of the octahedral cell.{{Sfn|van Ittersum|2020|p=79}} Each of them is an edge of a different 16-cell. Two of them are the face diagonals of the square face between two cubes; each is a {{sqrt|2}} chord that connects two vertices of those 8-cell cubes across a square face, connects two vertices of two 16-cell tetrahedra (inscribed in the cubes), and connects two opposite vertices of a 24-cell octahedron (diagonally across two of the three orthogonal square central sections).{{Efn|name=root 2 chords}} The third perpendicular long diameter of the octahedron does exactly the same (by symmetry); so it also connects two vertices of a pair of cubes across their common square face: but a different pair of cubes, from one of the other tesseracts in the 24-cell.{{Efn|name=vertex-bonded octahedra}}|name=octahedral diameters}} As we saw [[#Relationships among interior polytopes|above]], 16-cell {{sqrt|2}} tetrahedral cells are inscribed in tesseract {{sqrt|1}} cubic cells, sharing the same volume. 24-cell {{sqrt|1}} octahedral cells overlap their volume with {{sqrt|1}} cubic cells: they are bisected by a square face into two square pyramids,{{sfn|Coxeter|1973|page=150|postscript=: "Thus the 24 cells of the {3, 4, 3} are dipyramids based on the 24 squares of the <math>\gamma_4</math>. (Their centres are the mid-points of the 24 edges of the <math>\beta_4</math>.)"}} the apexes of which also lie at a vertex of a cube.{{Efn|This might appear at first to be angularly impossible, and indeed it would be in a flat space of only three dimensions. If two cubes rest face-to-face in an ordinary 3-dimensional space (e.g. on the surface of a table in an ordinary 3-dimensional room), an octahedron will fit inside them such that four of its six vertices are at the four corners of the square face between the two cubes; but then the other two octahedral vertices will not lie at a cube corner (they will fall within the volume of the two cubes, but not at a cube vertex). In four dimensions, this is no less true! The other two octahedral vertices do ''not'' lie at a corner of the adjacent face-bonded cube in the same tesseract. However, in the 24-cell there is not just one inscribed tesseract (of 8 cubes), there are three overlapping tesseracts (of 8 cubes each). The other two octahedral vertices ''do'' lie at the corner of a cube: but a cube in another (overlapping) tesseract.{{Efn|name=octahedra both on and in cubes}}}} The octahedra share volume not only with the cubes, but with the tetrahedra inscribed in them; thus the 24-cell, tesseracts, and 16-cells all share some boundary volume.{{Efn|name=octahedra both on and in cubes}} === As a configuration === This [[W:Regular 4-polytope#As configurations|configuration matrix]]{{Sfn|Coxeter|1973|p=12|loc=§1.8. Configurations}} represents the 24-cell. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole 24-cell. The non-diagonal numbers say how many of the column's element occur in or at the row's element. {| class=wikitable |- align=center |\||style="background-color:#808080;"|v||style="background-color:#E6194B;"|e||style="background-color:#3CB44B;"|f||style="background-color:#FFE119;"|c |- align=right |align=left style="background-color:#808080;"|v||style="background-color:#E0F0FF"|'''24'''||8||12||6 |- align=right |align=left style="background-color:#E6194B;"|e||2||style="background-color:#f0FFE0"|'''96'''||3||3 |- align=right |align=left style="background-color:#3CB44B;"|f||3||3||style="background-color:#f0FFE0"|'''96'''||2 |- align=right |align=left style="background-color:#FFE119;"|c||6||12||8||style="background-color:#f0FFE0"|'''24''' |} Since the 24-cell is self-dual, its matrix is identical to its 180 degree rotation. In the [[W:uniform 4-polytope|uniform]] D₄ construction, {{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}}, the face and cell rows and columns split into 3 partitions.<ref>[https://bendwavy.org/klitzing/incmats/ico.htm 24-cell: o3x3o *b3o]</ref> The dual of this construction will have 3 partitions of vertices and edges, and 1 class each of faces and cells. {| class=wikitable |\||style="background-color:#808080;"|v||style="background-color:#E6194B;"|e||style="background-color:#3CB44B;"|f1||style="background-color:#3CB44B;"|f2||style="background-color:#3CB44B;"|f3||style="background-color:#FFE119;"|c1||style="background-color:#FFE119;"|c2||style="background-color:#FFE119;"|c3 |- align=right |align=left style="background-color:#808080;"|v||style="background-color:#E0F0FF"|'''24'''||8||4||4||4||2||2||2 |- align=right |align=left style="background-color:#E6194B;"|e||2||style="background-color:#f0FFE0"|'''96'''||1||1||1||1||1||1 |- align=right |align=left style="background-color:#3CB44B;"|f1||3||3||style="background-color:#f0FFE0"|'''32'''||style="background-color:#f0FFE0"|*||style="background-color:#f0FFE0"|*||1||1||0 |- align=right |align=left style="background-color:#3CB44B;"|f2||3||3||style="background-color:#f0FFE0"|*||style="background-color:#f0FFE0"|'''32'''||style="background-color:#f0FFE0"|*||1||0||1 |- align=right |align=left style="background-color:#3CB44B;"|f3||3||3||style="background-color:#f0FFE0"|*||style="background-color:#f0FFE0"|*||style="background-color:#f0FFE0"|'''32'''||0||1||1 |- align=right |align=left style="background-color:#FFE119;"|c1||6||12||4||4||0||style="background-color:#f0FFE0"|'''8'''||style="background-color:#f0FFE0"|*||style="background-color:#f0FFE0"|* |- align=right |align=left style="background-color:#FFE119;"|c2||6||12||4||0||4||style="background-color:#f0FFE0"|*||style="background-color:#f0FFE0"|'''8'''||style="background-color:#f0FFE0"|* |- align=right |align=left style="background-color:#FFE119;"|c3||6||12||0||4||4||style="background-color:#f0FFE0"|*||style="background-color:#f0FFE0"|*||style="background-color:#f0FFE0"|'''8''' |} ==Symmetries, root systems, and tessellations== [[File:F4 roots by 24-cell duals.svg|thumb|upright|The compound of the 24 vertices of the 24-cell (red nodes), and its unscaled dual (yellow nodes), represent the 48 root vectors of the [[W:F4 (mathematics)|F₄]] group, as shown in this F₄ Coxeter plane projection]] The 24 root vectors of the [[W:D4 (root system)|D₄ root system]] of the [[W:Simple Lie group|simple Lie group]] [[W:SO(8)|SO(8)]] form the vertices of a 24-cell. The vertices can be seen in 3 [[W:Hyperplane|hyperplane]]s,{{Efn|One way to visualize the ''n''-dimensional [[W:Hyperplane|hyperplane]]s is as the ''n''-spaces which can be defined by ''n + 1'' points. A point is the 0-space which is defined by 1 point. A line is the 1-space which is defined by 2 points which are not coincident. A plane is the 2-space which is defined by 3 points which are not colinear (any triangle). In 4-space, a 3-dimensional hyperplane is the 3-space which is defined by 4 points which are not coplanar (any tetrahedron). In 5-space, a 4-dimensional hyperplane is the 4-space which is defined by 5 points which are not cocellular (any 5-cell). These [[W:Simplex|simplex]] figures divide the hyperplane into two parts (inside and outside the figure), but in addition they divide the enclosing space into two parts (above and below the hyperplane). The ''n'' points ''bound'' a finite simplex figure (from the outside), and they ''define'' an infinite hyperplane (from the inside).{{Sfn|Coxeter|1973|loc=§7.2.|p=120|ps=: "... any ''n''+1 points which do not lie in an (''n''-1)-space are the vertices of an ''n''-dimensional ''simplex''.... Thus the general simplex may alternatively be defined as a finite region of ''n''-space enclosed by ''n''+1 ''hyperplanes'' or (''n''-1)-spaces."}} These two divisions are orthogonal, so the defining simplex divides space into six regions: inside the simplex and in the hyperplane, inside the simplex but above or below the hyperplane, outside the simplex but in the hyperplane, and outside the simplex above or below the hyperplane.|name=hyperplanes|group=}} with the 6 vertices of an [[W:Octahedron|octahedron]] cell on each of the outer hyperplanes and 12 vertices of a [[W:Cuboctahedron|cuboctahedron]] on a central hyperplane. These vertices, combined with the 8 vertices of the [[16-cell]], represent the 32 root vectors of the B₄ and C₄ simple Lie groups. The 48 vertices (or strictly speaking their radius vectors) of the union of the 24-cell and its dual form the [[W:Root system|root system]] of type [[W:F4 (mathematics)|F₄]].{{Sfn|van Ittersum|2020|loc=§4.2.5|p=78}} The 24 vertices of the original 24-cell form a root system of type D₄; its size has the ratio {{sqrt|2}}:1. This is likewise true for the 24 vertices of its dual. The full [[W:Symmetry group|symmetry group]] of the 24-cell is the [[W:Weyl group|Weyl group]] of F₄, which is generated by [[W:Reflection (mathematics)|reflections]] through the hyperplanes orthogonal to the F₄ roots. This is a [[W:Solvable group|solvable group]] of order 1152. The rotational symmetry group of the 24-cell is of order 576. ===Quaternionic interpretation=== [[File:Binary tetrahedral group elements.png|thumb|The 24 quaternion{{Efn|name=quaternions}} elements of the [[W:Binary tetrahedral group|binary tetrahedral group]] match the vertices of the 24-cell. Seen in 4-fold symmetry projection: * 1 order-1: 1 * 1 order-2: -1 * 6 order-4: ±i, ±j, ±k * 8 order-6: (+1±i±j±k)/2 * 8 order-3: (-1±i±j±k)/2.]]When interpreted as the [[W:Quaternion|quaternion]]s,{{Efn|In [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]], a [[W:Quaternion|quaternion]] is simply a (w, x, y, z) Cartesian coordinate. [[W:William Rowan Hamilton|Hamilton]] did not see them as such when he [[W:History of quaternions|discovered the quaternions]]. [[W:Ludwig Schläfli|Schläfli]] would be the first to consider [[W:4-dimensional space|four-dimensional Euclidean space]], publishing his discovery of the regular [[W:Polyscheme|polyscheme]]s in 1852, but Hamilton would never be influenced by that work, which remained obscure into the 20th century. Hamilton found the quaternions when he realized that a fourth dimension, in some sense, would be necessary in order to model rotations in three-dimensional space.{{Sfn|Stillwell|2001|p=18-21}} Although he described a quaternion as an ''ordered four-element multiple of real numbers'', the quaternions were for him an extension of the complex numbers, not a Euclidean space of four dimensions.|name=quaternions}} the F₄ [[W:root lattice|root lattice]] (which is the integral span of the vertices of the 24-cell) is closed under multiplication and is therefore a [[W:ring (mathematics)|ring]]. This is the ring of [[W:Hurwitz integral quaternion|Hurwitz integral quaternion]]s. The vertices of the 24-cell form the [[W:Group of units|group of units]] (i.e. the group of invertible elements) in the Hurwitz quaternion ring (this group is also known as the [[W:Binary tetrahedral group|binary tetrahedral group]]). The vertices of the 24-cell are precisely the 24 Hurwitz quaternions with norm squared 1, and the vertices of the dual 24-cell are those with norm squared 2. The D₄ root lattice is the [[W:Dual lattice|dual]] of the F₄ and is given by the subring of Hurwitz quaternions with even norm squared.{{Sfn|Egan|2021|ps=; quaternions, the binary tetrahedral group and the binary octahedral group, with rotating illustrations.}} Viewed as the 24 unit [[W:Hurwitz quaternion|Hurwitz quaternion]]s, the [[#Great hexagons|unit radius coordinates]] of the 24-cell represent (in antipodal pairs) the 12 rotations of a regular tetrahedron.{{Sfn|Stillwell|2001|p=22}} Vertices of other [[W:Convex regular 4-polytope|convex regular 4-polytope]]s also form multiplicative groups of quaternions, but few of them generate a root lattice.{{Sfn|Koca et. al.|2007}} ===Voronoi cells=== The [[W:Voronoi cell|Voronoi cell]]s of the [[W:D4 (root system)|D₄]] root lattice are regular 24-cells. The corresponding Voronoi tessellation gives the [[W:Tessellation|tessellation]] of 4-dimensional [[W:Euclidean space|Euclidean space]] by regular 24-cells, the [[W:24-cell honeycomb|24-cell honeycomb]]. The 24-cells are centered at the D₄ lattice points (Hurwitz quaternions with even norm squared) while the vertices are at the F₄ lattice points with odd norm squared. Each 24-cell of this tessellation has 24 neighbors. With each of these it shares an octahedron. It also has 24 other neighbors with which it shares only a single vertex. Eight 24-cells meet at any given vertex in this tessellation. The [[W:Schläfli symbol|Schläfli symbol]] for this tessellation is {3,4,3,3}. It is one of only three regular tessellations of '''R'''⁴. The unit [[W:Ball (mathematics)|balls]] inscribed in the 24-cells of this tessellation give rise to the densest known [[W:lattice packing|lattice packing]] of [[W:Hypersphere|hypersphere]]s in 4 dimensions. The vertex configuration of the 24-cell has also been shown to give the [[W:24-cell honeycomb#Kissing number|highest possible kissing number in 4 dimensions]]. ===Radially equilateral honeycomb=== The dual tessellation of the [[W:24-cell honeycomb|24-cell honeycomb {3,4,3,3}]] is the [[W:16-cell honeycomb|16-cell honeycomb {3,3,4,3}]]. The third regular tessellation of four dimensional space is the [[W:Tesseractic honeycomb|tesseractic honeycomb {4,3,3,4}]], whose vertices can be described by 4-integer Cartesian coordinates.{{Efn|name=quaternions}} The congruent relationships among these three tessellations can be helpful in visualizing the 24-cell, in particular the radial equilateral symmetry which it shares with the tesseract.{{Efn||name=radially equilateral}} A honeycomb of unit edge length 24-cells may be overlaid on a honeycomb of unit edge length tesseracts such that every vertex of a tesseract (every 4-integer coordinate) is also the vertex of a 24-cell (and tesseract edges are also 24-cell edges), and every center of a 24-cell is also the center of a tesseract.{{Sfn|Coxeter|1973|p=163|ps=: Coxeter notes that [[W:Thorold Gosset|Thorold Gosset]] was apparently the first to see that the cells of the 24-cell honeycomb {3,4,3,3} are concentric with alternate cells of the tesseractic honeycomb {4,3,3,4}, and that this observation enabled Gosset's method of construction of the complete set of regular polytopes and honeycombs.}} The 24-cells are twice as large as the tesseracts by 4-dimensional content (hypervolume), so overall there are two tesseracts for every 24-cell, only half of which are inscribed in a 24-cell. If those tesseracts are colored black, and their adjacent tesseracts (with which they share a cubical facet) are colored red, a 4-dimensional checkerboard results.{{Sfn|Coxeter|1973|p=156|loc=|ps=: "...the chess-board has an n-dimensional analogue."}} Of the 24 center-to-vertex radii{{Efn|It is important to visualize the radii only as invisible interior features of the 24-cell (dashed lines), since they are not edges of the honeycomb. Similarly, the center of the 24-cell is empty (not a vertex of the honeycomb).}} of each 24-cell, 16 are also the radii of a black tesseract inscribed in the 24-cell. The other 8 radii extend outside the black tesseract (through the centers of its cubical facets) to the centers of the 8 adjacent red tesseracts. Thus the 24-cell honeycomb and the tesseractic honeycomb coincide in a special way: 8 of the 24 vertices of each 24-cell do not occur at a vertex of a tesseract (they occur at the center of a tesseract instead). Each black tesseract is cut from a 24-cell by truncating it at these 8 vertices, slicing off 8 cubic pyramids (as in reversing Gosset's construction,{{Sfn|Coxeter|1973|p=150|loc=Gosset}} but instead of being removed the pyramids are simply colored red and left in place). Eight 24-cells meet at the center of each red tesseract: each one meets its opposite at that shared vertex, and the six others at a shared octahedral cell. <!-- illustration needed: the red/black checkerboard of the combined 24-cell honeycomb and tesseractic honeycomb; use a vertex-first projection of the 24-cells, and outline the edges of the rhombic dodecahedra as blue lines --> The red tesseracts are filled cells (they contain a central vertex and radii); the black tesseracts are empty cells. The vertex set of this union of two honeycombs includes the vertices of all the 24-cells and tesseracts, plus the centers of the red tesseracts. Adding the 24-cell centers (which are also the black tesseract centers) to this honeycomb yields a 16-cell honeycomb, the vertex set of which includes all the vertices and centers of all the 24-cells and tesseracts. The formerly empty centers of adjacent 24-cells become the opposite vertices of a unit edge length 16-cell. 24 half-16-cells (octahedral pyramids) meet at each formerly empty center to fill each 24-cell, and their octahedral bases are the 6-vertex octahedral facets of the 24-cell (shared with an adjacent 24-cell).{{Efn|Unlike the 24-cell and the tesseract, the 16-cell is not radially equilateral; therefore 16-cells of two different sizes (unit edge length versus unit radius) occur in the unit edge length honeycomb. The twenty-four 16-cells that meet at the center of each 24-cell have unit edge length, and radius {{sfrac|{{radic|2}}|2}}. The three 16-cells inscribed in each 24-cell have edge length {{radic|2}}, and unit radius.}} Notice the complete absence of pentagons anywhere in this union of three honeycombs. Like the 24-cell, 4-dimensional Euclidean space itself is entirely filled by a complex of all the polytopes that can be built out of regular triangles and squares (except the 5-cell), but that complex does not require (or permit) any of the pentagonal polytopes.{{Efn|name=pentagonal polytopes}} == Rotations == The [[#Geometry|regular convex 4-polytopes]] are an [[W:Group action|expression]] of their underlying [[W:Symmetry (geometry)|symmetry]] which is known as [[W:SO(4)|SO(4)]],{{Sfn|Goucher|2019|loc=Spin Groups}} the [[W:Orthogonal group|group]] of rotations{{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹₄.}} about a fixed point in 4-dimensional Euclidean space.{{Efn|[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] may occur around a plane, as when adjacent cells are folded around their plane of intersection (by analogy to the way adjacent faces are folded around their line of intersection).{{Efn|Three dimensional [[W:Rotation (mathematics)#In Euclidean geometry|rotations]] occur around an axis line. [[W:Rotations in 4-dimensional Euclidean space|Four dimensional rotations]] may occur around a plane. So in three dimensions we may fold planes around a common line (as when folding a flat net of 6 squares up into a cube), and in four dimensions we may fold cells around a common plane (as when [[W:Tesseract#Geometry|folding a flat net of 8 cubes up into a tesseract]]). Folding around a square face is just folding around ''two'' of its orthogonal edges ''at the same time''; there is not enough space in three dimensions to do this, just as there is not enough space in two dimensions to fold around a line (only enough to fold around a point).|name=simple rotations|group=}} But in four dimensions there is yet another way in which rotations can occur, called a '''[[W:Rotations in 4-dimensional Euclidean space#Geometry of 4D rotations|double rotation]]'''. Double rotations are an emergent phenomenon in the fourth dimension and have no analogy in three dimensions: folding up square faces and folding up cubical cells are both examples of '''simple rotations''', the only kind that occur in fewer than four dimensions. In 3-dimensional rotations, the points in a line remain fixed during the rotation, while every other point moves. In 4-dimensional simple rotations, the points in a plane remain fixed during the rotation, while every other point moves. ''In 4-dimensional double rotations, a point remains fixed during rotation, and every other point moves'' (as in a 2-dimensional rotation!).{{Efn|There are (at least) two kinds of correct [[W:Four-dimensional space#Dimensional analogy|dimensional analogies]]: the usual kind between dimension ''n'' and dimension ''n'' + 1, and the much rarer and less obvious kind between dimension ''n'' and dimension ''n'' + 2. An example of the latter is that rotations in 4-space may take place around a single point, as do rotations in 2-space. Another is the [[W:n-sphere#Other relations|''n''-sphere rule]] that the ''surface area'' of the sphere embedded in ''n''+2 dimensions is exactly 2''π r'' times the ''volume'' enclosed by the sphere embedded in ''n'' dimensions, the most well-known examples being that the circumference of a circle is 2''π r'' times 1, and the surface area of the ordinary sphere is 2''π r'' times 2''r''. Coxeter cites{{Sfn|Coxeter|1973|p=119|loc=§7.1. Dimensional Analogy|ps=: "For instance, seeing that the circumference of a circle is 2''π r'', while the surface of a sphere is 4''π r ''², ... it is unlikely that the use of analogy, unaided by computation, would ever lead us to the correct expression [for the hyper-surface of a hyper-sphere], 2''π'' ²''r'' ³."}} this as an instance in which dimensional analogy can fail us as a method, but it is really our failure to recognize whether a one- or two-dimensional analogy is the appropriate method.|name=two-dimensional analogy}}|name=double rotations}} === The 3 Cartesian bases of the 24-cell === There are three distinct orientations of the tesseractic honeycomb which could be made to coincide with the 24-cell [[#Radially equilateral honeycomb|honeycomb]], depending on which of the 24-cell's three disjoint sets of 8 orthogonal vertices (which set of 4 perpendicular axes, or equivalently, which inscribed basis 16-cell){{Efn|name=three basis 16-cells}} was chosen to align it, just as three tesseracts can be inscribed in the 24-cell, rotated with respect to each other.{{Efn|name=three 8-cells}} The distance from one of these orientations to another is an [[#Isoclinic rotations|isoclinic rotation]] through 60 degrees (a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] of 60 degrees in each pair of completely orthogonal invariant planes, around a single fixed point).{{Efn|name=Clifford displacement}} This rotation can be seen most clearly in the hexagonal central planes, where every hexagon rotates to change which of its three diameters is aligned with a coordinate system axis.{{Efn|name=non-orthogonal hexagons|group=}} === Planes of rotation === [[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.{{Sfn|Kim|Rote|2016|p=6|loc=§5. Four-Dimensional Rotations}} Thus the general rotation in 4-space is a ''double rotation''.{{Sfn|Perez-Gracia & Thomas|2017|loc=§7. Conclusions|ps=; "Rotations in three dimensions are determined by a rotation axis and the rotation angle about it, where the rotation axis is perpendicular to the plane in which points are being rotated. The situation in four dimensions is more complicated. In this case, rotations are determined by two orthogonal planes and two angles, one for each plane. Cayley proved that a general 4D rotation can always be decomposed into two 4D rotations, each of them being determined by two equal rotation angles up to a sign change."}} There are two important special cases, called a ''simple rotation'' and an ''isoclinic rotation''.{{Efn|A [[W:Rotations in 4-dimensional Euclidean space|rotation in 4-space]] is completely characterized by choosing an invariant plane and an angle and direction (left or right) through which it rotates, and another angle and direction through which its one completely orthogonal invariant plane rotates. Two rotational displacements are identical if they have the same pair of invariant planes of rotation, through the same angles in the same directions (and hence also the same chiral pairing of directions). Thus the general rotation in 4-space is a '''double rotation''', characterized by ''two'' angles. A '''simple rotation''' is a special case in which one rotational angle is 0.{{Efn|Any double rotation (including an isoclinic rotation) can be seen as the composition of two simple rotations ''a'' and ''b'': the ''left'' double rotation as ''a'' then ''b'', and the ''right'' double rotation as ''b'' then ''a''. Simple rotations are not commutative; left and right rotations (in general) reach different destinations. The difference between a double rotation and its two composing simple rotations is that the double rotation is 4-dimensionally diagonal: each moving vertex reaches its destination ''directly'' without passing through the intermediate point touched by ''a'' then ''b'', or the other intermediate point touched by ''b'' then ''a'', by rotating on a single helical geodesic (so it is the shortest path).{{Efn|name=helical geodesic}} Conversely, any simple rotation can be seen as the composition of two ''equal-angled'' double rotations (a left isoclinic rotation and a right isoclinic rotation),{{Efn|name=one true circle}} as discovered by [[W:Arthur Cayley|Cayley]]; perhaps surprisingly, this composition ''is'' commutative, and is possible for any double rotation as well.{{Sfn|Perez-Gracia & Thomas|2017}}|name=double rotation}} An '''isoclinic rotation''' is a different special case,{{Efn|name=Clifford displacement}} similar but not identical to two simple rotations through the ''same'' angle.{{Efn|name=plane movement in rotations}}|name=identical rotations}} ==== Simple rotations ==== [[Image:24-cell.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Efn|name=planes through vertices}}]]In 3 dimensions a spinning polyhedron has a single invariant central ''plane of rotation''. The plane is an [[W:Invariant set|invariant set]] because each point in the plane moves in a circle but stays within the plane. Only ''one'' of a polyhedron's central planes can be invariant during a particular rotation; the choice of invariant central plane, and the angular distance and direction it is rotated, completely specifies the rotation. Points outside the invariant plane also move in circles (unless they are on the fixed ''axis of rotation'' perpendicular to the invariant plane), but the circles do not lie within a [[#Geodesics|''central'' plane]]. When a 4-polytope is rotating with only one invariant central plane, the same kind of [[W:Rotations in 4-dimensional Euclidean space#Simple rotations|simple rotation]] is happening that occurs in 3 dimensions. One difference is that instead of a fixed axis of rotation, there is an entire fixed central plane in which the points do not move. The fixed plane is the one central plane that is [[W:Completely orthogonal|completely orthogonal]] to the invariant plane of rotation. In the 24-cell, there is a simple rotation which will take any vertex ''directly'' to any other vertex, also moving most of the other vertices but leaving at least 2 and at most 6 other vertices fixed (the vertices that the fixed central plane intersects). The vertex moves along a great circle in the invariant plane of rotation between adjacent vertices of a great hexagon, a great square or a great [[W:Digon|digon]], and the completely orthogonal fixed plane is a digon, a square or a hexagon, respectively.{{Efn|In the 24-cell each great square plane is [[W:Completely orthogonal|completely orthogonal]] to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two antipodal vertices: a great [[W:Digon|digon]] plane.|name=pairs of completely orthogonal planes}} ==== Double rotations ==== [[Image:24-cell-orig.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|double rotation]].]]The points in the completely orthogonal central plane are not ''constrained'' to be fixed. It is also possible for them to be rotating in circles, as a second invariant plane, at a rate independent of the first invariant plane's rotation: a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] in two perpendicular non-intersecting planes{{Efn|name=how planes intersect at a single point}} of rotation at once.{{Efn|name=double rotation}} In a double rotation there is no fixed plane or axis: every point moves except the center point. The angular distance rotated may be different in the two completely orthogonal central planes, but they are always both invariant: their circularly moving points remain within the plane ''as the whole plane tilts sideways'' in the completely orthogonal rotation. A rotation in 4-space always has (at least) ''two'' completely orthogonal invariant planes of rotation, although in a simple rotation the angle of rotation in one of them is 0. Double rotations come in two [[W:Chiral|chiral]] forms: ''left'' and ''right'' rotations.{{Efn|The adjectives ''left'' and ''right'' are commonly used in two different senses, to distinguish two distinct kinds of pairing. They can refer to alternate directions: the hand on the left side of the body, versus the hand on the right side. Or they can refer to a [[W:Chiral|chiral]] pair of enantiomorphous objects: a left hand is the mirror image of a right hand (like an inside-out glove). In the case of hands the sense intended is rarely ambiguous, because of course the hand on your left side ''is'' the mirror image of the hand on your right side: a hand is either left ''or'' right in both senses. But in the case of double-rotating 4-dimensional objects, only one sense of left versus right properly applies: the enantiomorphous sense, in which the left and right rotation are inside-out mirror images of each other. There ''are'' two directions, which we may call positive and negative, in which moving vertices may be circling on their isoclines, but it would be ambiguous to label those circular directions "right" and "left", since a rotation's direction and its chirality are independent properties: a right (or left) rotation may be circling in either the positive or negative direction. The left rotation is not rotating "to the left", the right rotation is not rotating "to the right", and unlike your left and right hands, double rotations do not lie on the left or right side of the 4-polytope. If double rotations must be analogized to left and right hands, they are better thought of as a pair of clasped hands, centered on the body, because of course they have a common center.|name=clasped hands}} In a double rotation each vertex moves in a spiral along two orthogonal great circles at once.{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in their places in the plane ''as the plane moves'', rotating ''and'' tilting sideways by the angle that the ''other'' plane rotates.|name=helical geodesic}} Either the path is right-hand [[W:Screw thread#Handedness|threaded]] (like most screws and bolts), moving along the circles in the "same" directions, or it is left-hand threaded (like a reverse-threaded bolt), moving along the circles in what we conventionally say are "opposite" directions (according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is "up" on each of the 4 coordinate axes).{{Sfn|Perez-Gracia & Thomas|2017|loc=§5. A useful mapping|pp=12−13}} In double rotations of the 24-cell that take vertices to vertices, one invariant plane of rotation contains either a great hexagon, a great square, or only an axis (two vertices, a great digon). The completely orthogonal invariant plane of rotation will necessarily contain a great digon, a great square, or a great hexagon, respectively. The selection of an invariant plane of rotation, a rotational direction and angle through which to rotate it, and a rotational direction and angle through which to rotate its completely orthogonal plane, completely determines the nature of the rotational displacement. In the 24-cell there are several noteworthy kinds of double rotation permitted by these parameters.{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|pp=30-32|ps=; §3. The Dodecagonal Aspect;{{Efn|name=Petrie and Clifford dodecagram}} Coxeter considers the 150°/30° double rotation of period 12 which locates 12 of the 225 distinct 24-cells inscribed in the [[120-cell]], a regular 4-polytope with 120 dodecahedral cells that is the convex hull of the compound of 25 disjoint 24-cells.}} ==== Isoclinic rotations ==== When the angles of rotation in the two completely orthogonal invariant planes are exactly the same, a [[W:Rotations in 4-dimensional Euclidean space#Special property of SO(4) among rotation groups in general|remarkably symmetric]] [[W:Geometric transformation|transformation]] occurs:{{Sfn|Perez-Gracia & Thomas|2017|loc=§2. Isoclinic rotations|pp=2−3}} all the great circle planes Clifford parallel{{Efn|name=Clifford parallels}} to the pair of invariant planes become pairs of invariant planes of rotation themselves, through that same angle, and the 4-polytope rotates [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] in many directions at once.{{Sfn|Kim|Rote|2016|loc=§6. Angles between two Planes in 4-Space|pp=7-10}} Each vertex moves an equal distance in four orthogonal directions at the same time.{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance|Pythagorean distance]] equal to the square root of four times the square of that distance. (In the 4-dimensional case, the orthogonal distance equals half the total Pythagorean distance.) All vertices are displaced to a vertex more than one edge length away.{{Efn|name=missing the nearest vertices}} For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} ≈ 0.866 (half the {{radic|3}} chord length) in four orthogonal directions.{{Efn|{{radic|3/4}} ≈ 0.866 is the long radius of the {{radic|2}}-edge regular tetrahedron (the unit-radius 16-cell's cell). Those four tetrahedron radii are not orthogonal, and they radiate symmetrically compressed into 3 dimensions (not 4). The four orthogonal {{radic|3/4}} ≈ 0.866 displacements summing to a 120° degree displacement in the 24-cell's characteristic isoclinic rotation{{Efn|name=isoclinic 4-dimensional diagonal}} are not as easy to visualize as radii, but they can be imagined as successive orthogonal steps in a path extending in all 4 dimensions, along the orthogonal edges of a [[5-cell#Orthoschemes|4-orthoscheme]]. In an actual left (or right) isoclinic rotation the four orthogonal {{radic|3/4}} ≈ 0.866 steps of each 120° displacement are concurrent, not successive, so they ''are'' actually symmetrical radii in 4 dimensions. In fact they are four orthogonal [[#Characteristic orthoscheme|mid-edge radii of a unit-radius 24-cell]] centered at the rotating vertex. Finally, in 2 dimensional units, {{radic|3/4}} ≈ 0.866 is the area of the equilateral triangle face of the unit-edge, unit-radius 24-cell. The area of the radial equilateral triangles in a unit-radius radially equilateral polytope{{Efn|name=radially equilateral}} is {{radic|3/4}} ≈ 0.866.|name=root 3/4}}|name=isoclinic 4-dimensional diagonal}} In the 24-cell any isoclinic rotation through 60 degrees in a hexagonal plane takes each vertex to a vertex two edge lengths away, rotates ''all 16'' hexagons by 60 degrees, and takes ''every'' great circle polygon (square,{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} hexagon or triangle) to a Clifford parallel great circle polygon of the same kind 120 degrees away. An isoclinic rotation is also called a ''Clifford displacement'', after its [[W:William Kingdon Clifford|discoverer]].{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle in the completely orthogonal rotation.{{Efn|name=one true circle}} A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways.{{Efn|name=plane movement in rotations}} All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon 120 degrees away. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 120 degrees away.|name=Clifford displacement}} The 24-cell in the ''double'' rotation animation appears to turn itself inside out.{{Efn|That a double rotation can turn a 4-polytope inside out is even more noticeable in the [[W:Rotations in 4-dimensional Euclidean space#Double rotations|tesseract double rotation]].}} It appears to, because it actually does, reversing the [[W:Chirality|chirality]] of the whole 4-polytope just the way your bathroom mirror reverses the chirality of your image by a 180 degree reflection. Each 360 degree isoclinic rotation is as if the 24-cell surface had been stripped off like a glove and turned inside out, making a right-hand glove into a left-hand glove (or vice versa).{{Sfn|Coxeter|1973|p=141|loc=§7.x. Historical remarks|ps=; "[[W:August Ferdinand Möbius|Möbius]] realized, as early as 1827, that a four-dimensional rotation would be required to bring two enantiomorphous solids into coincidence. This idea was neatly deployed by [[W:H. G. Wells|H. G. Wells]] in ''The Plattner Story''."}} In a simple rotation of the 24-cell in a hexagonal plane, each vertex in the plane rotates first along an edge to an adjacent vertex 60 degrees away. But in an isoclinic rotation in ''two'' completely orthogonal planes one of which is a great hexagon,{{Efn|name=pairs of completely orthogonal planes}} each vertex rotates first to a non-adjacent vertex {{radic|3}} and 120° distant. The double 60-degree rotation's helical geodesics pass through every other vertex, missing the vertices in between.{{Efn|In an isoclinic rotation vertices move diagonally, like the [[W:bishop (chess)|bishop]]s in [[W:Chess|chess]]. Vertices in an isoclinic rotation ''cannot'' reach their orthogonally nearest neighbor vertices{{Efn|name=8 nearest vertices}} by double-rotating directly toward them (and also orthogonally to that direction), because that double rotation takes them diagonally between their nearest vertices, missing them, to a vertex farther away in a larger-radius surrounding shell of vertices,{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} the way bishops are confined to the white or black squares of the [[W:Chessboard|chessboard]] and cannot reach squares of the opposite color, even those immediately adjacent.{{Efn|Isoclinic rotations{{Efn|name=isoclinic geodesic}} partition the 24 cells (and the 24 vertices) of the 24-cell into two disjoint subsets of 12 cells (and 12 vertices), even and odd (or black and white), which shift places among themselves, in a manner dimensionally analogous to the way the [[W:Bishop (chess)|bishops]]' diagonal moves{{Efn|name=missing the nearest vertices}} restrict them to the black or white squares of the [[W:Chessboard|chessboard]].{{Efn|Left and right isoclinic rotations partition the 24 cells (and 24 vertices) into black and white in the same way.{{Sfn|Coxeter|1973|p=156|loc=|ps=: "...the chess-board has an n-dimensional analogue."}} The rotations of all fibrations of the same kind of great polygon use the same chessboard, which is a convention of the coordinate system based on even and odd coordinates. ''Left and right are not colors:'' in either a left (or right) rotation half the moving vertices are black, running along black isoclines through black vertices, and the other half are white vertices, also rotating among themselves.{{Efn|Chirality and even/odd parity are distinct flavors. Things which have even/odd coordinate parity are '''''black or white:''''' the squares of the [[W:Chessboard|chessboard]],{{Efn|Since it is difficult to color points and lines white, we sometimes use black and red instead of black and white. In particular, isocline chords are sometimes shown as black or red ''dashed'' lines.{{Efn|name=interior features}}|name=black and red}} '''cells''', '''vertices''' and the '''isoclines''' which connect them by isoclinic rotation.{{Efn|name=isoclinic geodesic}} Everything else is '''''black and white:''''' e.g. adjacent '''face-bonded cell pairs''', or '''edges''' and '''chords''' which are black at one end and white at the other. Things which have [[W:Chirality|chirality]] come in '''''right or left''''' enantiomorphous forms: '''[[#Isoclinic rotations|isoclinic rotations]]''' and '''chiral objects''' which include '''[[#Characteristic orthoscheme|characteristic orthoscheme]]s''', '''[[#Chiral symmetry operations|sets of Clifford parallel great polygon planes]]''',{{Efn|name=completely orthogonal Clifford parallels are special}} '''[[W:Fiber bundle|fiber bundle]]s''' of Clifford parallel circles (whether or not the circles themselves are chiral), and the chiral cell rings of tetrahedra found in the [[16-cell#Helical construction|16-cell]] and [[600-cell#Boerdijk–Coxeter helix rings|600-cell]]. Things which have '''''neither''''' an even/odd parity nor a chirality include all '''edges''' and '''faces''' (shared by black and white cells), '''[[#Geodesics|great circle polygons]]''' and their '''[[W:Hopf fibration|fibration]]s''', and non-chiral cell rings such as the 24-cell's [[#Cell rings|cell rings of octahedra]]. Some things are associated with '''''both''''' an even/odd parity and a chirality: '''isoclines''' are black or white because they connect vertices which are all of the same color, and they ''act'' as left or right chiral objects when they are vertex paths in a left or right rotation, although they have no inherent chirality themselves. Each left (or right) rotation traverses an equal number of black and white isoclines.{{Efn|name=Clifford polygon}}|name=left-right versus black-white}}|name=isoclinic chessboard}}|name=black and white}} Things moving diagonally move farther than 1 unit of distance in each movement step ({{radic|2}} on the chessboard, {{radic|3}} in the 24-cell), but at the cost of ''missing'' half the destinations.{{Efn|name=one true circle}} However, in an isoclinic rotation of a rigid body all the vertices rotate at once, so every destination ''will'' be reached by some vertex. Moreover, there is another isoclinic rotation in hexagon invariant planes which does take each vertex to an adjacent (nearest) vertex. A 24-cell can displace each vertex to a vertex 60° away (a nearest vertex) by rotating isoclinically by 30° in two completely orthogonal invariant planes (one of them a hexagon), ''not'' by double-rotating directly toward the nearest vertex (and also orthogonally to that direction), but instead by double-rotating directly toward a more distant vertex (and also orthogonally to that direction). This helical 30° isoclinic rotation takes the vertex 60° to its nearest-neighbor vertex by a ''different path'' than a simple 60° rotation would. The path along the helical isocline and the path along the simple great circle have the same 60° arc-length, but they consist of disjoint sets of points (except for their endpoints, the two vertices). They are both geodesic (shortest) arcs, but on two alternate kinds of geodesic circle. One is doubly curved (through all four dimensions), and one is simply curved (lying in a two-dimensional plane).|name=missing the nearest vertices}} Each {{radic|3}} chord of the helical geodesic{{Efn|Although adjacent vertices on the isoclinic geodesic are a {{radic|3}} chord apart, a point on a rigid body under rotation does not travel along a chord: it moves along an arc between the two endpoints of the chord (a longer distance). In a ''simple'' rotation between two vertices {{radic|3}} apart, the vertex moves along the arc of a hexagonal great circle to a vertex two great hexagon edges away, and passes through the intervening hexagon vertex midway. But in an ''isoclinic'' rotation between two vertices {{radic|3}} apart the vertex moves along a helical arc called an isocline (not a planar great circle),{{Efn|name=isoclinic geodesic}} which does ''not'' pass through an intervening vertex: it misses the vertex nearest to its midpoint.{{Efn|name=missing the nearest vertices}}|name=isocline misses vertex}} crosses between two Clifford parallel hexagon central planes, and lies in another hexagon central plane that intersects them both.{{Efn|Departing from any vertex V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V₀ and V₂ are ''two'' {{radic|3}} chords apart,{{Efn|V₀ and V₂ are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V₀ and V₂ are ''one'' {{radic|3}} chord apart on some other isocline, and just {{radic|1}} apart on some great hexagon. Between V₀ and V₂, the isoclinic rotation has gone the long way around the 24-cell over two {{radic|3}} chords to reach a vertex that was only {{radic|1}} away. More generally, isoclines are geodesics because the distance between their successive vertices is the shortest distance between those two vertices in some rotation connecting them, but on the 3-sphere there may be another rotation which is shorter. A path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}} P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. V₀ and V₃ are adjacent vertices, {{radic|1}} apart. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 180° isoclinic rotation, and one quarter of the 24-cell's double-loop decagram₅ Clifford polygon.{{Efn|name=Clifford polygon}}|name=360 degree geodesic path visiting 3 hexagonal planes}} The {{radic|3}} chords meet at a 60° angle, but since they lie in different planes they form a [[W:Helix|helix]] not a [[#Great triangles|triangle]]. The helix of {{radic|3}} chords closes into a loop only after twelve {{radic|3}} chords: a 720° isoclinic rotation{{Efn|An isoclinic rotation by 60° is two simple rotations by 60° at the same time.{{Efn|The composition of two simple 60° rotations in a pair of completely orthogonal invariant planes is a 60° isoclinic rotation in ''four'' pairs of completely orthogonal invariant planes.{{Efn|name=double rotation}} Thus the isoclinic rotation is the compound of four simple rotations, and all 24 vertices rotate in invariant hexagon planes, versus just 6 vertices in a simple rotation.}} It moves all the vertices 120° at the same time, in various different directions. Six successive diagonal rotational increments, of 60°x60° each, move each vertex through 720° on a Möbius double loop called an ''isocline'', ''twice'' around the 24-cell and back to its point of origin, in the ''same time'' (six rotational units) that it would take a simple rotation to take the vertex ''once'' around the 24-cell on an ordinary great circle.{{Efn|name=double threaded}} The helical double loop 4𝝅 isocline is just another kind of ''single'' full circle, of the same time interval and period (6 chords) as the simple great circle. The isocline is ''one'' true circle,{{Efn|name=4-dimensional great circles}} as perfectly round and geodesic as the simple great circle, even through its chords are {{radic|3}} longer, its circumference is 4𝝅 instead of 2𝝅,{{Efn|All 3-sphere isoclines of the same circumference are directly or enantiomorphously congruent circles.{{Efn|name=not all isoclines are circles}} An ordinary great circle is an isocline of circumference <math>2\pi r</math>; simple rotations of unit-radius polytopes take place on 2𝝅 isoclines. Double rotations may have isoclines of other than <math>2\pi r</math> circumference. The ''characteristic rotation'' of a regular 4-polytope is the isoclinic rotation in which the central planes containing its edges are invariant planes of rotation. The 16-cell and 24-cell edge-rotate on isoclines of 4𝝅 circumference. The 600-cell edge-rotates on isoclines of 5𝝅 circumference.|name=isocline circumference}} it circles through four dimensions instead of two,{{Efn|name=Villarceau circles}} and it has two chiral forms (left and right).{{Efn|name=Clifford polygon}} Nevertheless, to avoid confusion we always refer to it as an ''isocline'' and reserve the term ''great circle'' for an ordinary great circle in the plane.{{Efn|name=isocline}}|name=one true circle}} over a [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] {12/5} dodecagram with {{radic|3}} edges. All 24 vertices rotate at once, on two Clifford parallel dodecagon isoclines. Each vertex visits half the 24 vertex positions. Although each isocline is a circular spiral through all 4 dimensions, not a 2-dimensional circle in the plane, like an ordinary great circle it is a geodesic, because it is the shortest circle through those 12 vertices.{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''.{{Efn||name=double rotation}} A '''[[W:Geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:Helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:Screw threads|screw threads]] either, because they form a closed loop like any circle.{{Efn|name=double threaded}} Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in ''two'' orthogonal great circles at once.{{Efn|Isoclinic geodesics or ''isoclines'' are 4-dimensional great circles in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two orthogonal great circles at once.{{Efn|name=not all isoclines are circles}} They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of great circles (great 1-spheres).{{Efn|name=great 2-spheres}} Discrete isoclines are polygons;{{Efn|name=Clifford polygon}} discrete great 2-spheres are polyhedra.|name=4-dimensional great circles}} They are true circles,{{Efn|name=one true circle}} and even form [[W:Hopf fibration|fibrations]] like ordinary 2-dimensional great circles.{{Efn|name=hexagonal fibrations}}{{Efn|name=square fibrations}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are [[W:Geodesics|geodesics]], and isoclines on the [[W:3-sphere|3-sphere]] are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.|name=not all isoclines are circles}} they always occur in pairs{{Efn|Isoclines on the 3-sphere occur in non-intersecting pairs of even/odd coordinate parity.{{Efn|name=black and white}} A single black or white isocline forms a [[W:Möbius loop|Möbius loop]] called the {1,1} torus knot or Villarceau circle{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut."}} in which each of two "circles" linked in a Möbius "figure eight" loop traverses through all four dimensions.{{Efn|name=Clifford polygon}} The double loop is a true circle in four dimensions.{{Efn|name=one true circle}} Even and odd isoclines are also linked, not in a Möbius loop but as a [[W:Hopf link|Hopf link]] of two non-intersecting circles,{{Efn|name=Clifford parallels}} as are all the Clifford parallel isoclines of a [[W:Hopf fibration|Hopf fiber bundle]].|name=Villarceau circles}} as [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]], the geodesic paths traversed by vertices in an [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] around the 3-sphere through the non-adjacent vertices{{Efn|name=missing the nearest vertices}} of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] '''Clifford polygon'''.{{Efn|name=Clifford polygon}}|name=isoclinic geodesic}} A 360 degree isoclinic rotation moves each vertex only halfway around its circuit. After six 60° rotational displacements each vertex has departed from six vertex positions and reached a seventh vertex position adjacent to its antipodal vertex. Each central plane (every hexagon or square in the 24-cell) has rotated 360 degrees and been tilted sideways all the way around 360 degrees back to its original position (like a coin flipping twice), but its [[W:Orientation entanglement|orientation]] in the 4-space in which it is embedded is now different.{{Sfn|Mebius|2015|loc=Motivation|pp=2-3|ps=; "This research originated from ... the desire to construct a computer implementation of a specific motion of the human arm, known among folk dance experts as the ''Philippine wine dance'' or ''Binasuan'' and performed by physicist [[W:Richard P. Feynman|Richard P. Feynman]] during his [[W:Dirac|Dirac]] memorial lecture 1986<ref>{{Cite book|title=Elementary particles and the laws of physics|chapter=The reason for antiparticles|last1=Feynman|first1=Richard|last2=Weinberg|first2=Steven|publisher=Cambridge University Press|year=1987|ref={{SfnRef|Feynman & Weinberg|1987}}}}</ref> to show that a single rotation (2𝝅) is not equivalent in all respects to no rotation at all, whereas a double rotation (4𝝅) is."}} Because the 24-cell is now inside-out, if the isoclinic rotation is continued in the same rotational direction through six more 60° isoclinic displacements, the 24 moving vertices will pass through the other half of the vertices, and each vertex will arrive back at the vertex position it departed from, after tracing a closed helical loop over twelve {{radic|3}} chords. It takes a 720 degree isoclinic rotation for each vertex to traverse a geodesic circle of circumference <math>8\pi</math>, [[W:Winding number|winding]] around the 24-cell 5 times and returning the 24-cell to its original orientation.{{Efn|In a 720° isoclinic rotation of a rigid 24-cell the 24 vertices rotate along two Clifford parallel dodecagram₅ geodesic loops (12 vertices circling in each loop) and return to their original positions.{{Efn|name=Villarceau circles}}}} The twin dodecagram winding paths that the vertices take as they loop five times around the 24-cell form a double helix bent into a ring.{{Efn|The 24-cell's helical dodecagram₅ geodesic is bent into a twisted ring in the fourth dimension. Its [[W:Screw thread|screw thread]] maintains the same chirality{{Efn|name=Clifford polygon}} and even/odd parity of rotation (black or white) throughout.{{Efn|name=black and white}} Two Clifford parallel 12-vertex circular helixes form a Möbius strip one edge wide, a 4-dimensional circular double helix.{{Efn|A strip of paper can form a [[W:Möbius strip#Polyhedral surfaces and flat foldings|flattened Möbius strip]] in the plane by folding it at <math>60^\circ</math> angles so that its center line lies along an equilateral triangle, and attaching the ends. The shortest strip for which this is possible consists of three equilateral paper triangles, folded at the edges where two triangles meet. Since the loop traverses both sides of each paper triangle, it is a hexagonal loop over six equilateral triangles. Its [[W:Aspect ratio|aspect ratio]]{{snd}}the ratio of the strip's length{{efn|The length of a strip can be measured at its centerline, or by cutting the resulting Möbius strip perpendicularly to its boundary so that it forms a rectangle.}} to its width{{snd}}is {{nowrap|<math>\sqrt 3\approx 1.73</math>.}}}} This 60° isocline is a [[W:Skew polygon|skewed]] instance of the [[W:Polygram (geometry)#Regular compound polygons|regular compound polygon]] denoted {12/5} or dodecagram₅. Successive {{radic|3}} edges belong to different [[#8-cell|8-cells]], as the 720° isoclinic rotation takes each hexagon through all six hexagons in the [[#6-cell rings|6-cell ring]], and each 8-cell through all three 8-cells twice.{{Efn|name=three 8-cells}}|name=double threaded}} === Clifford parallel polytopes === Two planes are also called ''isoclinic'' if an isoclinic rotation will bring them together.{{Efn|name=two angles between central planes}} The isoclinic planes are precisely those central planes with Clifford parallel geodesic great circles.{{Sfn|Kim|Rote|2016|loc=Relations to Clifford parallelism|pp=8-9}} Clifford parallel great circles do not intersect,{{Efn|name=Clifford parallels}} so isoclinic great circle polygons have disjoint vertices. In the 24-cell every hexagonal central plane is isoclinic to three others, and every square central plane is isoclinic to five others. We can pick out 4 mutually isoclinic (Clifford parallel) great hexagons (four different ways) covering all 24 vertices of the 24-cell just once (a hexagonal fibration).{{Efn|The 24-cell has four sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]]{{Efn|name=Clifford parallels}} great circles each passing through 6 vertices (a great hexagon), with only one great hexagon in each set passing through each vertex, and the 4 hexagons in each set reaching all 24 vertices.{{Efn|name=four hexagonal fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of non-intersecting linked great circles. The 24-cell can also be divided (eight different ways) into 2 disjoint subsets of 12 vertices (dodecagrams), each skew [[#Helical hdodecagrams and their isoclines|dodecagram forming an isoclinic geodesic or ''isocline'']] that is the rotational circle traversed by those 12 vertices in one particular left or right [[#Isoclinic rotations|isoclinic rotation]]. Each of these sets of two Clifford parallel isoclines belongs to one of the four discrete Hopf fibrations of hexagonal great circles as either its left or right rotation.{{Efn|Each set of four [[W:Clifford parallel|Clifford parallel]] [[#Geodesics|great circle]] polygons is a different bundle of fibers than the corresponding set of two Clifford parallel isocline{{Efn|name=isoclinic geodesic}} polygrams, but the two [[W:Fiber bundles|fiber bundles]] together constitute the same discrete [[W:Hopf fibration|Hopf fibration]], because they enumerate the 24 vertices together by their intersection in the same distinct (left or right) isoclinic rotation. They are the [[W:Warp and woof|warp and woof]] of the same woven fabric that is the fibration.|name=great circles and isoclines are same fibration}}|name=hexagonal fibrations}} We can pick out 6 mutually isoclinic (Clifford parallel) great squares{{Efn|Each great square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal). There is also another way in which completely orthogonal planes are in a distinguished category of Clifford parallel planes: they are not [[W:Chiral|chiral]], or strictly speaking they possess both chiralities. A pair of isoclinic (Clifford parallel) planes is either a ''left pair'' or a ''right pair'', unless they are separated by two angles of 90° (completely orthogonal planes) or 0° (coincident planes).{{Sfn|Kim|Rote|2016|p=8|loc=Left and Right Pairs of Isoclinic Planes}} Most isoclinic planes are brought together only by a left isoclinic rotation or a right isoclinic rotation, respectively. Completely orthogonal planes are special: the pair of planes is both a left and a right pair, so either a left or a right isoclinic rotation will bring them together. This occurs because isoclinic square planes are 180° apart at all vertex pairs: not just Clifford parallel but completely orthogonal. The isoclines (chiral vertex paths){{Efn|name=isoclinic geodesic}} of 90° isoclinic rotations are special for the same reason. Left and right isoclines loop through the same set of antipodal vertices (hitting both ends of each [[16-cell#Helical construction|16-cell axis]]), instead of looping through disjoint left and right subsets of black or white antipodal vertices (hitting just one end of each axis), as the left and right isoclines of all other fibrations do.|name=completely orthogonal Clifford parallels are special}} (three different ways) covering all 24 vertices of the 24-cell just once (a square fibration).{{Efn|The 24-cell has three sets of 6 non-intersecting Clifford parallel great circles each passing through 4 vertices (a great square), with only one great square in each set passing through each vertex, and the 6 squares in each set reaching all 24 vertices.{{Efn|name=three square fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of 6 non-intersecting linked great squares, which is simply the compound of the three inscribed 16-cell's discrete Hopf fibrations of 2 great squares. The 24-cell can also be divided (six different ways) into 3 disjoint subsets of 8 vertices (octagrams) that do ''not'' lie in a square central plane, but comprise a 16-cell and lie on a skew [[#Helical octagrams and thei isoclines|octagram₃ forming an isoclinic geodesic or ''isocline'']] that is the rotational cirle traversed by those 8 vertices in one particular left or right [[16-cell#Rotations|isoclinic rotation]] as they rotate positions within the 16-cell.|name=square fibrations}} Every isoclinic rotation taking vertices to vertices corresponds to a discrete fibration.{{Efn|name=fibrations are distinguished only by rotations}} Two dimensional great circle polygons are not the only polytopes in the 24-cell which are parallel in the Clifford sense.{{Sfn|Tyrrell & Semple|1971|pp=1-9|loc=§1. Introduction}} Congruent polytopes of 2, 3 or 4 dimensions can be said to be Clifford parallel in 4 dimensions if their corresponding vertices are all the same distance apart. The three 16-cells inscribed in the 24-cell are Clifford parallels. Clifford parallel polytopes are ''completely disjoint'' polytopes.{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or linage.|name=completely disjoint}} A 60 degree isoclinic rotation in hexagonal planes takes each 16-cell to a disjoint 16-cell. Like all [[#Double rotations|double rotations]], isoclinic rotations come in two [[W:Chiral|chiral]] forms: there is a disjoint 16-cell to the ''left'' of each 16-cell, and another to its ''right''.{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=Six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[#Great hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[#Great squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:Tesseract|hypercube (a tesseract or 8-cell)]], in [[#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells (as in [[#Reciprocal constructions from 8-cell and 16-cell|Gosset's construction of the 24-cell]]). The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[W:3-sphere|3-sphere]] symmetric: four [[#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' orthogonal great circles at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:Chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell (whose vertices are one {{radic|1}} edge away) by rotating toward it;{{Efn|name=missing the nearest vertices}} it can only reach the 16-cell ''beyond'' it (120° away). But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. If so, that was not an error in our visualization; there are two chiral images we can ascribe to the 24-cell, from mirror-image viewpoints which turn the 24-cell inside-out. But from either viewpoint, the 16-cell to the "left" is the one reached by the left isoclinic rotation, as that is the only [[#Double rotations|sense in which the two 16-cells are left or right]] of each other.{{Efn|name=clasped hands}}|name=three isoclinic 16-cells}} All Clifford parallel 4-polytopes are related by an isoclinic rotation,{{Efn|name=Clifford displacement}} but not all isoclinic polytopes are Clifford parallels (completely disjoint).{{Efn|All isoclinic ''planes'' are Clifford parallels (completely disjoint).{{Efn|name=completely disjoint}} Three and four dimensional cocentric objects may intersect (sharing elements) but still be related by an isoclinic rotation. Polyhedra and 4-polytopes may be isoclinic and ''not'' disjoint, if all of their corresponding planes are either Clifford parallel, or cocellular (in the same hyperplane) or coincident (the same plane).}} The three 8-cells in the 24-cell are isoclinic but not Clifford parallel. Like the 16-cells, they are rotated 60 degrees isoclinically with respect to each other, but their vertices are not all disjoint (and therefore not all equidistant). Each vertex occurs in two of the three 8-cells (as each 16-cell occurs in two of the three 8-cells).{{Efn|name=three 8-cells}} Isoclinic rotations relate the convex regular 4-polytopes to each other. An isoclinic rotation of a single 16-cell will generate{{Efn|By ''generate'' we mean simply that some vertex of the first polytope will visit each vertex of the generated polytope in the course of the rotation.}} a 24-cell. A simple rotation of a single 16-cell will not, because its vertices will not reach either of the other two 16-cells' vertices in the course of the rotation. An isoclinic rotation of the 24-cell will generate the 600-cell, and an isoclinic rotation of the 600-cell will generate the 120-cell. (Or they can all be generated directly by an isoclinic rotation of the 16-cell, generating isoclinic copies of itself.) The different convex regular 4-polytopes nest inside each other, and multiple instances of the same 4-polytope hide next to each other in the Clifford parallel subspaces that comprise the 3-sphere.{{Sfn|Tyrrell & Semple|1971|loc=Clifford Parallel Spaces and Clifford Reguli|pp=20-33}} For an object of more than one dimension, the only way to reach these parallel subspaces directly is by isoclinic rotation. Like a key operating a four-dimensional lock, an object must twist in two completely perpendicular tumbler cylinders at once in order to move the short distance between Clifford parallel subspaces. === Rings === In the 24-cell there are sets of rings of six different kinds, described separately in detail in other sections of this article. This section describes how the different kinds of rings are [[#Relationships among interior polytopes|intertwined]]. The 24-cell contains four kinds of [[#Geodesics|geodesic fibers]] (polygonal rings running through vertices): [[#Great squares|great circle squares]] and their [[16-cell#Helical construction|isoclinic helix octagrams]],{{Efn|name=square fibrations}} and [[#Great hexagons|great circle hexagons]] and their [[#Isoclinic rotations|isoclinic helix dodecagrams]].{{Efn|name=hexagonal fibrations}} It also contains two kinds of [[#Cell rings|cell rings]] (chains of octahedra bent into a ring in the fourth dimension): four octahedra connected vertex-to-vertex and bent into a square, and six octahedra connected face-to-face and bent into a hexagon. ==== 4-cell rings ==== Four unit-edge-length octahedra can be connected vertex-to-vertex along a common axis of length 4{{radic|2}}. The axis can then be bent into a square of edge length {{radic|2}}. Although it is possible to do this in a space of only three dimensions, that is not how it occurs in the 24-cell. Although the {{radic|2}} axes of the four octahedra occupy the same plane, forming one of the 18 {{radic|2}} great squares of the 24-cell, each octahedron occupies a different 3-dimensional hyperplane,{{Efn|Just as each face of a [[W:Polyhedron|polyhedron]] occupies a different (2-dimensional) face plane, each cell of a [[W:Polychoron|polychoron]] occupies a different (3-dimensional) cell [[W:Hyperplane|hyperplane]].{{Efn|name=hyperplanes}}}} and all four dimensions are utilized. The 24-cell can be partitioned into 6 such 4-cell rings (three different ways), mutually interlinked like adjacent links in a chain (but these [[W:Link (knot theory)|links]] all have a common center). An [[#Isoclinic rotations|isoclinic rotation]] in a great square plane by a multiple of 90° takes each octahedron in the ring to an octahedron in the ring. ==== 6-cell rings ==== [[File:Six face-bonded octahedra.jpg|thumb|400px|A 4-dimensional ring of 6 face-bonded octahedra, bounded by two intersecting sets of three Clifford parallel great hexagons of different colors, cut and laid out flat in 3 dimensional space.{{Efn|name=6-cell ring}}]]Six regular octahedra can be connected face-to-face along a common axis that passes through their centers of volume, forming a stack or column with only triangular faces. In a space of four dimensions, the axis can then be bent 60° in the fourth dimension at each of the six octahedron centers, in a plane orthogonal to all three orthogonal central planes of each octahedron, such that the top and bottom triangular faces of the column become coincident. The column becomes a ring around a hexagonal axis. The 24-cell can be partitioned into 4 such rings (four different ways), mutually interlinked. Because the hexagonal axis joins cell centers (not vertices), it is not a great hexagon of the 24-cell.{{Efn|The axial hexagon of the 6-octahedron ring does not intersect any vertices or edges of the 24-cell, but it does hit faces. In a unit-edge-length 24-cell, it has edges of length 1/2.{{Efn|When unit-edge octahedra are placed face-to-face the distance between their centers of volume is {{radic|2/3}} ≈ 0.816.{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(i): Octahedron}} When 24 face-bonded octahedra are bent into a 24-cell lying on the 3-sphere, the centers of the octahedra are closer together in 4-space. Within the curved 3-dimensional surface space filled by the 24 cells, the cell centers are still {{radic|2/3}} apart along the curved geodesics that join them. But on the straight chords that join them, which dip inside the 3-sphere, they are only 1/2 edge length apart.}} Because it joins six cell centers, the axial hexagon is a great hexagon of the smaller dual 24-cell that is formed by joining the 24 cell centers.{{Efn|name=common core}}}} However, six great hexagons can be found in the ring of six octahedra, running along the edges of the octahedra. In the column of six octahedra (before it is bent into a ring) there are six spiral paths along edges running up the column: three parallel helices spiraling clockwise, and three parallel helices spiraling counterclockwise. Each clockwise helix intersects each counterclockwise helix at two vertices three edge lengths apart. Bending the column into a ring changes these helices into great circle hexagons.{{Efn|There is a choice of planes in which to fold the column into a ring, but they are equivalent in that they produce congruent rings. Whichever folding planes are chosen, each of the six helices joins its own two ends and forms a simple great circle hexagon. These hexagons are ''not'' helices: they lie on ordinary flat great circles. Three of them are Clifford parallel{{Efn|name=Clifford parallels}} and belong to one [[#Great hexagons|hexagonal]] fibration. They intersect the other three, which belong to another hexagonal fibration. The three parallel great circles of each fibration spiral around each other in the sense that they form a [[W:Link (knot theory)|link]] of three ordinary circles, but they are not twisted: the 6-cell ring has no [[W:Torsion of a curve|torsion]], either clockwise or counterclockwise.{{Efn|name=6-cell ring is not chiral}}|name=6-cell ring}} The ring has two sets of three great hexagons, each on three Clifford parallel great circles.{{Efn|The three great hexagons are Clifford parallel, which is different than ordinary parallelism.{{Efn|name=Clifford parallels}} Clifford parallel great hexagons pass through each other like adjacent links of a chain, forming a [[W:Hopf link|Hopf link]]. Unlike links in a 3-dimensional chain, they share the same center point. In the 24-cell, Clifford parallel great hexagons occur in sets of four, not three. The fourth parallel hexagon lies completely outside the 6-cell ring; its 6 vertices are completely disjoint from the ring's 18 vertices.}} The great hexagons in each parallel set of three do not intersect, but each intersects the other three great hexagons (to which it is not Clifford parallel) at two antipodal vertices. A [[#Simple rotations|simple rotation]] in any of the great hexagon planes by a multiple of 60° rotates only that hexagon invariantly, taking each vertex in that hexagon to a vertex in the same hexagon. An [[#Isoclinic rotations|isoclinic rotation]] by 60° in any of the six great hexagon planes rotates all three Clifford parallel great hexagons invariantly, and takes each octahedron in the ring to a ''non-adjacent'' octahedron in the ring.{{Efn|An isoclinic rotation by a multiple of 60° takes even-numbered octahedra in the ring to even-numbered octahedra, and odd-numbered octahedra to odd-numbered octahedra.{{Efn|In the column of 6 octahedral cells, we number the cells 0-5 going up the column. We also label each vertex with an integer 0-5 based on how many edge lengths it is up the column.}} It is impossible for an even-numbered octahedron to reach an odd-numbered octahedron, or vice versa, by a left or a right isoclinic rotation alone.{{Efn|name=black and white}}|name=black and white octahedra}} Each isoclinically displaced octahedron is also rotated itself. After a 360° isoclinic rotation each octahedron is back in the same position, but in a different orientation. In a 720° isoclinic rotation, its vertices are returned to their original [[W:Orientation entanglement|orientation]]. Four Clifford parallel great hexagons comprise a discrete fiber bundle covering all 24 vertices in a [[W:Hopf fibration|Hopf fibration]]. The 24-cell has four such [[#Great hexagons|discrete hexagonal fibrations]] <math>F_a, F_b, F_c, F_d</math>. Each great hexagon belongs to just one fibration, and the four fibrations are defined by disjoint sets of four great hexagons each.{{Sfn|Kim|Rote|2016|loc=§8.3 Properties of the Hopf Fibration|pp=14-16|ps=; Corollary 9. Every great circle belongs to a unique right [(and left)] Hopf bundle.}} Each fibration is the domain (container) of a unique left-right pair of isoclinic rotations (left and right Hopf fiber bundles).{{Efn|The choice of a partitioning of a regular 4-polytope into cell rings (a fibration) is arbitrary, because all of its cells are identical. No particular fibration is distinguished, ''unless'' the 4-polytope is rotating. Each fibration corresponds to a left-right pair of isoclinic rotations in a particular set of Clifford parallel invariant central planes of rotation. In the 24-cell, distinguishing a hexagonal fibration{{Efn|name=hexagonal fibrations}} means choosing a cell-disjoint set of four 6-cell rings that is the unique container of a left-right pair of isoclinic rotations in four Clifford parallel hexagonal invariant planes. The left and right rotations take place in chiral subspaces of that container,{{Sfn|Kim|Rote|2016|p=12|loc=§8 The Construction of Hopf Fibrations; 3}} but the fibration and the octahedral cell rings themselves are not chiral objects.{{Efn|name=6-cell ring is not chiral}}|name=fibrations are distinguished only by rotations}} Four cell-disjoint 6-cell rings also comprise each discrete fibration defined by four Clifford parallel great hexagons. Each 6-cell ring contains only 18 of the 24 vertices, and only 6 of the 16 great hexagons, which we see illustrated above running along the cell ring's edges: 3 spiraling clockwise and 3 counterclockwise. Those 6 hexagons running along the cell ring's edges are not among the set of four parallel hexagons which define the fibration. For example, one of the four 6-cell rings in fibration <math>F_a</math> contains 3 parallel hexagons running clockwise along the cell ring's edges from fibration <math>F_b</math>, and 3 parallel hexagons running counterclockwise along the cell ring's edges from fibration <math>F_c</math>, but that cell ring contains no great hexagons from fibration <math>F_a</math> or fibration <math>F_d</math>. The 24-cell contains 16 great hexagons, divided into four disjoint sets of four hexagons, each disjoint set uniquely defining a fibration. Each fibration is also a distinct set of four cell-disjoint 6-cell rings. The 24-cell has exactly 16 distinct 6-cell rings. Each 6-cell ring belongs to just one of the four fibrations.{{Efn|The dual polytope of the 24-cell is another 24-cell. It can be constructed by placing vertices at the 24 cell centers. Each 6-cell ring corresponds to a great hexagon in the dual 24-cell, so there are 16 distinct 6-cell rings, as there are 16 distinct great hexagons, each belonging to just one fibration.}} ==== Helical dodecagrams and their isoclines ==== Another kind of geodesic fiber, the [[#Isoclinic rotations|helical dodecagram isoclines]], can be found within a 6-cell ring of octahedra. Each of these geodesics runs through every ''fifth'' vertex of a skew [[W:Dodecagon#Related figures|dodecagram]]₅, which in the unit-radius, unit-edge-length 24-cell has twelve {{radic|3}} edges. The dodagram does not lie in a single central plane, but is composed of twelve linked {{radic|3}} chords from different hexagon great circles. The isocline geodesic fiber is the path of an isoclinic rotation,{{Efn|name=isoclinic geodesic}} a helical rather than simply circular path around the 24-cell linking non-adjacent vertices, that winds five times around the 24-cell before completing its twelve-vertex loop.{{Efn|The chord-path of an isocline (the geodesic along which a vertex moves under isoclinic rotation) may be called the 4-polytope's '''Clifford polygon''', as it is the skew polygonal shape of the rotational circles traversed by the 4-polytope's vertices in its characteristic [[W:Clifford displacement|Clifford displacement]].{{Sfn|Tyrrell & Semple|1971|loc=Linear Systems of Clifford Parallels|pp=34-57}} The isocline is a helical Möbius double loop which reverses its chirality twice in the course of a full double circuit. The double loop is entirely contained within a single [[#Cell rings|cell ring]], where it follows chords connecting even (odd) vertices: typically opposite vertices of adjacent cells, two edge lengths apart.{{Efn|name=black and white}} Both "halves" of the double loop pass through each cell in the cell ring, but intersect only two even (odd) vertices in each even (odd) cell. Each pair of intersected vertices in an even (odd) cell lie opposite each other on the [[W:Möbius strip|Möbius strip]], exactly one edge length apart. Thus each cell has both helices passing through it, which are Clifford parallels{{Efn|name=Clifford parallels}} of opposite chirality at each pair of parallel points. Globally these two helices are a single connected circle of ''both'' chiralities, with no net [[W:Torsion of a curve|torsion]]. An isocline acts as a left (or right) isocline when traversed by a left (or right) rotation (of different fibrations).{{Efn|name=one true circle}}|name=Clifford polygon}} Rather than a flat hexagon, it forms a [[W:Skew polygon|skew]] {12/5} dodecagram.{{Efn|name=double threaded}} Each fibration of four 6-cell rings contains four such dodecagram isoclines, two black and two white, that connect even and odd vertices respectively.{{Efn|Only one kind of 6-cell ring exists, not two different chiral kinds (right-handed and left-handed), because octahedra have opposing faces and form untwisted cell rings. Two chiral sets of three Clifford parallel{{Efn|name=Clifford parallels}} [[#Great hexagons|great hexagons]] run through each [[#6-cell rings|6-cell ring]].{{Efn|name=hexagonal fibrations}} Each of the skew dodecagrams lies on a different kind of circle called an ''isocline'',{{Efn|name=not all isoclines are circles}} a helical circle [[W:Winding number|winding]] through all four dimensions instead of lying in a single plane.{{Efn|name=isoclinic geodesic}} These helical great circles occur in Clifford parallel [[W:Hopf fibration|fiber bundles]] just as ordinary planar great circles do. In the 6-cell ring, black and white dodecagrams pass through even and odd vertices respectively, and miss the vertices in between, so the isoclines are disjoint.{{Efn|name=black and white}}|name=6-cell ring is not chiral}} The fibration's right (or left) rotation traverses a black isocline and a white isocline in parallel, rotating all 24 vertices.{{Efn|name=missing the nearest vertices}} Beginning at any vertex at one end of the column of six octahedra, we can follow an isoclinic path of {{radic|3}} chords of an isocline from octahedron to octahedron. In the 24-cell the {{radic|1}} edges are [[#Great hexagons|great hexagon]] edges (and octahedron edges); in the column of six octahedra we see six great hexagons running along the octahedra's edges. The {{radic|3}} chords are great hexagon diagonals, joining great hexagon vertices two {{radic|1}} edges apart. We find them in the ring of six octahedra running from a vertex in one octahedron to a vertex in the next octahedron, passing through the face shared by the two octahedra (but not touching any of the face's 3 vertices). Each {{radic|3}} chord is a chord of just one great hexagon (an edge of a [[#Great triangles|great triangle]] inscribed in that great hexagon), but successive {{radic|3}} chords belong to different great hexagons.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} At each vertex the isoclinic path of {{radic|3}} chords bends 60 degrees in two central planes{{Efn|Two central planes in which the path bends 60° at the vertex are (a) the great hexagon plane that the chord ''before'' the vertex belongs to, and (b) the great hexagon plane that the chord ''after'' the vertex belongs to. Plane (b) contains the 120° isocline chord joining the original vertex to a vertex in great hexagon plane (c), Clifford parallel to (a); the vertex moves over this chord to this next vertex. The angle of inclination between the Clifford parallel (isoclinic) great hexagon planes (a) and (c) is also 60°. In this 60° interval of the isoclinic rotation, great hexagon plane (a) rotates 60° within itself ''and'' tilts 60° in an orthogonal plane (not plane (b)) to become great hexagon plane (c). The three great hexagon planes (a), (b) and (c) are not orthogonal (they are inclined at 60° to each other), but (a) and (b) are two central hexagons in the same cuboctahedron, and (b) and (c) likewise in an orthogonal cuboctahedron.{{Efn|name=cuboctahedral hexagons}}}} at once: 60 degrees around the great hexagon that the chord before the vertex belongs to, and 60 degrees into the plane of a different great hexagon entirely, that the chord after the vertex belongs to.{{Efn|At each vertex there is only one adjacent great hexagon plane that the isocline can bend 60 degrees into: the isoclinic path is ''deterministic'' in the sense that it is linear, not branching, because each vertex in the cell ring is a place where just two of the six great hexagons contained in the cell ring cross. If each great hexagon is given edges and chords of a particular color (as in the 6-cell ring illustration), we can name each great hexagon by its color, and each kind of vertex by a hyphenated two-color name. The cell ring contains 18 vertices named by the 9 unique two-color combinations; each vertex and its antipodal vertex have the same two colors in their name, since when two great hexagons intersect they do so at antipodal vertices. Each isoclinic skew dodecagram contains one {{radic|3}} chord of each color, and visits all 9 different color-pairs of vertex.{{Efn|Each vertex of the 6-cell ring is intersected by two skew dodecagrams of the same parity (black or white) belonging to different fibrations.{{Efn|name=6-cell ring is not chiral}}|name=dodecagrams hitting vertex of 6-cell ring}}}} The path follows one great hexagon from each octahedron to the next, but switches to another of the six great hexagons in the next link of the dodecagram₅ path. <s>Followed along the column of six octahedra (and "around the end" where the column is bent into a ring) the path may at first appear to be zig-zagging between three adjacent parallel hexagonal central planes (like a [[W:Petrie polygon|Petrie polygon]]), but it is not: any isoclinic path we can pick out always zig-zags between ''two sets'' of three adjacent parallel hexagonal central planes, intersecting only every even (or odd) vertex and never changing its inherent even/odd parity, as it visits all six of the great hexagons in the 6-cell ring in rotation.{{Efn|The 24-cell's [[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Petrie polygon]] is a skew [[W:Skew polygon#Regular skew polygons in four dimensions|dodecagon]] {12} and also (orthogonally) a skew [[W:Dodecagram|dodecagram]] {12/5} which zig-zags 90° left and right like the edges dividing the black and white squares on the [[W:Chessboard|chessboard]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell ''h₁ is {12}, h₂ is {12/5}''}} In contrast, the skew dodecagram₅ isocline does not zig-zag, and stays on one side or the other of the dividing line between black and white, like the [[W:Bishop (chess)|bishop]]s' paths along the diagonals of either the black or white squares of the chessboard.{{Efn|name=missing the nearest vertices}} The Petrie dodecagon is a circular helix of {{radic|1}} edges that zig-zag 90° left and right along 12 edges of 6 different octahedra (with 3 consecutive edges in each octahedron) in a 360° rotation. In contrast, the isoclinic dodecagram₅ has {{radic|3}} edges which all bend either left or right at every fifth vertex along a geodesic spiral of potentially either chirality (left or right){{Efn|name=Clifford polygon}} but only one color (black or white),{{Efn|name=black and white}} visiting two verticies of each of those same 6 octahedra in a 720° rotation.|name=Petrie and Clifford dodecagram}} When it has traversed one chord from each of the six great hexagons, after 720 degrees of isoclinic rotation (either left or right), it closes its skew dodecagram and begins to repeat itself, circling again through the black (or white) vertices and cells.</s> At each vertex, there are four great hexagons{{Efn|Each pair of adjacent edges of a great hexagon has just one isocline curving alongside it, missing the vertex between the two edges (but not the way the {{radic|3}} edge of the great triangle inscribed in the great hexagon misses the vertex,{{Efn|The {{radic|3}} chord passes through the mid-edge of one of the 24-cell's {{radic|1}} radii. Since the 24-cell can be constructed, with its long radii, from {{radic|1}} triangles which meet at its center,{{Efn|name=radially equilateral}} this is a mid-edge of one of the six {{radic|1}} triangles in a great hexagon, as seen in the [[#Hypercubic chords|chord diagram]].|name=root 3 chord hits a mid-radius}} because the isocline is an arc on the surface not a chord). If we number the vertices around the hexagon 0-5, the hexagon has three pairs of adjacent edges connecting even vertices (one inscribed great triangle), and three pairs connecting odd vertices (the other inscribed great triangle). Even and odd pairs of edges have the arc of a black and a white isocline respectively curving alongside.{{Efn|name=black and white}} The black and white isoclines belong to the same fibration.|name=isoclines at hexagons}} and four dodecagram isoclines (all black or all white) that cross at the vertex.{{Efn|Each dodecagram isocline hits only one end of an axis, unlike a great circle in the plane which hits both ends. Clifford parallel pairs of black and white isoclines from the same left-right pair of isoclinic rotations (the same fibration) do not intersect, but they hit opposite (antipodal) vertices of one of the 24-cell's 12 axes.|name=dodecagram isoclines at an axis}} Two dodecagram isoclines (one black and one white) comprise a unique (left or right) fiber bundle of isoclines covering all 24 vertices in each distinct (left or right) isoclinic rotation. Each fibration has a unique left and right isoclinic rotation, and corresponding unique left and right fiber bundles of isoclines.{{Efn|The isoclines themselves are not left or right, only the bundles are. Each isocline is left ''and'' right.{{Efn|name=Clifford polygon}}}} There are 8 distinct dedecagram isoclines in the 24-cell (4 black and 4 white). Each dodecagram is a skew ''Clifford polygon'' of no inherent chirality, that acts as a left (or right) isocline when traversed by a left (or right) rotation in different fibrations.{{Efn|name=Clifford polygon}} ==== Helical octagrams and their isoclines ==== The 24-cell contains 18 helical {8/3} [[W:Octagram|octagram]] isoclines (9 black and 9 white). Three pairs of octagram edge-helices are found in each of the three inscribed 16-cells, described elsewhere as the [[16-cell#Helical construction|helical construction of the 16-cell]]. In summary, each 16-cell can be decomposed (three different ways) into a left-right pair of 8-cell rings of {{radic|2}}-edged tetrahedral cells. Each 8-cell ring twists either left or right around an axial octagram helix of eight chords. In each 16-cell there are exactly 6 distinct helices, identical octagrams which each circle through all eight vertices. Each acts as either a left helix or a right helix or a zig-zag Petrie polygon in each of the six distinct isoclinic rotations (three left and three right), and has no inherent chirality except in the context of a particular rotation. Adjacent vertices on the {8/3} octagram isoclines are {{radic|2}} = 90° apart, so the circumference of the isocline is 4𝝅. An isoclinic rotation by 90° in great square invariant planes takes each great square to its completely orthogonal great square in a twisting displacement, and each vertex to a vertex 90° away over a rotational curve. The rotational curve over each {{radic|2}} chord of the {8/3} octagram makes three 90° left (or right) turns. Each of the 3 fibrations of the 24-cell's 18 great squares corresponds to a distinct left (and right) isoclinic rotation in great square invariant planes. Each 60° step of the rotation takes 6 disjoint great squares (2 from each 16-cell) to great squares in a neighboring 16-cell, on [[16-cell#Helical construction|8-chord helical isoclines characteristic of the 16-cell]].{{Efn|As [[16-cell#Helical construction|in the 16-cell, the isocline is an octagram]] which intersects only 8 vertices, even though the 24-cell has more vertices closer together than the 16-cell. The isocline curve misses the additional vertices in between. As in the 16-cell, the first vertex it intersects is {{radic|2}} away. The 24-cell employs more octagram isoclines (3 in parallel in each rotation) than the 16-cell does (1 in each rotation). The 3 helical isoclines are Clifford parallel;{{Efn|name=Clifford parallels}} they spiral around each other in a triple helix, with the disjoint helices' corresponding vertex pairs joined by {{radic|1}} {{=}} 60° chords. The triple helix of 3 isoclines contains 24 disjoint {{radic|2}} edges (6 disjoint great squares) and 24 vertices, and constitutes a discrete fibration of the 24-cell, just as the 4-cell ring does.|name=octagram isoclines}} In the 24-cell, these 18 helical octagram isoclines can be found within the six orthogonal [[#4-cell rings|4-cell rings]] of octahedra. Each 4-cell ring has cells bonded vertex-to-vertex around a great square axis, and we find antipodal vertices at opposite vertices of the great square. A {{radic|4}} chord (the diameter of the great square and of the isocline) connects them. [[#Boundary cells|Boundary cells]] describes how the {{radic|2}} axes of the 24-cell's octahedral cells are the edges of the 16-cell's tetrahedral cells, each tetrahedron is inscribed in a (tesseract) cube, and each octahedron is inscribed in a pair of cubes (from different tesseracts), bridging them.{{Efn|name=octahedral diameters}} The vertex-bonded octahedra of the 4-cell ring also lie in different tesseracts.{{Efn|Two tesseracts share only vertices, not any edges, faces, cubes (with inscribed tetrahedra), or octahedra (whose central square planes are square faces of cubes). An octahedron that touches another octahedron at a vertex (but not at an edge or a face) is touching an octahedron in another tesseract, and a pair of adjacent cubes in the other tesseract whose common square face the octahedron spans, and a tetrahedron inscribed in each of those cubes.|name=vertex-bonded octahedra}} The isocline's four {{radic|4}} diameter chords form an [[W:Octagram#Star polygon compounds|octagram_8{4}=4{2}]] with {{radic|4}} edges that each run from the vertex of one cube and octahedron and tetrahedron, to the vertex of another cube and octahedron and tetrahedron (in a different tesseract), straight through the center of the 24-cell on one of the 12 {{radic|4}} axes. The octahedra in the 4-cell rings are vertex-bonded to more than two other octahedra, because three 4-cell rings (and their three axial great squares, which belong to different 16-cells) cross at 90° at each bonding vertex. At that vertex the octagram makes two right-angled turns at once: 90° around the great square, and 90° orthogonally into a different 4-cell ring entirely. The 180° four-edge arc joining two ends of each {{radic|4}} diameter chord of the octagram runs through the volumes and opposite vertices of two face-bonded {{radic|2}} tetrahedra (in the same 16-cell), which are also the opposite vertices of two vertex-bonded octahedra in different 4-cell rings (and different tesseracts). The [[W:Octagram|720° octagram]] isocline runs through 8 vertices of the four-cell ring and through the volumes of 16 tetrahedra. At each vertex, there are three great squares and six octagram isoclines (three black-white pairs) that cross at the vertex.{{Efn|name=completely orthogonal Clifford parallels are special}} This is the characteristic rotation of the 16-cell, ''not'' the 24-cell's characteristic rotation, and it does not take whole 16-cells ''of the 24-cell'' to each other the way the [[#Helical dodecagrams and their isoclines|24-cell's rotation in great hexagon planes]] does.{{Efn|The [[600-cell#Squares and 4𝝅 octagrams|600-cell's isoclinic rotation in great square planes]] takes whole 16-cells to other 16-cells in different 24-cells.}} {| class="wikitable" width=610 !colspan=5|Five ways of looking at a [[W:Skew polygon|skew]] [[W:24-gon#Related polygons|24-gram]] |- ![[16-cell#Rotations|Edge path]] ![[W:Petrie polygon|Petrie polygon]]s ![[600-cell#Squares and 4𝝅 octagrams|In a 600-cell]] ![[#Great squares|Discrete fibration]] ![[16-cell#Helical construction|Diameter chords]] |- ![[16-cell#Helical construction|16-cells]]_3{3/8} ![[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Dodecagons]]_2{12} ![[W:24-gon#Related polygons|24-gram]]_{24/5} ![[#Great squares|Squares]]_6{4} ![[W:24-gon#Related polygons|_{{24/12}={12/2}}]] |- |align=center|[[File:Regular_star_figure_3(8,3).svg|120px]] |align=center|[[File:Regular_star_figure_2(12,1).svg|120px]] |align=center|[[File:Regular_star_polygon_24-5.svg|120px]] |align=center|[[File:Regular_star_figure_6(4,1).svg|120px]] |align=center|[[File:Regular_star_figure_12(2,1).svg|120px]] |- |The 24-cell's three inscribed Clifford parallel 16-cells revealed as disjoint 8-point 4-polytopes with {{radic|2}} edges.{{Efn|name=octagram isoclines}} |2 [[W:Skew polygon|skew polygon]]s of 12 {{radic|1}} edges each. The 24-cell can be decomposed into 2 disjoint zig-zag [[W:Dodecagon|dodecagon]]s (4 different ways).{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon ''h₁'' is {12} }} |In [[600-cell#Hexagons|compounds of 5 24-cells]], isoclines with [[600-cell#Golden chords|golden chords]] of length <big>φ</big> {{=}} {{radic|2.𝚽}} connect all 24-cells in [[600-cell#Squares and 4𝝅 octagrams|24-chord circuits]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon orthogonal ''h₂'' is [[W:Dodecagon#Related figures|{12/5}]], half of [[W:24-gon#Related polygons|{24/5}]] as each Petrie polygon is half the 24-cell}} |Their isoclinic rotation takes 6 Clifford parallel (disjoint) great squares with {{radic|2}} edges to each other. |Two vertices four {{radic|2}} chords apart on a Petrie polygon are antipodal vertices joined by a {{radic|4}} axis. |} ===Characteristic orthoscheme=== {| class="wikitable floatright" !colspan=6|Characteristics of the 24-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); "24-cell"}} |- !align=right| !align=center|edge{{Sfn|Coxeter|1973|p=139|loc=§7.9 The characteristic simplex}} !colspan=2 align=center|arc !colspan=2 align=center|dihedral{{Sfn|Coxeter|1973|p=290|loc=Table I(ii); "dihedral angles"}} |- !align=right|𝒍 |align=center|<small><math>1</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |align=center|<small>120°</small> |align=center|<small><math>\tfrac{2\pi}{3}</math></small> |- | | | | | |- !align=right|𝟀 |align=center|<small><math>\sqrt{\tfrac{1}{3}} \approx 0.577</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |- !align=right|𝝉{{Efn|{{Harv|Coxeter|1973}} uses the greek letter 𝝓 (phi) to represent one of the three ''characteristic angles'' 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the [[W:Golden ratio|golden ratio]] constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.|name=reversed greek symbols}} |align=center|<small><math>\sqrt{\tfrac{1}{4}} = 0.5</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |- !align=right|𝟁 |align=center|<small><math>\sqrt{\tfrac{1}{12}} \approx 0.289</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |- | | | | | |- !align=right|<small><math>_0R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{2}} \approx 0.707</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- !align=right|<small><math>_1R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{4}} = 0.5</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- !align=right|<small><math>_2R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{6}} \approx 0.408</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- | | | | | |- !align=right|<small><math>_0R^4/l</math></small> |align=center|<small><math>1</math></small> |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_1R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{3}{4}} \approx 0.866</math></small>{{Efn|name=root 3/4}} |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_2R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{2}{3}} \approx 0.816</math></small> |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_3R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{2}} \approx 0.707</math></small> |align=center| |align=center| |align=center| |align=center| |} Every regular 4-polytope has its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic 4-orthoscheme]], an [[5-cell#Irregular 5-cells|irregular 5-cell]].{{Efn|name=characteristic orthoscheme}} The '''characteristic 5-cell of the regular 24-cell''' is represented by the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, which can be read as a list of the dihedral angles between its mirror facets.{{Efn|For a regular ''k''-polytope, the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] of the characteristic ''k-''orthoscheme is the ''k''-polytope's diagram without the [[W:Coxeter-Dynkin diagram#Application with uniform polytopes|generating point ring]]. The regular ''k-''polytope is subdivided by its symmetry (''k''-1)-elements into ''g'' instances of its characteristic ''k''-orthoscheme that surround its center, where ''g'' is the ''order'' of the ''k''-polytope's [[W:Coxeter group|symmetry group]].{{Sfn|Coxeter|1973|pp=130-133|loc=§7.6 The symmetry group of the general regular polytope}}}} It is an irregular [[W:Hyperpyramid|tetrahedral pyramid]] based on the [[W:Octahedron#Characteristic orthoscheme|characteristic tetrahedron of the regular octahedron]]. The regular 24-cell is subdivided by its symmetry hyperplanes into 1152 instances of its characteristic 5-cell that all meet at its center.{{Sfn|Kim|Rote|2016|pp=17-20|loc=§10 The Coxeter Classification of Four-Dimensional Point Groups}} The characteristic 5-cell (4-orthoscheme) has four more edges than its base characteristic tetrahedron (3-orthoscheme), joining the four vertices of the base to its apex (the fifth vertex of the 4-orthoscheme, at the center of the regular 24-cell).{{Efn|The four edges of each 4-orthoscheme which meet at the center of the regular 4-polytope are of unequal length, because they are the four characteristic radii of the regular 4-polytope: a vertex radius, an edge center radius, a face center radius, and a cell center radius. The five vertices of the 4-orthoscheme always include one regular 4-polytope vertex, one regular 4-polytope edge center, one regular 4-polytope face center, one regular 4-polytope cell center, and the regular 4-polytope center. Those five vertices (in that order) comprise a path along four mutually perpendicular edges (that makes three right angle turns), the characteristic feature of a 4-orthoscheme. The 4-orthoscheme has five dissimilar 3-orthoscheme facets.|name=characteristic radii}} If the regular 24-cell has radius and edge length 𝒍 = 1, its characteristic 5-cell's ten edges have lengths <small><math>\sqrt{\tfrac{1}{3}}</math></small>, <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{12}}</math></small> around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus <small><math>\sqrt{\tfrac{1}{2}}</math></small>, <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{6}}</math></small> (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the octahedron), plus <small><math>1</math></small>, <small><math>\sqrt{\tfrac{3}{4}}</math></small>, <small><math>\sqrt{\tfrac{2}{3}}</math></small>, <small><math>\sqrt{\tfrac{1}{2}}</math></small> (edges which are the characteristic radii of the 24-cell). The 4-edge path along orthogonal edges of the orthoscheme is <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{12}}</math></small>, <small><math>\sqrt{\tfrac{1}{6}}</math></small>, <small><math>\sqrt{\tfrac{1}{2}}</math></small>, first from a 24-cell vertex to a 24-cell edge center, then turning 90° to a 24-cell face center, then turning 90° to a 24-cell octahedral cell center, then turning 90° to the 24-cell center. === Reflections === The 24-cell can be [[#Tetrahedral constructions|constructed by the reflections of its characteristic 5-cell]] in its own facets (its tetrahedral mirror walls).{{Efn|The reflecting surface of a (3-dimensional) polyhedron consists of 2-dimensional faces; the reflecting surface of a (4-dimensional) [[W:Polychoron|polychoron]] consists of 3-dimensional cells.}} Reflections and rotations are related: a reflection in an ''even'' number of ''intersecting'' mirrors is a rotation.{{Sfn|Coxeter|1973|pp=33-38|loc=§3.1 Congruent transformations}} Consequently, regular polytopes can be generated by reflections or by rotations. For example, any [[#Isoclinic rotations|720° isoclinic rotation]] of the 24-cell in a great hexagon invariant plane takes each of the 24 vertices to and through eleven other vertices and back to itself, on a skew [[#Helical dodecagrams and their isoclines|dodecagram₅ geodesic isocline]] that winds five times around the 3-sphere on every fifth vertex of the dodecagram. Any pair of antipodal vertices performing such an orbit visits 2 * 12 = 24 distinct vertices and [[#Clifford parallel polytopes|generates the 24-cell]] sequentially in the twelve steps of a single 720° isoclinic rotation, just as any single characteristic 5-cell reflecting itself in its own mirror walls generates the 24 vertices simultaneously by reflection. Tracing the orbit of one vertex during the 720° isoclinic rotation reveals more about the relationship between reflections and rotations as generative operations.{{Efn|<blockquote>Let Q denote a rotation, R a reflection, T a translation, and let Q^''q'' R^''r'' T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q² is a double rotation (in four dimensions).<br><br>Every orthogonal transformation is expressible as {{indent|12}}Q^''q'' R^''r''<br>where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as {{indent|12}}Q^''q'' R^''r'' T<br>where 2''q'' + ''r'' + 1 ≤ ''n''.<br><br>For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q², or a screw-displacement QT (where the rotation component Q is a simple rotation). Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}</blockquote>|name=transformations}} The vertex follows an [[#Helical dodecagrams and their isoclines|isocline]] (a doubly curved geodesic circle) rather than an ordinary great circle.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} The isocline connects non-adjacent vertices , but curves away from the great circle path over the two edges connecting those vertices, missing the vertex in between.{{Efn|name=isocline misses vertex}} Although the isocline does not follow a great circle in the plane, it is a great circle of another kind that curves in two completely orthogonal directions at once, and winds through all four dimensions. === Chiral symmetry operations === A [[W:Symmetry operation|symmetry operation]] is a rotation or reflection which leaves the object indistinguishable from itself before the transformation. The 24-cell has 1152 distinct symmetry operations (576 rotations and 576 reflections). Each rotation is equivalent to two [[#Reflections|reflections]], in a distinct pair of non-parallel mirror planes.{{Efn|name=transformations}} Pictured are sets of disjoint [[#Geodesics|great circle polygons]], each in a distinct central plane of the 24-cell. For example, {24/4}=4{6} is an orthogonal projection of the 24-cell picturing 4 of its [16] great hexagon planes.{{Efn|name=four hexagonal fibrations}} The 4 planes lie Clifford parallel to the projection plane and to each other, and their great polygons collectively constitute a discrete [[W:Hopf fibration|Hopf fibration]] of 4 non-intersecting great circles which visit all 24 vertices just once. Each row of the table describes a class of distinct rotational displacements. Each '''rotation class''' takes the '''left planes''' pictured to the corresponding '''right planes''' pictured.{{Efn|The left planes are Clifford parallel, and the right planes are Clifford parallel; each set of planes is a fibration. Each left plane is Clifford parallel to its corresponding right plane in an isoclinic rotation,{{Efn|In an ''isoclinic'' rotation each invariant plane is Clifford parallel to the plane it moves to, and they do not intersect at any time (except at the central point). In a ''simple'' rotation the invariant plane intersects the plane it moves to in a line, and moves to it by rotating around that line.|name=plane movement in rotations}} but the two sets of planes are not all mutually Clifford parallel; they are different fibrations, except in table rows where the left and right planes are the same set.}} The 24 vertices of the moving planes move in parallel between the left and right planes over the '''isocline''' chord paths pictured. For example, the <math>[32]R_{q7,q8}</math> rotation class consists of [32] vertex displacements by an arc-distance of {{sfrac|2𝝅|3}} = 120° between 16 great hexagon planes represented by quaternion group <math>q7</math> and a corresponding set of 16 great hexagon planes represented by quaternion group <math>q8</math>.{{Efn|A quaternion group <math>\pm{q_n}</math> corresponds to a distinct set of Clifford parallel great circle polygons, e.g. <math>q7</math> corresponds to a set of four disjoint great hexagons.{{Efn|[[File:Regular_star_figure_4(6,1).svg|thumb|200px|The 24-cell as a compound of four non-intersecting great hexagons {24/4}=4{6}.]]There are 4 sets of 4 disjoint great hexagons in the 24-cell (of a total of [16] distinct great hexagons), designated <math>q7</math>, <math>-q7</math>, <math>q8</math> and <math>-q8</math>.{{Efn|name=union of q7 and q8}} Each named set of 4 Clifford parallel{{Efn|name=Clifford parallels}} hexagons comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=four hexagonal fibrations}} Note that <math>q_n</math> and <math>-{q_n}</math> generally are distinct sets. The corresponding vertices of the <math>q_n</math> planes and the <math>-{q_n}</math> planes are 180° apart.{{Efn|name=two angles between central planes}}|name=quaternion group}} One of the [32] vertex displacements in this class moves the representative [[#Great hexagons|vertex coordinate]] <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> to the vertex coordinate <math>(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math>.{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in standard (vertex-up) orientation is <math>(0,0,1,0)</math>, the Cartesian "north pole". Thus e.g. <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> designates a {{radic|1}} chord of 60° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great hexagons|great hexagon]], intersecting the north and south poles. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the north and south poles. This quaternion coordinate <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> is thus representative of the 4 disjoint great hexagons pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [16] great hexagons (four fibrations of great hexagons) that occur in the 24-cell.{{Efn|name=four hexagonal fibrations}}|name=north pole relative coordinate}} Corresponding vertices in the left and right hexagon planes are 5 vertices apart on a Petrie polygon of the 24-cell, so the {{radic|3}} displacement chords of the 24 moving vertices form 2 disjoint skew {12/5} dodecagram helixes, pictured in the isocline column. {| class=wikitable style="white-space:nowrap;text-align:center" !colspan=15|Proper [[W:SO(4)|rotations]] of the 24-cell [[W:F4 (mathematics)|symmetry group ''F₄'']]{{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2, Symmetry operations|pp=1438-1439}} |- !Isocline{{Efn|An ''isocline'' is the circular geodesic path taken by a vertex that lies in an invariant plane of rotation, during a complete revolution. In an [[#Isoclinic rotations|isoclinic rotation]] every vertex lies in an invariant plane of rotation, and the isocline it rotates on is a helical geodesic circle that winds through all four dimensions, not a simple geodesic great circle in the plane. In a [[#Simple rotations|simple rotation]] there is only one invariant plane of rotation, and each vertex that lies in it rotates on a simple geodesic great circle in the plane. Both the helical geodesic isocline of an isoclinic rotation and the simple geodesic isocline of a simple rotation are great circles, but to avoid confusion between them we generally reserve the term ''isocline'' for the former, and reserve the term ''great circle'' for the latter, an ordinary great circle in the plane. Strictly, however, the latter is an isocline of circumference <math>2\pi r</math>, and the former is an isocline of circumference greater than <math>2\pi r</math>.{{Efn|name=isoclinic geodesic}}|name=isocline}} !colspan=4|Rotation class{{Efn|Each class of rotational displacements (each table row) corresponds to a distinct rigid left (and right) [[#Isoclinic rotations|isoclinic rotation]] in multiple invariant planes concurrently.{{Efn|name=invariant planes of an isoclinic rotation}} The '''Isocline''' is the path followed by a vertex,{{Efn|name=isocline}} which is a helical geodesic circle that does not lie in any one central plane. Each rotational displacement takes one invariant '''Left plane''' to the corresponding invariant '''Right plane''', with all the left (or right) displacements taking place concurrently.{{Efn|name=plane movement in rotations}} Each left plane is separated from the corresponding right plane by two equal angles,{{Efn|name=two angles between central planes}} each equal to one half of the arc-angle by which each vertex is displaced (the angle and distance that appears in the '''Rotation class''' column).|name=isoclinic rotation}} !colspan=5|Left planes <math>ql</math>{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], all the '''Left planes''' move together, remain Clifford parallel while moving, and carry all their points with them to the '''Right planes''' as they move: they are invariant planes.{{Efn|name=plane movement in rotations}} Because the left (and right) set of central polygons are a fibration covering all the vertices, every vertex is a point carried along in an invariant plane.|name=invariant planes of an isoclinic rotation}} !colspan=5|Right planes <math>qr</math> |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/10}=2{12/5}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. Each disjoint triangle can be seen as a skew {12/5} [[W:Dodecagon|Related figures]] with {{radic|3}} edges and a circumference of 8𝝅. The 4 disjoint skew [[#Helical hdodecagrams and their isoclines|dodecagram isoclines]] are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 60° like wheels ''and'' 60° orthogonally like coins flipping, displacing each vertex by 120°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only three skew dodecagram isoclines, not six, because opposite vertices of each hexagon ride on opposing rails of the same Clifford dodecagram, in the same (not opposite) rotational direction.{{Efn|name=Clifford polygon}}}} |name=dodecagram}}<br>[[File:Regular_star_figure_2(12,5).svg|100px]]<br><math>^{q7,q8}</math><br>[8] 8𝝅 {12/5} |colspan=4|<math>[32]R_{q7,q8}</math>{{Efn|The <math>[32]R_{q7,q8}</math> isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q8}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=four hexagonal fibrations}}<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math>{{Efn|name=north pole relative coordinate}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. The 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Great triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented, as in the 4 left planes of this rotation class (table row).{{Efn|name=four hexagonal fibrations}}|name=great triangles}}<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q8}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math> |- style="background: white;"| |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/2}=2{6} dodecagram]], each point represents two vertices, and each line represents multiple 24-cell edges. Each disjoint hexagon can be seen as a skew {12} [[W:Dodecagon|dodecagon]], a Petrie polygon of the 24-cell, by viewing it as two open skew hexagons with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The dodecagon projects to a single hexagon in two dimensions because it skews through all four dimensions. Those 2 disjoint skew dodecagons are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 30° like wheels ''and'' 30° orthogonally like coins flipping, displacing each vertex by 60°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only two parallel dodecagon isoclines, not six, because only alternate vertices of each hexagon ride on different dodecagon rails; the three vertices of each great triangle inscribed in the great hexagon occupy the same dodecagon Petrie polygon, four vertices apart, and they circulate on that isocline.{{Efn|name=Clifford polygon}}}} Alternatively, the 2 hexagons can be seen as 4 disjoint hexagons: 2 pairs of Clifford parallel great hexagons, so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 2 dodecagon isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=dodecagon}}<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7,-q8}</math><br>[16] 2𝝅 {6} |colspan=4|<math>[32]R_{q7,-q8}</math>{{Efn|The <math>[32]R_{q7,-q8}</math> isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (30° away) it passes directly over the mid-point of a 24-cell edge.}} Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q8}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{-q8}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: white;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q7,q7}</math><br>[24] 0𝝅 {1} |colspan=4|<math>[32]R_{q7,q7}</math>{{Efn|The <math>[32]R_{q7,q7}</math> isoclinic rotation in great hexagon invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left hexagon rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q7}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q7,-q7}</math><br>[12] 1𝝅 {2} |colspan=4|<math>[32]R_{q7,-q7}</math>{{Efn|The <math>[32]R_{q7,-q7}</math> isoclinic rotation in hexagon invariant planes takes each vertex to a vertex three vertices away (180° {{=}} {{radic|4}} away),{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left hexagon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q7}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=great triangles}}<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{-q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}<br>[[File:Regular_star_figure_2(6,1).svg|100px]]<br><math>^{q7,q1}</math><br>[8] 4𝝅 {12} |colspan=4|<math>[16]R_{q7,q1}</math>{{Efn|The <math>[16]R_{q7,q1}</math> isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|This ''hybrid isoclinic rotation'' carries the two kinds of [[#Geodesics|central planes]] to each other: great square planes [[16-cell#Coordinates|characteristic of the 16-cell]] and great hexagon (great triangle) planes [[#Great hexagons|characteristic of the 24-cell]].{{Efn|The edges and 4𝝅 characteristic [[16-cell#Rotations|rotations of the 16-cell]] lie in the great square central planes. Rotations of this type are an expression of the [[W:Hyperoctahedral group|<math>B_4</math> symmetry group]]. The edges and 4𝝅 characteristic [[#Rotations|rotations of the 24-cell]] lie in the great hexagon (great triangle) central planes. Rotations of this type are an expression of the [[W:F4 (mathematics)|<math>F_4</math> symmetry group]].|name=edge rotation planes}} This is possible because some great hexagon planes lie Clifford parallel to some great square planes.{{Efn|Two great circle polygons either intersect in a common axis, or they are Clifford parallel (isoclinic) and share no vertices.{{Efn||name=two angles between central planes}} Three great squares and four great hexagons intersect at each 24-cell vertex. Each great hexagon intersects 9 distinct great squares, 3 in each of its 3 axes, and lies Clifford parallel to the other 9 great squares. Each great square intersects 8 distinct great hexagons, 4 in each of its 2 axes, and lies Clifford parallel to the other 8 great hexagons.|name=hybrid isoclinic planes}}|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[8] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]{{Efn|[[File:Regular_star_figure_6(4,1).svg|thumb|200px|The 24-cell as a compound of six non-intersecting great squares {24/6}=6{4}.]]There are 3 sets of 6 disjoint great squares in the 24-cell (of a total of [18] distinct great squares),{{Efn|The 24-cell has 18 great squares, in 3 disjoint sets of 6 mutually orthogonal great squares comprising a 16-cell.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Within each 16-cell are 3 sets of 2 completely orthogonal great squares, so each great square is disjoint not only from all the great squares in the other two 16-cells, but also from one other great square in the same 16-cell. Each great square is disjoint from 13 others, and shares two vertices (an axis) with 4 others (in the same 16-cell).|name=unions of q1 q2 q3}} designated <math>\pm q1</math>, <math>\pm q2</math>, and <math>\pm q3</math>. Each named set{{Efn|Because in the 24-cell each great square is completely orthogonal to another great square, the quaternion groups <math>q1</math> and <math>-{q1}</math> (for example) correspond to the same set of great square planes. That distinct set of 6 disjoint great squares <math>\pm q1</math> has two names, used in the left (or right) rotational context, because it constitutes both a left and a right fibration of great squares.|name=two quaternion group names for square fibrations}} of 6 Clifford parallel{{Efn|name=Clifford parallels}} squares comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=three square fibrations}}<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q1}</math><br>[8] 2𝝅 {4} |colspan=4|<math>(1,0,0,0)</math> |- style="background: #E6FFEE;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/10}=2{12/5}]]{{Efn|name=dodecagram}}<br>[[File:Regular_star_figure_2(12,5).svg|100px]]<br><math>^{q7,-q1}</math><br>[8] 4𝝅 {6/2} |colspan=4|<math>[16]R_{q7,-q1}</math>{{Efn|The <math>[16]R_{q7,-q1}</math> isoclinic rotation in hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[8] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q1}</math><br>[8] 2𝝅 {4} |colspan=4|<math>(-1,0,0,0)</math> |- style="background: #E6FFEE;"| |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q6,q6}</math><br>[24] 0𝝅 {1} |colspan=4|<math>[36]R_{q6,q6}</math>{{Efn|The <math>[36]R_{q6,q6}</math> isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,q6}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math>{{Efn|The representative coordinate <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> is not a vertex of the unit-radius 24-cell in standard (vertex-up) orientation, it is the center of an octahedral cell. Some of the 24-cell's lines of symmetry (Coxeter's "reflecting circles") run through cell centers rather than through vertices, and quaternion group <math>q6</math> corresponds to a set of those. However, <math>q6</math> also corresponds to the set of great squares pictured, which lie orthogonal to those cells (completely disjoint from the cell).{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in ''cell-first'' orientation is <math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math>. Thus e.g. <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> designates a {{radic|2}} chord of 90° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great squares|great square]], intersecting the top vertex. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the top vertex. This quaternion coordinate <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> is thus representative of the 6 disjoint great squares pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [18] great squares (three fibrations of great squares) that occur in the 24-cell.{{Efn|name=three square fibrations}}|name=north cell relative coordinate}}|name=lines of symmetry}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q6,-q6}</math><br>[12] 1𝝅 {2} |colspan=4|<math>[36]R_{q6,-q6}</math>{{Efn|The <math>[36]R_{q6,-q6}</math> isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q6}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2},0,0)</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/9}=3{8/3}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/3}{{=}}3{4} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|2}} chords. Each disjoint square can be seen as a skew {8/3} [[W:Octagram|octagram]] with {{radic|2}} edges: two open skew squares with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅, visible in the {24/9}{{=}}3{8/3} orthogonal projection.{{Efn|[[File:Regular_star_figure_3(8,3).svg|thumb|200px|Icositetragon {24/9}{{=}}3{8/3} is a compound of three octagrams {8/3}, as the 24-cell is a compound of three 16-cells.]]This orthogonal projection of a 24-cell to a 24-gram {24/9}{{=}}3{8/3} exhibits 3 disjoint [[16-cell#Helical construction|octagram {8/3} isoclines of a 16-cell]], each of which is a circular isocline path through the 8 vertices of one of the 3 disjoint 16-cells inscribed in the 24-cell.}} The octagram projects to a single square in two dimensions because it skews through all four dimensions. Those 3 disjoint [[16-cell#Helical construction|skew octagram isoclines]] are the circular vertex paths characteristic of an [[#Helical octagrams and their isoclines|isoclinic rotation in great square planes]], in which the 6 Clifford parallel great squares are invariant rotation planes. The great squares rotate 90° like wheels ''and'' 90° orthogonally like coins flipping, displacing each vertex by 180°, so each vertex exchanges places with its antipodal vertex. Each octagram isocline circles through the 8 vertices of a disjoint 16-cell. Alternatively, the 3 squares can be seen as a fibration of 6 Clifford parallel squares.{{Efn|name=three square fibrations}} This illustrates that the 3 octagram isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 6 squares.|name=octagram}}<br>[[File:Regular_star_figure_3(8,3).svg|100px]]<br><math>^{q6,-q4}</math><br>[36] 4𝝅 {8/3} |colspan=4|<math>[144]R_{q6,-q4}</math>{{Efn|The <math>[144]R_{q6,-q4}</math> isoclinic rotation in great square invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left square rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right square plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q4}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[72] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q4}</math><br>[72] 2𝝅 {4} |colspan=4|<math>(0,0,-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2})</math> |- style="background: white;"| |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |𝝅 |180° |{{radic|4}} |2 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q4,q4}</math><br>[24] 0𝝅 {1} |colspan=4|<math>[72]R_{q4,q4}</math>{{Efn|The <math>[72]R_{q4,q4}</math> isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq4,q4}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q4}</math><br>[36] 2𝝅 {4} |colspan=4|<math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q4}</math><br>[36] 2𝝅 {4} |colspan=4|<math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}<br>[[File:Regular_star_figure_2(6,1).svg|100px]]<br><math>^{q2,q7}</math><br>[48] 4𝝅 {12} |colspan=4|<math>[96]R_{q2,q7}</math>{{Efn|The <math>[96]R_{q2,q7}</math> isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left square rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q7}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q2}</math><br>[48] 2𝝅 {4} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[48] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: #E6FFEE;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q2,-q2}</math><br>[9] 4𝝅 {2} |colspan=4|<math>[18]R_{q2,-q2}</math>{{Efn|The <math>[18]R_{q2,-q2}</math> isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,-q2}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q2}</math><br>[9] 2𝝅 {4} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q2}</math><br>[9] 2𝝅 {4} |colspan=4|<math>(0,0,0,-1)</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q2,q1}</math><br>[12] 4𝝅 {2} |colspan=4|<math>[12]R_{q2,q1}</math>{{Efn|The <math>[12]R_{q2,q1}</math> isoclinic rotation in great digon invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left digon rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right digon plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q2}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |- style="background: white;"| |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q1,q1}</math><br>[0] 0𝝅 {1} |colspan=4|<math>[1]R_{q1,q1}</math>{{Efn|The <math>[1]R_{q1,q1}</math> rotation is the ''identity operation'' of the 24-cell, in which no points move.|name=Rq1,q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[0] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[0] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |- style="background: white;"| |0 |0° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1,-q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>[1]R_{q1,-q1}</math>{{Efn|The <math>[1]R_{q1,-q1}</math> rotation is the ''central inversion'' of the 24-cell. This isoclinic rotation in great digon invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left digon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right digon plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq1,-q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{-q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(-1,0,0,0)</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |} In a rotation class <math>[d]{R_{ql,qr}}</math> each quaternion group <math>\pm{q_n}</math> may be representative not only of its own fibration of Clifford parallel planes{{Efn|name=quaternion group}} but also of the other congruent fibrations.{{Efn|name=four hexagonal fibrations}} For example, rotation class <math>[4]R_{q7,q8}</math> takes the 4 hexagon planes of <math>q7</math> to the 4 hexagon planes of <math>q8</math> which are 120° away, in an isoclinic rotation. But in a rigid rotation of this kind,{{Efn|name=invariant planes of an isoclinic rotation}} all [16] hexagon planes move in congruent rotational displacements, so this rotation class also includes <math>[4]R_{-q7,-q8}</math>, <math>[4]R_{q8,q7}</math> and <math>[4]R_{-q8,-q7}</math>. The name <math>[16]R_{q7,q8}</math> is the conventional representation for all [16] congruent plane displacements. These rotation classes are all subclasses of <math>[32]R_{q7,q8}</math> which has [32] distinct rotational displacements rather than [16] because there are two [[W:Chiral|chiral]] ways to perform any class of rotations, designated its ''left rotations'' and its ''right rotations''. The [16] left displacements of this class are not congruent with the [16] right displacements, but enantiomorphous like a pair of shoes.{{Efn|A ''right rotation'' is performed by rotating the left and right planes in the "same" direction, and a ''left rotation'' is performed by rotating left and right planes in "opposite" directions, according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is "up" on each of the 4 coordinate axes. Left and right rotations are [[W:chiral|chiral]] enantiomorphous ''shapes'' (like a pair of shoes), not opposite rotational ''directions''. Both left and right rotations can be performed in either the positive or negative rotational direction (from left planes to right planes, or right planes to left planes), but that is an additional distinction.{{Efn|name=clasped hands}}|name=chirality versus direction}} Each left (or right) isoclinic rotation takes [16] left planes to [16] right planes, but the left and right planes correspond differently in the left and right rotations. The left and right rotational displacements of the same left plane take it to different right planes. Each rotation class (table row) describes a distinct left (and right) [[24-cell#Double rotations|double rotation]]. The left (or right) rotations carry the left planes to the right planes simultaneously,{{Efn|name=plane movement in rotations}} through a characteristic twisting rotational displacement.{{Efn|name=two angles between central planes}} For example, the <math>[32]R_{q7,q8}</math> rotation moves all [16] hexagonal planes at once by {{sfrac|2𝝅|3}} = 120° each. Repeated 12 times, this left (or right) isoclinic rotation moves each plane 720° and back to itself in the same [[W:Orientation entanglement|orientation]], <s>passing through all 4 planes of the <math>q7</math> left set and all 4 planes of the <math>q8</math> right set once each</s>.{{Efn|The <math>\pm q7</math> and <math>\pm q8</math> sets of planes are not disjoint; the union of any two of these four sets is a set of 6 planes. The left (versus right) isoclinic rotation of each of these rotation classes (table rows) visits a distinct left (versus right) circular sequence of the same set of 6 Clifford parallel planes.|name=union of q7 and q8}} The picture in the isocline column represents the helical paths of the vertices as they move between planes in the left and right plane sets. In the <math>[32]R_{q7,q8}</math> example it can be seen as a set of 2 Clifford parallel skew {12/5} dodecagrams, <s>each having one edge in each great hexagon plane, and</s> circular helixes which skew to the left (or right) at each vertex throughout the left (or right) double rotation.{{Efn|name=clasped hands}} The 24 vertices circulate on the two parallel {12/5} isoclines. == Visualization == [[File:OctacCrop.jpg|thumb|[[W:Octacube (sculpture)|Octacube steel sculpture]] at Pennsylvania State University]] === Cell rings === The 24-cell is bounded by 24 [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]]. For visualization purposes, it is convenient that the octahedron has opposing parallel [[W:Face (geometry)|faces]] (a trait it shares with the cells of the [[W:Tesseract|tesseract]] and the [[120-cell]]). One can stack octahedrons face to face in a straight line bent in the 4th direction into a [[W:Great circle|great circle]] with a [[W:Circumference|circumference]] of 6 cells.{{Sfn|Coxeter|1970|loc=§8. The simplex, cube, cross-polytope and 24-cell|p=18|ps=; Coxeter studied cell rings in the general case of their geometry and [[W:Group theory|group theory]], identifying each cell ring as a [[W:Polytope|polytope]] in its own right which fills a three-dimensional manifold (such as the [[W:3-sphere|3-sphere]]) with its corresponding [[W:Honeycomb (geometry)|honeycomb]]. He found that cell rings follow [[W:Petrie polygon|Petrie polygon]]s{{Efn|name=Petrie and Clifford dodecagram}} and some (but not all) cell rings and their honeycombs are ''twisted'', occurring in left- and right-handed [[W:chiral|chiral]] forms. Specifically, he found that since the 24-cell's octahedral cells have opposing faces, the cell rings in the 24-cell are of the non-chiral (directly congruent) kind.{{Efn|name=6-cell ring is not chiral}} Each of the 24-cell's cell rings has its corresponding honeycomb in Euclidean (rather than hyperbolic) space, so the 24-cell tiles 4-dimensional Euclidean space by translation to form the [[W:24-cell honeycomb|24-cell honeycomb]].}}{{Sfn|Banchoff|2013|ps=, studied the decomposition of regular 4-polytopes into honeycombs of tori tiling the [[W:Clifford torus|Clifford torus]], showed how the honeycombs correspond to [[W:Hopf fibration|Hopf fibration]]s, and made a particular study of the [[#6-cell rings|24-cell's 4 rings of 6 octahedral cells]] with illustrations.}} The cell locations lend themselves to a [[W:3-sphere|hyperspherical]] description. Pick an arbitrary cell and label it the "[[W:North Pole|North Pole]]". Eight great circle meridians (two cells long) radiate out in 3 dimensions, converging at the 3rd "[[W:South Pole|South Pole]]" cell. This skeleton accounts for 18 of the 24 cells (2&nbsp;+&nbsp;{{gaps|8|×|2}}). See the table below. There is another related [[#Geodesics|great circle]] in the 24-cell, the dual of the one above. A path that traverses 6 vertices solely along edges resides in the dual of this polytope, which is itself since it is self dual. These are the [[#Great hexagons|hexagonal]] geodesics [[#Geodesics|described above]].{{Efn|name=hexagonal fibrations}} One can easily follow this path in a rendering of the equatorial [[W:Cuboctahedron|cuboctahedron]] cross-section. Starting at the North Pole, we can build up the 24-cell in 5 latitudinal layers. With the exception of the poles, each layer represents a separate 2-sphere, with the equator being a great 2-sphere.{{Efn|name=great 2-spheres}} The cells labeled equatorial in the following table are interstitial to the meridian great circle cells. The interstitial "equatorial" cells touch the meridian cells at their faces. They touch each other, and the pole cells at their vertices. This latter subset of eight non-meridian and pole cells has the same relative position to each other as the cells in a [[W:Tesseract|tesseract]] (8-cell), although they touch at their vertices instead of their faces. {| class="wikitable" |- ! Layer # ! Number of Cells ! Description ! Colatitude ! Region |- | style="text-align: center" | 1 | style="text-align: center" | 1 cell | North Pole | style="text-align: center" | 0° | rowspan="2" | Northern Hemisphere |- | style="text-align: center" | 2 | style="text-align: center" | 8 cells | First layer of meridian cells | style="text-align: center" | 60° |- | style="text-align: center" | 3 | style="text-align: center" | 6 cells | Non-meridian / interstitial | style="text-align: center" | 90° | style="text-align: center" |Equator |- | style="text-align: center" | 4 | style="text-align: center" | 8 cells | Second layer of meridian cells | style="text-align: center" | 120° | rowspan="2" | Southern Hemisphere |- | style="text-align: center" | 5 | style="text-align: center" | 1 cell | South Pole | style="text-align: center" | 180° |- ! Total ! 24 cells ! colspan="3" | |} [[File:24-cell-6 ring edge center perspective.png|thumb|An edge-center perspective projection, showing one of four rings of 6 octahedra around the equator]] The 24-cell can be partitioned into cell-disjoint sets of four of these 6-cell great circle rings, forming a discrete [[W:Hopf fibration|Hopf fibration]] of four non-intersecting linked rings.{{Efn|name=fibrations are distinguished only by rotations}} One ring is "vertical", encompassing the pole cells and four meridian cells. The other three rings each encompass two equatorial cells and four meridian cells, two from the northern hemisphere and two from the southern.{{sfn|Banchoff|2013|p=|pp=265-266|loc=}} Note this hexagon great circle path implies the interior/dihedral angle between adjacent cells is 180 - 360/6 = 120 degrees. This suggests you can adjacently stack exactly three 24-cells in a plane and form a 4-D honeycomb of 24-cells as described previously. One can also follow a [[#Geodesics|great circle]] route, through the octahedrons' opposing vertices, that is four cells long. These are the [[#Great squares|square]] geodesics along four {{sqrt|2}} chords [[#Geodesics|described above]]. This path corresponds to traversing diagonally through the squares in the cuboctahedron cross-section. The 24-cell is the only regular polytope in more than two dimensions where you can traverse a great circle purely through opposing vertices (and the interior) of each cell. This great circle is self dual. This path was touched on above regarding the set of 8 non-meridian (equatorial) and pole cells. The 24-cell can be equipartitioned into three 8-cell subsets, each having the organization of a tesseract. Each of these subsets can be further equipartitioned into two non-intersecting linked great circle chains, four cells long. Collectively these three subsets now produce another, six ring, discrete Hopf fibration. === Parallel projections === [[Image:Orthogonal projection envelopes 24-cell.png|thumb|Projection envelopes of the 24-cell. (Each cell is drawn with different colored faces, inverted cells are undrawn)]] The ''vertex-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Rhombic dodecahedron|rhombic dodecahedral]] [[W:Projection envelope|envelope]]. Twelve of the 24 octahedral cells project in pairs onto six square dipyramids that meet at the center of the rhombic dodecahedron. The remaining 12 octahedral cells project onto the 12 rhombic faces of the rhombic dodecahedron. The ''cell-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Cuboctahedron|cuboctahedral]] envelope. Two of the octahedral cells, the nearest and farther from the viewer along the ''w''-axis, project onto an octahedron whose vertices lie at the center of the cuboctahedron's square faces. Surrounding this central octahedron lie the projections of 16 other cells, having 8 pairs that each project to one of the 8 volumes lying between a triangular face of the central octahedron and the closest triangular face of the cuboctahedron. The remaining 6 cells project onto the square faces of the cuboctahedron. This corresponds with the decomposition of the cuboctahedron into a regular octahedron and 8 irregular but equal octahedra, each of which is in the shape of the convex hull of a cube with two opposite vertices removed. The ''edge-first'' parallel projection has an [[W:Elongated hexagonal dipyramidelongated hexagonal dipyramid|Elongated hexagonal dipyramidelongated hexagonal dipyramid]]al envelope, and the ''face-first'' parallel projection has a nonuniform hexagonal bi-[[W:Hexagonal antiprism|antiprismic]] envelope. === Perspective projections === The ''vertex-first'' [[W:Perspective projection|perspective projection]] of the 24-cell into 3-dimensional space has a [[W:Tetrakis hexahedron|tetrakis hexahedral]] envelope. The layout of cells in this image is similar to the image under parallel projection. The following sequence of images shows the structure of the cell-first perspective projection of the 24-cell into 3 dimensions. The 4D viewpoint is placed at a distance of five times the vertex-center radius of the 24-cell. {|class="wikitable" width=660 !colspan=3|Cell-first perspective projection |- valign=top |[[Image:24cell-perspective-cell-first-01.png|220px]]<BR>In this image, the nearest cell is rendered in red, and the remaining cells are in edge-outline. For clarity, cells facing away from the 4D viewpoint have been culled. |[[Image:24cell-perspective-cell-first-02.png|220px]]<BR>In this image, four of the 8 cells surrounding the nearest cell are shown in green. The fourth cell is behind the central cell in this viewpoint (slightly discernible since the red cell is semi-transparent). |[[Image:24cell-perspective-cell-first-03.png|220px]]<BR>Finally, all 8 cells surrounding the nearest cell are shown, with the last four rendered in magenta. |- |colspan=3|Note that these images do not include cells which are facing away from the 4D viewpoint. Hence, only 9 cells are shown here. On the far side of the 24-cell are another 9 cells in an identical arrangement. The remaining 6 cells lie on the "equator" of the 24-cell, and bridge the two sets of cells. |} {| class="wikitable" width=440 |[[Image:24cell section anim.gif|220px]]<br>Animated cross-section of 24-cell |- |colspan=2 valign=top|[[Image:3D stereoscopic projection icositetrachoron.PNG|450px]]<br>A [[W:Stereoscopy|stereoscopic]] 3D projection of an icositetrachoron (24-cell). |- |colspan=3|[[File:Cell24Construction.ogv|450px]]<br>Isometric Orthogonal Projection of: 8 Cell(Tesseract) + 16 Cell = 24 Cell |} == Related polytopes == === Three Coxeter group constructions === There are two lower symmetry forms of the 24-cell, derived as a [[W:Rectification (geometry)|rectified]] 16-cell, with B₄ or [3,3,4] symmetry drawn bicolored with 8 and 16 [[W:Octahedron|octahedral]] cells. Lastly it can be constructed from D₄ or [3^1,1,1] symmetry, and drawn tricolored with 8 octahedra each.<!-- it would be nice to illustrate another of these lower-symmetry decompositions of the 24-cell, into 4 different-colored helixes of 6 face-bonded octahedral cells, as those are the cell rings of its fibration described in /* Visualization */ --> {| class="wikitable collapsible collapsed" !colspan=12| Three [[W:Net (polytope)|nets]] of the ''24-cell'' with cells colored by D₄, B₄, and F₄ symmetry |- ![[W:Rectified demitesseract|Rectified demitesseract]] ![[W:Rectified demitesseract|Rectified 16-cell]] !Regular 24-cell |- !D₄, [3^1,1,1], order 192 !B₄, [3,3,4], order 384 !F₄, [3,4,3], order 1152 |- |colspan=3 align=center|[[Image:24-cell net 3-symmetries.png|659px]] |- valign=top |width=213|Three sets of 8 [[W:Rectified tetrahedron|rectified tetrahedral]] cells |width=213|One set of 16 [[W:Rectified tetrahedron|rectified tetrahedral]] cells and one set of 8 [[W:Octahedron|octahedral]] cells. |width=213|One set of 24 [[W:Octahedron|octahedral]] cells |- |colspan=3 align=center|'''[[W:Vertex figure|Vertex figure]]'''<br>(Each edge corresponds to one triangular face, colored by symmetry arrangement) |- align=center |[[Image:Rectified demitesseract verf.png|120px]] |[[Image:Rectified 16-cell verf.png|120px]] |[[Image:24 cell verf.svg|120px]] |} === Related complex polygons === The [[W:Regular complex polygon|regular complex polygon]] ₄{3}₄, {{Coxeter–Dynkin diagram|4node_1|3|4node}} or {{Coxeter–Dynkin diagram|node_h|6|4node}} contains the 24 vertices of the 24-cell, and 24 4-edges that correspond to central squares of 24 of 48 octahedral cells. Its symmetry is ₄[3]₄, order 96.{{Sfn|Coxeter|1991|p=}} The regular complex polytope ₃{4}₃, {{Coxeter–Dynkin diagram|3node_1|4|3node}} or {{Coxeter–Dynkin diagram|node_h|8|3node}}, in <math>\mathbb{C}^2</math> has a real representation as a 24-cell in 4-dimensional space. ₃{4}₃ has 24 vertices, and 24 3-edges. Its symmetry is ₃[4]₃, order 72. {| class=wikitable width=600 |+ Related figures in orthogonal projections |- !Name !{3,4,3}, {{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}} !₄{3}₄, {{Coxeter–Dynkin diagram|4node_1|3|4node}} !₃{4}₃, {{Coxeter–Dynkin diagram|3node_1|4|3node}} |- !Symmetry ![3,4,3], {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, order 1152 !₄[3]₄, {{Coxeter–Dynkin diagram|4node|3|4node}}, order 96 !₃[4]₃, {{Coxeter–Dynkin diagram|3node|4|3node}}, order 72 |- align=center !Vertices |24||24||24 |- align=center !Edges |96 2-edges||24 4-edge||24 3-edges |- valign=top !valign=center|Image |[[File:24-cell t0 F4.svg|200px]]<BR>24-cell in F4 Coxeter plane, with 24 vertices in two rings of 12, and 96 edges. |[[File:Complex polygon 4-3-4.png|200px]]<BR>₄{3}₄, {{Coxeter–Dynkin diagram|4node_1|3|4node}} has 24 vertices and 32 4-edges, shown here with 8 red, green, blue, and yellow square 4-edges. |[[File:Complex polygon 3-4-3-fill1.png|200px]]<BR>₃{4}₃ or {{Coxeter–Dynkin diagram|3node_1|4|3node}} has 24 vertices and 24 3-edges, shown here with 8 red, 8 green, and 8 blue square 3-edges, with blue edges filled. |} === Related 4-polytopes === Several [[W:Uniform 4-polytope|uniform 4-polytope]]s can be derived from the 24-cell via [[W:Truncation (geometry)|truncation]]: * truncating at 1/3 of the edge length yields the [[W:Truncated 24-cell|truncated 24-cell]]; * truncating at 1/2 of the edge length yields the [[W:Rectified 24-cell|rectified 24-cell]]; * and truncating at half the depth to the dual 24-cell yields the [[W:Bitruncated 24-cell|bitruncated 24-cell]], which is [[W:Cell-transitive|cell-transitive]]. The 96 edges of the 24-cell can be partitioned into the [[W:Golden ratio|golden ratio]] to produce the 96 vertices of the [[W:Snub 24-cell|snub 24-cell]]. This is done by first placing vectors along the 24-cell's edges such that each two-dimensional face is bounded by a cycle, then similarly partitioning each edge into the golden ratio along the direction of its vector. An analogous modification to an [[W:Octahedron|octahedron]] produces an [[W:Regular icosahedron|icosahedron]], or "[[W:Regular icosahedron#Uniform colorings and subsymmetries|snub octahedron]]." The 24-cell is the unique convex self-dual regular Euclidean polytope that is neither a [[W:Polygon|polygon]] nor a [[W:simplex (geometry)|simplex]]. Relaxing the condition of convexity admits two further figures: the [[W:Great 120-cell|great 120-cell]] and [[W:Grand stellated 120-cell|grand stellated 120-cell]]. With itself, it can form a [[W:Polytope compound|polytope compound]]: the [[#Symmetries, root systems, and tessellations|compound of two 24-cells]]. === Related uniform polytopes === {{Demitesseract family}} {{24-cell_family}} The 24-cell can also be derived as a rectified 16-cell: {{Tesseract family}} {{Symmetric_tessellations}} ==See also== *[[W:Octacube (sculpture)|Octacube (sculpture)]] *[[W:Uniform 4-polytope#The F4 family|Uniform 4-polytope § The F4 family]] == Notes == {{Regular convex 4-polytopes Notelist|wiki=W:}} == Citations == {{Regular convex 4-polytopes Reflist|wiki=W:}} == References == {{Refbegin}} {{Regular convex 4-polytopes Refs|wiki=W:}} <br> * {{cite book|last=Ghyka|first=Matila|title=The Geometry of Art and Life|date=1977|place=New York|publisher=Dover Publications|isbn=978-0-486-23542-4|ref={{SfnRef|Ghyka|1977}}}} * {{cite journal|last1=Itoh|first1=Jin-ichi|last2=Nara|first2=Chie|doi=10.1007/s00022-021-00575-6|doi-access=free|issue=13|journal=[[W:Journal of Geometry|Journal of Geometry]]|title=Continuous flattening of the 2-dimensional skeleton of a regular 24-cell|volume=112|year=2021|ref=SfnRef|Itoh & Nara|2021}}}} {{Refend}} ==External links== * [https://bendwavy.org/klitzing/incmats/ico.htm ico], at [https://bendwavy.org/klitzing/home.htm Klitzing polytopes] * [https://polytope.miraheze.org/wiki/Icositetrachoron Icositetrachoron], at [https://polytope.miraheze.org/wiki/Main_Page Polytope wiki] * [http://hi.gher.space/wiki/Xylochoron Xylochoron], at [http://hi.gher.space/wiki/Main_Page Higher space] * [https://www.qfbox.info/4d/24-cell The 24-cell], at [https://www.qfbox.info/4d/index 4D Euclidean Space] * [https://web.archive.org/web/20051118135108/http://valdostamuseum.org/hamsmith/24anime.html 24-cell animations] * [http://members.home.nl/fg.marcelis/24-cell.htm 24-cell in stereographic projections] * [http://eusebeia.dyndns.org/4d/24-cell.html 24-cell description and diagrams] {{Webarchive|url=https://web.archive.org/web/20070715053230/http://eusebeia.dyndns.org/4d/24-cell.html |date=2007-07-15 }} * [https://web.archive.org/web/20071204034724/http://www.xs4all.nl/~jemebius/Ab4help.htm Petrie dodecagons in the 24-cell: mathematics and animation software] [[Category:Geometry]] [[Category:Polyscheme]] 272lac8zjbpouz93ggcmeht87nrdr4l 0 322684 2818511 2756859 2026-07-19T09:04:22Z ~2026-40651-77 3101990 2818511 wikitext text/x-wiki {{title|Neural mechanisms of delayed gratification:<br> What brain systems are involved in the ability to delay gratification }} __TOC__ ==Overview== {{RoundBoxTop|theme=5}} [[File:William Kidd (1790-1863) - Indulging - NG 2547 - National Galleries of Scotland.jpg|William_Kidd_(1790-1863)_-_Indulging_-_NG_2547_-_National_Galleries_of_Scotland|thumb|190px|right|'''Figure 1'''. A depiction of immediate gratification]] ; Scenario It is Friday afternoon, and you have just come home from a long and tiring day at work dealing with frustrating clients and coworkers. All you want to do is sit down and order McDonald’s from Uber Eats and watch your favourite movie. However, despite the gravitational pull of your amazingly comfortable couch, you push through the temptations and go upstairs to study for your upcoming exam. The decision to forgo an instantly gratifying experience for a task requiring deliberate effort without the immediate satisfaction, but which has positive long-term implications, is the perfect depiction of the war waging within your brain when choosing between immediate rewards versus long-term rewards. {{RoundBoxBottom}} [[w:Delayed_gratification|Delayed gratification]], a facet of motivation, is the ability to forego immediate temptations in favour of attaining a larger reward later. The ability to delay immediate gratification is foundational for achieving long-term goals and has been linked to academic achievement, socioemotional success, and positive health (Bembenutty, 2021; Yanaoka et al., 2022). Delayed gratification relies on a dynamic neurological interplay between high-order executive regions, such as the [[w:Pre-frontal_cortex|prefrontal cortex]] and [[w:Subcortical|subcortical]] regions involved in reward processing and emotional reactivity (Gao et al., 2021). Understanding this intricate circuitry has real-world applications to clinical practice concerning obesity, addiction, financial stability, and legal accountability (Hare et al., 2009; Xu, 2021). This chapter discusses brain systems involved in the ability to delay gratification. The neural basis of motivation is broad and complex, so prominent neural systems known to play a central role in delayed gratification are discussed. {{RoundBoxTop|theme=5}} '''Focus questions''' * What is delayed gratification? * What areas of the brain are involved in delayed gratification? * What factors enhance or impair the capacity for delayed gratification? {{RoundBoxBottom}} ==What is delayed gratification?== [[w:Motivation|Motivation]] is the driving force behind actions. It guides behaviour and sustains focus, intensity, and the duration of these behaviours, whether they are adaptive or maladaptive (Bandu et al., 2024). Delayed gratification is a unique form of motivation, operating on the ability to suppress the impulse to attain immediate rewards in favour of larger, more salient future rewards (Yanaoka et al., 2022). People practice delayed gratification daily, whether it be suppressing the urge to buy a daily morning coffee in the hopes to save for a house, or avoiding a fast-food restaurant to remain loyal to a weight loss program. However, the ability to delay gratification requires sustained mental effort and adaptive functioning of neural systems, however, humans are fallible, and these mechanisms can act against what is beneficial{{rewrite}}. Research evaluating factors involved in delayed gratification is assessed using psychological, neuroscientific assessments, and rodent-based studies (Gao et al., 2021) == Neural systems underpinning delayed gratification == [[File:Prefrontal cortex (left) - medial view.png|thumb|'''Figure 2.''' Sagittal view of the prefrontal cortex.]] With over 100 trillion neural connections in the brain, the mechanism of delayed gratification involves many regions of the brain, all playing a particular role in the strategic orchestration of balancing the impulsivity of the reward system and the calculated and analytical executive functions that steer behaviour toward long-term goals (Gao et al., 2021). It is important to note that the multiple cortices comprising the prefrontal cortex (see Figure 2) do not operate in isolation, rather a synchrony of neural impulses all playing a vital role in regulating and suppressing impulses from [[w:Limbic|limbic]] structures (Gao et al., 2021). === Memory === Arguably, the most important structure involved in delayed gratification is the [[w:Hippocampus|hippocampus]]. Located deep within the [[w:Temporal_lobes|temporal lobes]] (see Figure 3), the hippocampus plays a crucial role in memory formation, retrieval, and emotional processing (Lebreton et al., 2013). Lebreton et al. (2013) investigated the neural conflict the hippocampus faces when deliberating between temptations or imagined future satisfying scenarios. Using an [[w:Intertemporal_choice|intertemporal choice]] paradigm, the authors presented participants with concrete options (e.g., food, sports items, and culture) via two formats: visually (via images) or textually, requiring mental simulation. [[w:Functional_magnetic_resonance_imaging|Functional magnetic resonance imaging]] (fMRI) tracked hippocampal activity in twenty participants. The results revealed that the hippocampus is engaged when evaluating imagined outcomes, which reduces impulsivity when deliberating between immediate versus delayed gratification. Moreover, a study by Sasse et al. (2015) examined how episodic prospection (e.g., imagining future events) affects decision-making, particularly in the context of delayed gratification. While undergoing fMRI {{ic|explain abbreviation}} analysis, 23 participants were asked to choose between an immediate monetary reward or a larger reward after a delay. Before the choice, participants were told to imagine one of three scenarios: nothing (simply make the decision), meeting someone they know well, or meeting someone famous. The purpose of this was to anchor the delayed reward to a vivid future moment. Results showed that imagining future events, whether involving familiar or unfamiliar individuals, led participants to favour larger delayed rewards than immediate ones. This suggests that episodic prospection can reduce impulsivity in intertemporal choice. These findings highlight the crucial role the hippocampus plays in decision-making, particularly when imagination is vital for successful delay of gratification. [[File:Hippocampus.gif|thumb|'''Figure 3.''' Hippocampus located deep within the temporal lobes. ]] === Executive control === The [[w:Orbitofrontal_cortex|orbitofrontal cortex]] (OFC), situated in the ventral part of the frontal lobes (see Figure 4), operates as a neural valuation hub, with dense connections to limbic structures. It plays a key role in evaluating the relative value of different options based on prior experience to guide adaptive behaviours{{f}}. A study by Moro et al. (2023) investigated the OFC’s role in a [[w:Delay_discounting|delay discounting]] task, in which participants decided between an immediate reward and a larger delayed reward. [[w:Transcranial_direct_current_stimulation|Transcranial direct current stimulation]] (tCDS) was used to stimulate the OFC while participants performed an [[w:Intertemporal_choice|intertemporal choice]] task. Participants who received OFC tCDS during the intertemporal choice task exhibited a diminished tendency to discount delayed rewards, suggesting an enhanced valuation of future outcomes. Similarly, a review of neuroimaging and lesion studies by Sosa et al. (2021) identified the OFC as a key region involved in inhibitory control during delayed decision-making, particularly when participants are confronted with an immediate reward. This role is further supported by lesion studies in rodents, where OFC damage led to increased impulsivity, with smaller immediate rewards favoured over larger, delayed ones. However, it is important to acknowledge that the lesion methodology did not explicitly isolate the OFC, nor did it account for influence from adjacent cortical regions. Despite this, the OFC appears to play a critical role in evaluating the value of pursuing long-term goals when confronted with competing smaller immediate rewards. The [[w:Dorsolateral_prefrontal_cortex|dorsolateral prefrontal cortex]] (dlPFC), located in the outer lateral portion of the frontal lobe (see Figure 4), plays a crucial role in the top-down regulation of reward-driven impulses and is widely implicated in delayed gratification (Brosnan & Wiegand, 2017). A study conducted by Gbadeyan et al. (2016) utilised [https://soterixmedical.com/research/hd-tdcs high-definition transcranial direct current stimulation] (HD-tDCS) to causally manipulate activity in the dlPFC and examine hemispheric differences and behavioural effects during the [https://www.testable.org/experiment-guides/executive-function/flanker-task visual flanker task]. Participants who received HD-tDCS over either the left or right dlPFC exerted greater cognitive control during the visual flanker task than participants in the control condition, suggesting stimulation of either hemisphere enhances goal-directed behaviour. However, these findings are challenged in a meta-analysis conducted by Yongle et al. (2024), which investigated the hemispheric performance of the dlPFC during decision-making using [https://www.physio-pedia.com/Noninvasive_Brain_Stimulation_(NIBS) non-invasive brain stimulation] (NIBS). The analysis revealed stimulation of the left dlPFC significantly improves individuals' self-control and overrides impulsive responses elicited by tempting stimuli, while the right dlPFC modulates cognitive processes and emotional information, suggesting functional lateralisation of the dlPFC. These findings highlight that increased activity in the dlPFC (notably the left dlPFC) correlates with greater success in the ability to delay gratification when confronted with attractive immediate rewards by exerting greater cognitive control. [[File:Cortical midline structures.png|thumb|'''Figure 4.''' Yellow = vmPFC, blue = dlPFC, red = OFC. ]] Complementing the dlPFC's regulatory role, the [[w:Ventromedial_prefrontal_cortex|ventromedial prefrontal cortex]] (vmPFC), located at the base of the PFC (see Figure 4), supports balanced decision-making by synthesising reward valuation with emotional salience (Ciaramelli et al., 2021). A study conducted by Lamichhane et al. (2022) investigated the role of the vmPFC in a delay of gratification (DofG) task using an economic decision-making paradigm. Twenty-two participants from a local Washington university performed the DofG task while undergoing fMRI analysis. Results elucidate the vmPFC’s role in subjective valuation of rewards by integrating emotional, sensory, and motivational information to guide decisions that involve weighing immediate versus delayed outcomes. Furthermore, lesion studies have shown that individuals with vmPFC damage struggle with future-oriented evaluation, such as delayed gratification and subsequently succumb to impulsivity (Sellitto et al., 2010). Taken together, these findings suggest is that the vmPFC plays an integral role in an individual's ability to refrain from immediate gratification by evaluating the long-term consequences of actions and modulating impulsive drives to support goal-directed behaviour. === Reward seeking === Have you ever wondered why you cannot stop at just one bite of chocolate? [[w:Dopamine|Dopamine]] drives pleasure, motivation, and goal-directed behaviour, with the [[w:Ventral_tegmental_area|ventral tegmental area]] (VTA) as the key source. The VTA projects to regions such as the [[w:Nucleus_accumbens|nucleus accumbens]] and [[w:Basal_ganglia|basal ganglia]], translating motivation into action (Reeve, 2018, p. 53). Recent work by Gao et al. (2021) demonstrated that dopaminergic activity increased in the VTA during the waiting period of a delayed gratification task in mice. [[w:Optogenetic|Optogenetic]] manipulation revealed that activation of these neurons prolonged waiting, while silencing them decreased it, showing that increased dopaminergic activity in the VTA plays a causal role in sustaining patience and promoting goal-directed behaviour in delayed gratification. Supporting this, Bernosky-smith et al. (2021) found that rats with VTA dopamine suppression via [[w:ShRNA|shRNA]] preferred immediate rewards over delayed rewards, highlighting reduced capacity for delayed gratification. However, generalising these findings to humans is limited by genetic, anatomical, and behavioural differences in rodent models. Future research should prioritise human-based studies to enhance understanding of the neurobiological mechanisms underlying delayed gratification (Gao et al., 2021). [[File:Recolored Overview of reward structures in the human brain2.png|thumb|'''Figure 5.''' Reward pathways: Mesocortical (orange), mesolimbic (green), and nigrostriatal (red).]] When people experience something pleasurable, such as food, sex, or socialisation, dopamine projects from the VTA and floods the nucleus accumbens (NAcc) (see Figure 5). This neurological powerhouse plays a key role in the brain’s reward system. Studies have shown that decreased connectivity with the PFC and hyperactivity in NAcc can be linked to risk-taking, increased impulsivity, and addiction, all of which undermine delayed gratification (Basar et al., 2010). A study by Montag et al. (2017) examined the relationship between social media usage and grey matter volume in the NAcc, finding that decreased grey matter was associated with increased reward sensitivity and impulsivity. Furthermore, addiction studies show that the brain undergoes structural change due to excessive dopamine production from drug use, which reduces its sensitivity to dopamine and generates intense cravings and impulsivity (Chiara, 2002; Xu et al., 2024). These findings highlight the NAcc as a key driver of impulsive behaviour, where heightened dopamine activity and reduced grey matter volume bias individuals toward immediate rewards, undermining the capacity for delayed gratification.   {{Robelbox|title=Quiz|theme=5}}<quiz display=simple> {The prefrontal cortex regulates emotion and impulses, critical for delaying gratification: |type="()"} - True + False {The ventral tegmental area produces dopamine: |type="()"} + True - False </quiz> {{Robelbox/close}} == What enhances/impairs the ability to delay gratification == Understanding the neural mechanisms behind delayed gratification offers valuable insight into why some individuals gravitate toward short-term rewards, while others prioritise long-term goals. This variation can be influenced by neurobiological factors, such as variations in [[w:Dopaminergic_pathways|dopaminergic pathways]] and the development of the prefrontal cortex (Gao et al., 2021). Recognising the factors that enhance delayed gratification, such as mindfulness-based techniques and cognitive maturity, can empower individuals to make choices that support long-term well-being across domains including education, finances, and social relationships (Baumeister, 2007). Conversely, identifying factors that impair this mechanism, such as stress, a lack of future perspective, or impulsivity, can help individuals and practitioners address vulnerabilities before they solidify into maladaptive patterns. === Ego depletion === [[w:Ego_depletion|Ego depletion]] is a psychological phenomenon that posits that willpower, or self-control, draws from a finite supply of mental resources (Baumeister, 2007). Think of the brain as a battery: the more self-control is exerted on difficult tasks, the harder it becomes to exercise self-control and, consequently, delay gratification. With [[w:Freudian|Freudian]] roots, ego depletion operates on the notion that psychological forces are in a constant tug-of-war, exerting control over conflicting impulses (Baumeister, 2007). Ego depletion impairs the ability to delay gratification by weakening the brain's self-control systems. When mental resources are depleted, regions such as the dlPFC show reduced activity in fMRI analysis, making it harder to resist immediate gratification. In addition, evaluation systems such as the vmPFC may prioritise short-term gains over long-term goals. This results in an individual's propensity to choose instant gratification over long-term benefits (Inzlicht et al., 2016). A systematic review by Gissubel et al. (2018) examined 30 articles that investigated the role of ego depletion in undergraduate samples. The most common methods to induce ego depletion were the E-crossing task (cross out the letter “e” in written passages) and the [[w:Stroop_Test|Stroop Test]]. Results revealed that ego depletion increased susceptibility to low prosocial behaviour, binge drinking, emotional dysregulation, and decreased performance on memory tasks. These findings reflect a broader decline in self-regulatory capacity, including a diminished ability to delay gratification. However, psychological resources can be replenished after ego depletion. One effective method is to induce positive emotions. Gong and Li (2016) examined how inducing positive emotions after ego-depleting tasks in a student sample can enhance self-control in students. Their study method involved an E-crossing task, followed by a movie designed to elicit positive, negative or neutral emotions, and then a math puzzle. Taken together, these findings revealed that although ego depletion can undermine self-control, strategies such as emotion induction and mindfulness-based strategies offer a pathway to restoring the capacity for self-control and delayed gratification. === Maturation === [[File:Brain maturation ages 8-14.gif|thumb|'''Figure 6.''' Animation of the brain maturing between the ages 8-14. The last areas to develop are shown in pink.]] [[w:Adolescence|Adolescence]] is a developmental period characterised by profound physical, emotional, psychological and social changes due to the onset of [[w:Puberty|puberty]]. During puberty, the brain undergoes significant changes: the limbic system, which governs emotions and reward seeking (e.g., nucleus accumbens and ventral tegmental area), becomes highly active, while the PFC matures gradually into mid-to-late twenties (see Figure 6). This imbalance impacts an adolescent’s ability to attenuate risky and impulsive desires (Konrad et al., 2013). Galvan et al. (2006) examined the neurobiological development of neural systems implicated in reward-seeking in 37 participants aged 7-29. Using fMRI analysis, they measured neural activity in the PFC and NAcc during a reward-based decision-making and response inhibition task. Results revealed increased activation of the NAcc in adolescents during the task compared to children and adults, indicating a stronger propensity to favour immediate rewards. Additionally, the protracted development of the OFC in adolescents further biases decisions toward immediate rewards compared to delayed rewards. These findings highlight the impulsive nature of the adolescent brain, as higher executive systems such as the PFC are still developing, whilst the reward systems are overactive, leading to an impaired capacity to delay gratification. Fortunately, as the brain continues to develop, the capacity for self-control and delayed gratification increases (Achterberg et al., 2016). In a longitudinal study, Achterberg et al. (2016) investigated how maturation of the [[w:Frontostriatal_circuit|frontostriatal circuit]] can improve impulse control and, consequently, the ability to delay gratification. A sample of 192 healthy participants, aged between 8 and 26 years, completed a computerised version of the delay-discounting task while undergoing [[w:Diffusion_tensor_imaging|diffusion tensor imaging]] (DTI). A two-year interval separated completion of TDI and delay-discounting tasks to monitor maturation of the neural circuitry involved in delayed gratification. Results from the study found that around late adolescence is when the ability to delay gratification is at its strongest, with strengthened connectivity between the [[w:Striatum|striatum]] and the PFC, exerting top-down control over impulsive signals from the reward centre. These results provide valuable insight into the neural mechanisms involved in delayed gratification and how it can be enhanced through the development of the PFC, resulting in the regulation of impulsive drives. === Neurobiological differences === [[File:Phineas Gage Cased Daguerreotype WilgusPhoto2008-12-19 EnhancedRetouched Color.jpg|thumb|'''Figure 7.''' Phineas Gage pictured with the metal tampering rod that passed through his skull in 1848. ]] [[w:Attention-deficit_hyperactivity_disorder|Attention-deficit hyperactivity disorder]] (ADHD) is a [[w:Neurodevelopmental_disorder|neurodevelopmental disorder]] characterised by persistent patterns of impulsivity, inattention, and hyperactivity, stemming from distinct neurobiological differences that affect self-regulation and reward processing (Doidge et al., 2018). Individuals with ADHD often struggle with delayed gratification due to reduced activity in the PFC, making it difficult to suppress immediate urges and prioritise long-term goals. A meta-analysis by Plichta and Scheres (2014) examined behavioural and neurobiological mechanisms underlying impulsive decision-making in ADHD. [[w:Aggregate|Aggregate]] data from 32 studies revealed a consistent pattern: individuals with ADHD demonstrated impairments in tasks requiring delayed reward evaluation and consistently chose immediate rewards over delayed rewards. Participants also showed disrupted functioning in the [[w:Frontostriatal_circuit|frontostriatal circuit]], which plays a critical role in delayed gratification. Specifically, individuals with ADHD exhibited reduced activation in the PFC and hypoactivation in the NAcc. This weakened frontostriatal engagement impairs the ability to evaluate future rewards and regulate impulsive drives. Thus, highlighting the impact ADHD imposes on an individual's ability to delay gratification. While individuals with ADHD often struggle with delayed gratification due to executive dysfunction and impulsivity, emerging research suggests that [[w:Mindfulness|mindfulness]]-based techniques can serve as a powerful tool to enhance self-regulation (Bachmann et al., 2016). Bachmann et al. (2016) investigated multiple neuronal systems implicated in ADHD and examined the efficacy of [[w:Mindfulness-based_cognitive_therapy|mindfulness-based cognitive therapy]] (MCBT) using neuroimaging. Their findings showed that mindfulness practices such as body awareness, focused breathing, and non-judgmental attention can reduce the hyperactivity of the [[w:Default_mode_network|default mode network]] (DMN) in ADHD. This reduced activity in the DMN decreases mind wandering and distractibility in ADHD, thus enhancing the individual’s ability to self-monitor and exert cognitive control. These findings highlight the benefit of the neurobehavioural technique MCBT in regulating impaired executive functioning and, as a result, improving self-regulation of attention which can enhance the ability to delay gratification. It is also important to note that mindfulness-based techniques have shown to benefit [[w:Neurotypical|neurotypical]] individuals seeking to strengthen self-control and self-regulation. {{robelbox|theme=5|title=Case study}} <div style="{{Robelbox/pad}}"> Phineas Gage’s traumatic accident revolutionised the field of neuroscience, particularly how brain injury can significantly impact personality and behaviour. Gage was a railroad worker in the 1840s and suffered a catastrophic injury when a metal rod shot through his skull (see Figure 7). Gage remarkably survived, but not without significant changes to his personality and behaviour. Once a previously well-mannered and professional man, he was then regarded as profane, impulsive, and irreverent. Gage’s poignant case offers compelling evidence for the prefrontal cortex’s central function in executive regulation and impulsivity. </div> {{Robelbox/close}} ==Conclusion== The neural mechanisms underpinning delayed gratification involve a complex interplay of brain regions. Neuroimaging techniques, such as fMRI, have mapped brain function during tasks requiring delay of gratification and have provided critical evidence of specific neural structures involved, particularly the PFC and subcortical regions such as the VTA and NAcc. The PFC operates as an integrated system, coordinating impulse control, attention, and future planning to enable delayed gratification. Within it, the OFC, dlPFC, and vmPFC – responsible for reward evaluation, executive control, and value-based decision-making – work in synchrony to support the cognitive processes essential for delaying gratification. Meanwhile, the brain’s reward systems, particularly the VTA and NAcc, modulate dopaminergic signalling that shapes how individuals weigh immediate versus long-term rewards. Dysfunction in these systems can impair delayed gratification. Factors such as ego-depletion, which drains mental resources; immature cognitive development, a symptom of adolescence that limits executive function; and neurobiological differences, such as ADHD, which disrupts impulse regulation and attention, all impair the capacity to delay gratification. However, research shows that delayed gratification can be strengthened through positive emotions, which support self-regulation; cognitive maturation, which enhances future-oriented thinking and executive function; and mindfulness techniques, which reduce impulsivity by fostering self-awareness and prefrontal activation. In summary, prefrontal cortices and subcortical structures all play a vital role in an individual's capacity to delay gratification. ==See also== * [[Motivation and emotion/Book/2013/Delay of gratification|Delay of gratification]] (Book chapter, 2013) * [[Motivation and emotion/Book/2025/Executive function and motivation|Executive function and motivation]] (Book chapter, 2025) * [[Motivation and emotion/Book/2022/Reward system, motivation, and emotion|Reward system, motivation, and emotion]] (Book chapter, 2022) ==References== {{Hanging indent|1= Achterberg, M., Peper, J. S., Van Duijvenvoorde, A. C., Mandl, R. C., & Crone, E. A. (2016). Frontostriatal white matter integrity predicts development of delay of gratification: A longitudinal study. The Journal of Neuroscience, 36(6), 1954-1961. https://doi.org/10.1523/jneurosci.3459-15.2016 Bachmann, K., Lam, A. P., & Philipsen, A. (2016). Mindfulness-based cognitive therapy and the adult ADHD brain: A Neuropsychotherapeutic perspective. Frontiers in Psychiatry, 7. https://doi.org/10.3389/fpsyt.2016.00117 Baumeister, R. F., & Vohs, K. D. (2007). Self‐regulation, ego depletion, and motivation. Social and Personality Psychology Compass, 1(1), 115-128. https://doi.org/10.1111/j.1751-9004.2007.00001.x Bembenutty, H. (2021). Sustaining motivation and academic delay of gratification: Analysis and applications. Theory Into Practice, 61(1), 75-88. https://doi.org/10.1080/00405841.2021.1955555 Bernosky-Smith, K. A., Qiu, Y., Feja, M., Lee, Y. B., Loughlin, B., Li, J., & Bass, C. E. (2018). Ventral tegmental area D2 receptor knockdown enhances choice impulsivity in a delay-discounting task in rats. Behavioural Brain Research, 341, 129-134. https://doi.org/10.1016/j.bbr.2017.12.029 Galvan, A., Hare, T. A., Parra, C. E., Penn, J., Voss, H., Glover, G., & Casey, B. J. (2006). Earlier development of the Accumbens relative to Orbitofrontal cortex might underlie risk-taking behavior in adolescents. Journal of Neuroscience, 26(25), 6885-6892. https://doi.org/10.1523/jneurosci.1062-06.2006 Gao, Z., Wang, H., Lu, C., Lu, T., Froudist-Walsh, S., Chen, M., Wang, X., Hu, J., & Sun, W. (2021). The neural basis of delayed gratification. Science Advances, 7(49). https://doi.org/10.1126/sciadv.abg6611 Gbadeyan, O., McMahon, K., Steinhauser, M., & Meinzer, M. (2016). Stimulation of Dorsolateral prefrontal cortex enhances adaptive cognitive control: A high-definition Transcranial direct current stimulation study. The Journal of Neuroscience, 36(50), 12530-12536. https://doi.org/10.1523/jneurosci.2450-16.2016 Gong, S., & Li, Q. (2016). Rebuilding self-control after ego depletion: The role of positive emotions. Advances in Intelligent Systems and Computing, 1401-1411. https://doi.org/10.1007/978-981-10-1837-4_113 Hare, T., Camerer, C., & Rangel, A. (2009). Self-control in decision-making involves modulation of the vmPFC valuation system. NeuroImage, 47, S95. https://doi.org/10.1016/s1053-8119(09)70776-1 Konrad, K., Firk, C., & Uhlhaas, P. J. (2013). Brain development during adolescence. Deutsches Ärzteblatt international. https://doi.org/10.3238/arztebl.2013.0425 Lamichhane, B., Di Rosa, E., & Braver, T. S. (2022). Delay of gratification dissociates cognitive control and valuation brain regions in healthy young adults. Neuropsychologia, 173, 108303. https://doi.org/10.1016/j.neuropsychologia.2022.108303 Lebreton, M., Bertoux, M., Boutet, C., Lehericy, S., Dubois, B., Fossati, P., & Pessiglione, M. (2013). A critical role for the hippocampus in the valuation of imagined outcomes. PLoS Biology, 11(10), e1001684. https://doi.org/10.1371/journal.pbio.1001684 Plichta, M. M., & Scheres, A. (2014). Ventral–striatal responsiveness during reward anticipation in ADHD and its relation to trait impulsivity in the healthy population: A meta-analytic review of the fMRI literature. Neuroscience & Biobehavioral Reviews, 38, 125-134. https://doi.org/10.1016/j.neubiorev.2013.07.012 Reeve, J. (2018). Understanding motivation and emotion. Wiley. Sasse, L. K., Peters, J., Büchel, C., & Brassen, S. (2015). Effects of prospective thinking on intertemporal choice: The role of familiarity. Human Brain Mapping, 36(10), 4210-4221. https://doi.org/10.1002/hbm.22912 Sellitto, M., Ciaramelli, E., & Di Pellegrino, G. (2010). Myopic discounting of future rewards after medial Orbitofrontal damage in humans. The Journal of Neuroscience, 30(49), 16429-16436. https://doi.org/10.1523/jneurosci.2516-10.2010 Xu, H. (2021). Self-control in career decision-making: The roles of the hot and cool systems of gratification delay. Journal of Vocational Behavior, 131, 103652. https://doi.org/10.1016/j.jvb.2021.103652 Yanaoka, K., Michaelson, L. E., Guild, R. M., Dostart, G., Yonehiro, J., Saito, S., & Munakata, Y. (2022). Cultures crossing: The power of habit in delaying gratification. Psychological Science, 33(7), 1172-1181. https://doi.org/10.1177/09567976221074650 }} ==External links== * [https://neurosciencenews.com/delayed-gratification-hippocampus-2374/ Brain structures involved in delayed gratification identified] (neurosciencenews.com) * [https://www.simplypsychology.org/phineas-gage.html Phineas Gage: His accident and impact on psychology] (simplypsychology.org) * [https://positivepsychology.com/delayed-gratification/ What Is delayed gratification? 5 examples & definition] (positivepsychology.com) [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Delayed gratification]] [[Category:Motivation and emotion/Book/Neuroscience]] [[Category:Motivation and emotion/Book/Reward]] b14s01b864pqsfopjlmg59utpah0plc 2818512 2818511 2026-07-19T09:17:31Z MathXplore 2888076 Reverted edit by [[Special:Contributions/~2026-40651-77|~2026-40651-77]] ([[User_talk:~2026-40651-77|talk]]) to last version by [[User:Jtneill|Jtneill]] using [[Wikiversity:Rollback|rollback]] 2756859 wikitext text/x-wiki {{title|Neural mechanisms of delayed gratification:<br> What brain systems are involved in the ability to delay gratification?}} __TOC__ ==Overview== {{RoundBoxTop|theme=5}} [[File:William Kidd (1790-1863) - Indulging - NG 2547 - National Galleries of Scotland.jpg|William_Kidd_(1790-1863)_-_Indulging_-_NG_2547_-_National_Galleries_of_Scotland|thumb|190px|right|'''Figure 1'''. A depiction of immediate gratification]] ; Scenario It is Friday afternoon, and you have just come home from a long and tiring day at work dealing with frustrating clients and coworkers. All you want to do is sit down and order McDonald’s from Uber Eats and watch your favourite movie. However, despite the gravitational pull of your amazingly comfortable couch, you push through the temptations and go upstairs to study for your upcoming exam. The decision to forgo an instantly gratifying experience for a task requiring deliberate effort without the immediate satisfaction, but which has positive long-term implications, is the perfect depiction of the war waging within your brain when choosing between immediate rewards versus long-term rewards. {{RoundBoxBottom}} [[w:Delayed_gratification|Delayed gratification]], a facet of motivation, is the ability to forego immediate temptations in favour of attaining a larger reward later. The ability to delay immediate gratification is foundational for achieving long-term goals and has been linked to academic achievement, socioemotional success, and positive health (Bembenutty, 2021; Yanaoka et al., 2022). Delayed gratification relies on a dynamic neurological interplay between high-order executive regions, such as the [[w:Pre-frontal_cortex|prefrontal cortex]] and [[w:Subcortical|subcortical]] regions involved in reward processing and emotional reactivity (Gao et al., 2021). Understanding this intricate circuitry has real-world applications to clinical practice concerning obesity, addiction, financial stability, and legal accountability (Hare et al., 2009; Xu, 2021). This chapter discusses brain systems involved in the ability to delay gratification. The neural basis of motivation is broad and complex, so prominent neural systems known to play a central role in delayed gratification are discussed. {{RoundBoxTop|theme=5}} '''Focus questions''' * What is delayed gratification? * What areas of the brain are involved in delayed gratification? * What factors enhance or impair the capacity for delayed gratification? {{RoundBoxBottom}} ==What is delayed gratification?== [[w:Motivation|Motivation]] is the driving force behind actions. It guides behaviour and sustains focus, intensity, and the duration of these behaviours, whether they are adaptive or maladaptive (Bandu et al., 2024). Delayed gratification is a unique form of motivation, operating on the ability to suppress the impulse to attain immediate rewards in favour of larger, more salient future rewards (Yanaoka et al., 2022). People practice delayed gratification daily, whether it be suppressing the urge to buy a daily morning coffee in the hopes to save for a house, or avoiding a fast-food restaurant to remain loyal to a weight loss program. However, the ability to delay gratification requires sustained mental effort and adaptive functioning of neural systems, however, humans are fallible, and these mechanisms can act against what is beneficial{{rewrite}}. Research evaluating factors involved in delayed gratification is assessed using psychological, neuroscientific assessments, and rodent-based studies (Gao et al., 2021) == Neural systems underpinning delayed gratification == [[File:Prefrontal cortex (left) - medial view.png|thumb|'''Figure 2.''' Sagittal view of the prefrontal cortex.]] With over 100 trillion neural connections in the brain, the mechanism of delayed gratification involves many regions of the brain, all playing a particular role in the strategic orchestration of balancing the impulsivity of the reward system and the calculated and analytical executive functions that steer behaviour toward long-term goals (Gao et al., 2021). It is important to note that the multiple cortices comprising the prefrontal cortex (see Figure 2) do not operate in isolation, rather a synchrony of neural impulses all playing a vital role in regulating and suppressing impulses from [[w:Limbic|limbic]] structures (Gao et al., 2021). === Memory === Arguably, the most important structure involved in delayed gratification is the [[w:Hippocampus|hippocampus]]. Located deep within the [[w:Temporal_lobes|temporal lobes]] (see Figure 3), the hippocampus plays a crucial role in memory formation, retrieval, and emotional processing (Lebreton et al., 2013). Lebreton et al. (2013) investigated the neural conflict the hippocampus faces when deliberating between temptations or imagined future satisfying scenarios. Using an [[w:Intertemporal_choice|intertemporal choice]] paradigm, the authors presented participants with concrete options (e.g., food, sports items, and culture) via two formats: visually (via images) or textually, requiring mental simulation. [[w:Functional_magnetic_resonance_imaging|Functional magnetic resonance imaging]] (fMRI) tracked hippocampal activity in twenty participants. The results revealed that the hippocampus is engaged when evaluating imagined outcomes, which reduces impulsivity when deliberating between immediate versus delayed gratification. Moreover, a study by Sasse et al. (2015) examined how episodic prospection (e.g., imagining future events) affects decision-making, particularly in the context of delayed gratification. While undergoing fMRI {{ic|explain abbreviation}} analysis, 23 participants were asked to choose between an immediate monetary reward or a larger reward after a delay. Before the choice, participants were told to imagine one of three scenarios: nothing (simply make the decision), meeting someone they know well, or meeting someone famous. The purpose of this was to anchor the delayed reward to a vivid future moment. Results showed that imagining future events, whether involving familiar or unfamiliar individuals, led participants to favour larger delayed rewards than immediate ones. This suggests that episodic prospection can reduce impulsivity in intertemporal choice. These findings highlight the crucial role the hippocampus plays in decision-making, particularly when imagination is vital for successful delay of gratification. [[File:Hippocampus.gif|thumb|'''Figure 3.''' Hippocampus located deep within the temporal lobes. ]] === Executive control === The [[w:Orbitofrontal_cortex|orbitofrontal cortex]] (OFC), situated in the ventral part of the frontal lobes (see Figure 4), operates as a neural valuation hub, with dense connections to limbic structures. It plays a key role in evaluating the relative value of different options based on prior experience to guide adaptive behaviours{{f}}. A study by Moro et al. (2023) investigated the OFC’s role in a [[w:Delay_discounting|delay discounting]] task, in which participants decided between an immediate reward and a larger delayed reward. [[w:Transcranial_direct_current_stimulation|Transcranial direct current stimulation]] (tCDS) was used to stimulate the OFC while participants performed an [[w:Intertemporal_choice|intertemporal choice]] task. Participants who received OFC tCDS during the intertemporal choice task exhibited a diminished tendency to discount delayed rewards, suggesting an enhanced valuation of future outcomes. Similarly, a review of neuroimaging and lesion studies by Sosa et al. (2021) identified the OFC as a key region involved in inhibitory control during delayed decision-making, particularly when participants are confronted with an immediate reward. This role is further supported by lesion studies in rodents, where OFC damage led to increased impulsivity, with smaller immediate rewards favoured over larger, delayed ones. However, it is important to acknowledge that the lesion methodology did not explicitly isolate the OFC, nor did it account for influence from adjacent cortical regions. Despite this, the OFC appears to play a critical role in evaluating the value of pursuing long-term goals when confronted with competing smaller immediate rewards. The [[w:Dorsolateral_prefrontal_cortex|dorsolateral prefrontal cortex]] (dlPFC), located in the outer lateral portion of the frontal lobe (see Figure 4), plays a crucial role in the top-down regulation of reward-driven impulses and is widely implicated in delayed gratification (Brosnan & Wiegand, 2017). A study conducted by Gbadeyan et al. (2016) utilised [https://soterixmedical.com/research/hd-tdcs high-definition transcranial direct current stimulation] (HD-tDCS) to causally manipulate activity in the dlPFC and examine hemispheric differences and behavioural effects during the [https://www.testable.org/experiment-guides/executive-function/flanker-task visual flanker task]. Participants who received HD-tDCS over either the left or right dlPFC exerted greater cognitive control during the visual flanker task than participants in the control condition, suggesting stimulation of either hemisphere enhances goal-directed behaviour. However, these findings are challenged in a meta-analysis conducted by Yongle et al. (2024), which investigated the hemispheric performance of the dlPFC during decision-making using [https://www.physio-pedia.com/Noninvasive_Brain_Stimulation_(NIBS) non-invasive brain stimulation] (NIBS). The analysis revealed stimulation of the left dlPFC significantly improves individuals' self-control and overrides impulsive responses elicited by tempting stimuli, while the right dlPFC modulates cognitive processes and emotional information, suggesting functional lateralisation of the dlPFC. These findings highlight that increased activity in the dlPFC (notably the left dlPFC) correlates with greater success in the ability to delay gratification when confronted with attractive immediate rewards by exerting greater cognitive control. [[File:Cortical midline structures.png|thumb|'''Figure 4.''' Yellow = vmPFC, blue = dlPFC, red = OFC. ]] Complementing the dlPFC's regulatory role, the [[w:Ventromedial_prefrontal_cortex|ventromedial prefrontal cortex]] (vmPFC), located at the base of the PFC (see Figure 4), supports balanced decision-making by synthesising reward valuation with emotional salience (Ciaramelli et al., 2021). A study conducted by Lamichhane et al. (2022) investigated the role of the vmPFC in a delay of gratification (DofG) task using an economic decision-making paradigm. Twenty-two participants from a local Washington university performed the DofG task while undergoing fMRI analysis. Results elucidate the vmPFC’s role in subjective valuation of rewards by integrating emotional, sensory, and motivational information to guide decisions that involve weighing immediate versus delayed outcomes. Furthermore, lesion studies have shown that individuals with vmPFC damage struggle with future-oriented evaluation, such as delayed gratification and subsequently succumb to impulsivity (Sellitto et al., 2010). Taken together, these findings suggest is that the vmPFC plays an integral role in an individual's ability to refrain from immediate gratification by evaluating the long-term consequences of actions and modulating impulsive drives to support goal-directed behaviour. === Reward seeking === Have you ever wondered why you cannot stop at just one bite of chocolate? [[w:Dopamine|Dopamine]] drives pleasure, motivation, and goal-directed behaviour, with the [[w:Ventral_tegmental_area|ventral tegmental area]] (VTA) as the key source. The VTA projects to regions such as the [[w:Nucleus_accumbens|nucleus accumbens]] and [[w:Basal_ganglia|basal ganglia]], translating motivation into action (Reeve, 2018, p. 53). Recent work by Gao et al. (2021) demonstrated that dopaminergic activity increased in the VTA during the waiting period of a delayed gratification task in mice. [[w:Optogenetic|Optogenetic]] manipulation revealed that activation of these neurons prolonged waiting, while silencing them decreased it, showing that increased dopaminergic activity in the VTA plays a causal role in sustaining patience and promoting goal-directed behaviour in delayed gratification. Supporting this, Bernosky-smith et al. (2021) found that rats with VTA dopamine suppression via [[w:ShRNA|shRNA]] preferred immediate rewards over delayed rewards, highlighting reduced capacity for delayed gratification. However, generalising these findings to humans is limited by genetic, anatomical, and behavioural differences in rodent models. Future research should prioritise human-based studies to enhance understanding of the neurobiological mechanisms underlying delayed gratification (Gao et al., 2021). [[File:Recolored Overview of reward structures in the human brain2.png|thumb|'''Figure 5.''' Reward pathways: Mesocortical (orange), mesolimbic (green), and nigrostriatal (red).]] When people experience something pleasurable, such as food, sex, or socialisation, dopamine projects from the VTA and floods the nucleus accumbens (NAcc) (see Figure 5). This neurological powerhouse plays a key role in the brain’s reward system. Studies have shown that decreased connectivity with the PFC and hyperactivity in NAcc can be linked to risk-taking, increased impulsivity, and addiction, all of which undermine delayed gratification (Basar et al., 2010). A study by Montag et al. (2017) examined the relationship between social media usage and grey matter volume in the NAcc, finding that decreased grey matter was associated with increased reward sensitivity and impulsivity. Furthermore, addiction studies show that the brain undergoes structural change due to excessive dopamine production from drug use, which reduces its sensitivity to dopamine and generates intense cravings and impulsivity (Chiara, 2002; Xu et al., 2024). These findings highlight the NAcc as a key driver of impulsive behaviour, where heightened dopamine activity and reduced grey matter volume bias individuals toward immediate rewards, undermining the capacity for delayed gratification.   {{Robelbox|title=Quiz|theme=5}}<quiz display=simple> {The prefrontal cortex regulates emotion and impulses, critical for delaying gratification: |type="()"} - True + False {The ventral tegmental area produces dopamine: |type="()"} + True - False </quiz> {{Robelbox/close}} == What enhances/impairs the ability to delay gratification == Understanding the neural mechanisms behind delayed gratification offers valuable insight into why some individuals gravitate toward short-term rewards, while others prioritise long-term goals. This variation can be influenced by neurobiological factors, such as variations in [[w:Dopaminergic_pathways|dopaminergic pathways]] and the development of the prefrontal cortex (Gao et al., 2021). Recognising the factors that enhance delayed gratification, such as mindfulness-based techniques and cognitive maturity, can empower individuals to make choices that support long-term well-being across domains including education, finances, and social relationships (Baumeister, 2007). Conversely, identifying factors that impair this mechanism, such as stress, a lack of future perspective, or impulsivity, can help individuals and practitioners address vulnerabilities before they solidify into maladaptive patterns. === Ego depletion === [[w:Ego_depletion|Ego depletion]] is a psychological phenomenon that posits that willpower, or self-control, draws from a finite supply of mental resources (Baumeister, 2007). Think of the brain as a battery: the more self-control is exerted on difficult tasks, the harder it becomes to exercise self-control and, consequently, delay gratification. With [[w:Freudian|Freudian]] roots, ego depletion operates on the notion that psychological forces are in a constant tug-of-war, exerting control over conflicting impulses (Baumeister, 2007). Ego depletion impairs the ability to delay gratification by weakening the brain's self-control systems. When mental resources are depleted, regions such as the dlPFC show reduced activity in fMRI analysis, making it harder to resist immediate gratification. In addition, evaluation systems such as the vmPFC may prioritise short-term gains over long-term goals. This results in an individual's propensity to choose instant gratification over long-term benefits (Inzlicht et al., 2016). A systematic review by Gissubel et al. (2018) examined 30 articles that investigated the role of ego depletion in undergraduate samples. The most common methods to induce ego depletion were the E-crossing task (cross out the letter “e” in written passages) and the [[w:Stroop_Test|Stroop Test]]. Results revealed that ego depletion increased susceptibility to low prosocial behaviour, binge drinking, emotional dysregulation, and decreased performance on memory tasks. These findings reflect a broader decline in self-regulatory capacity, including a diminished ability to delay gratification. However, psychological resources can be replenished after ego depletion. One effective method is to induce positive emotions. Gong and Li (2016) examined how inducing positive emotions after ego-depleting tasks in a student sample can enhance self-control in students. Their study method involved an E-crossing task, followed by a movie designed to elicit positive, negative or neutral emotions, and then a math puzzle. Taken together, these findings revealed that although ego depletion can undermine self-control, strategies such as emotion induction and mindfulness-based strategies offer a pathway to restoring the capacity for self-control and delayed gratification. === Maturation === [[File:Brain maturation ages 8-14.gif|thumb|'''Figure 6.''' Animation of the brain maturing between the ages 8-14. The last areas to develop are shown in pink.]] [[w:Adolescence|Adolescence]] is a developmental period characterised by profound physical, emotional, psychological and social changes due to the onset of [[w:Puberty|puberty]]. During puberty, the brain undergoes significant changes: the limbic system, which governs emotions and reward seeking (e.g., nucleus accumbens and ventral tegmental area), becomes highly active, while the PFC matures gradually into mid-to-late twenties (see Figure 6). This imbalance impacts an adolescent’s ability to attenuate risky and impulsive desires (Konrad et al., 2013). Galvan et al. (2006) examined the neurobiological development of neural systems implicated in reward-seeking in 37 participants aged 7-29. Using fMRI analysis, they measured neural activity in the PFC and NAcc during a reward-based decision-making and response inhibition task. Results revealed increased activation of the NAcc in adolescents during the task compared to children and adults, indicating a stronger propensity to favour immediate rewards. Additionally, the protracted development of the OFC in adolescents further biases decisions toward immediate rewards compared to delayed rewards. These findings highlight the impulsive nature of the adolescent brain, as higher executive systems such as the PFC are still developing, whilst the reward systems are overactive, leading to an impaired capacity to delay gratification. Fortunately, as the brain continues to develop, the capacity for self-control and delayed gratification increases (Achterberg et al., 2016). In a longitudinal study, Achterberg et al. (2016) investigated how maturation of the [[w:Frontostriatal_circuit|frontostriatal circuit]] can improve impulse control and, consequently, the ability to delay gratification. A sample of 192 healthy participants, aged between 8 and 26 years, completed a computerised version of the delay-discounting task while undergoing [[w:Diffusion_tensor_imaging|diffusion tensor imaging]] (DTI). A two-year interval separated completion of TDI and delay-discounting tasks to monitor maturation of the neural circuitry involved in delayed gratification. Results from the study found that around late adolescence is when the ability to delay gratification is at its strongest, with strengthened connectivity between the [[w:Striatum|striatum]] and the PFC, exerting top-down control over impulsive signals from the reward centre. These results provide valuable insight into the neural mechanisms involved in delayed gratification and how it can be enhanced through the development of the PFC, resulting in the regulation of impulsive drives. === Neurobiological differences === [[File:Phineas Gage Cased Daguerreotype WilgusPhoto2008-12-19 EnhancedRetouched Color.jpg|thumb|'''Figure 7.''' Phineas Gage pictured with the metal tampering rod that passed through his skull in 1848. ]] [[w:Attention-deficit_hyperactivity_disorder|Attention-deficit hyperactivity disorder]] (ADHD) is a [[w:Neurodevelopmental_disorder|neurodevelopmental disorder]] characterised by persistent patterns of impulsivity, inattention, and hyperactivity, stemming from distinct neurobiological differences that affect self-regulation and reward processing (Doidge et al., 2018). Individuals with ADHD often struggle with delayed gratification due to reduced activity in the PFC, making it difficult to suppress immediate urges and prioritise long-term goals. A meta-analysis by Plichta and Scheres (2014) examined behavioural and neurobiological mechanisms underlying impulsive decision-making in ADHD. [[w:Aggregate|Aggregate]] data from 32 studies revealed a consistent pattern: individuals with ADHD demonstrated impairments in tasks requiring delayed reward evaluation and consistently chose immediate rewards over delayed rewards. Participants also showed disrupted functioning in the [[w:Frontostriatal_circuit|frontostriatal circuit]], which plays a critical role in delayed gratification. Specifically, individuals with ADHD exhibited reduced activation in the PFC and hypoactivation in the NAcc. This weakened frontostriatal engagement impairs the ability to evaluate future rewards and regulate impulsive drives. Thus, highlighting the impact ADHD imposes on an individual's ability to delay gratification. While individuals with ADHD often struggle with delayed gratification due to executive dysfunction and impulsivity, emerging research suggests that [[w:Mindfulness|mindfulness]]-based techniques can serve as a powerful tool to enhance self-regulation (Bachmann et al., 2016). Bachmann et al. (2016) investigated multiple neuronal systems implicated in ADHD and examined the efficacy of [[w:Mindfulness-based_cognitive_therapy|mindfulness-based cognitive therapy]] (MCBT) using neuroimaging. Their findings showed that mindfulness practices such as body awareness, focused breathing, and non-judgmental attention can reduce the hyperactivity of the [[w:Default_mode_network|default mode network]] (DMN) in ADHD. This reduced activity in the DMN decreases mind wandering and distractibility in ADHD, thus enhancing the individual’s ability to self-monitor and exert cognitive control. These findings highlight the benefit of the neurobehavioural technique MCBT in regulating impaired executive functioning and, as a result, improving self-regulation of attention which can enhance the ability to delay gratification. It is also important to note that mindfulness-based techniques have shown to benefit [[w:Neurotypical|neurotypical]] individuals seeking to strengthen self-control and self-regulation. {{robelbox|theme=5|title=Case study}} <div style="{{Robelbox/pad}}"> Phineas Gage’s traumatic accident revolutionised the field of neuroscience, particularly how brain injury can significantly impact personality and behaviour. Gage was a railroad worker in the 1840s and suffered a catastrophic injury when a metal rod shot through his skull (see Figure 7). Gage remarkably survived, but not without significant changes to his personality and behaviour. Once a previously well-mannered and professional man, he was then regarded as profane, impulsive, and irreverent. Gage’s poignant case offers compelling evidence for the prefrontal cortex’s central function in executive regulation and impulsivity. </div> {{Robelbox/close}} ==Conclusion== The neural mechanisms underpinning delayed gratification involve a complex interplay of brain regions. Neuroimaging techniques, such as fMRI, have mapped brain function during tasks requiring delay of gratification and have provided critical evidence of specific neural structures involved, particularly the PFC and subcortical regions such as the VTA and NAcc. The PFC operates as an integrated system, coordinating impulse control, attention, and future planning to enable delayed gratification. Within it, the OFC, dlPFC, and vmPFC – responsible for reward evaluation, executive control, and value-based decision-making – work in synchrony to support the cognitive processes essential for delaying gratification. Meanwhile, the brain’s reward systems, particularly the VTA and NAcc, modulate dopaminergic signalling that shapes how individuals weigh immediate versus long-term rewards. Dysfunction in these systems can impair delayed gratification. Factors such as ego-depletion, which drains mental resources; immature cognitive development, a symptom of adolescence that limits executive function; and neurobiological differences, such as ADHD, which disrupts impulse regulation and attention, all impair the capacity to delay gratification. However, research shows that delayed gratification can be strengthened through positive emotions, which support self-regulation; cognitive maturation, which enhances future-oriented thinking and executive function; and mindfulness techniques, which reduce impulsivity by fostering self-awareness and prefrontal activation. In summary, prefrontal cortices and subcortical structures all play a vital role in an individual's capacity to delay gratification. ==See also== * [[Motivation and emotion/Book/2013/Delay of gratification|Delay of gratification]] (Book chapter, 2013) * [[Motivation and emotion/Book/2025/Executive function and motivation|Executive function and motivation]] (Book chapter, 2025) * [[Motivation and emotion/Book/2022/Reward system, motivation, and emotion|Reward system, motivation, and emotion]] (Book chapter, 2022) ==References== {{Hanging indent|1= Achterberg, M., Peper, J. S., Van Duijvenvoorde, A. C., Mandl, R. C., & Crone, E. A. (2016). Frontostriatal white matter integrity predicts development of delay of gratification: A longitudinal study. The Journal of Neuroscience, 36(6), 1954-1961. https://doi.org/10.1523/jneurosci.3459-15.2016 Bachmann, K., Lam, A. P., & Philipsen, A. (2016). Mindfulness-based cognitive therapy and the adult ADHD brain: A Neuropsychotherapeutic perspective. Frontiers in Psychiatry, 7. https://doi.org/10.3389/fpsyt.2016.00117 Baumeister, R. F., & Vohs, K. D. (2007). Self‐regulation, ego depletion, and motivation. Social and Personality Psychology Compass, 1(1), 115-128. https://doi.org/10.1111/j.1751-9004.2007.00001.x Bembenutty, H. (2021). Sustaining motivation and academic delay of gratification: Analysis and applications. Theory Into Practice, 61(1), 75-88. https://doi.org/10.1080/00405841.2021.1955555 Bernosky-Smith, K. A., Qiu, Y., Feja, M., Lee, Y. B., Loughlin, B., Li, J., & Bass, C. E. (2018). Ventral tegmental area D2 receptor knockdown enhances choice impulsivity in a delay-discounting task in rats. Behavioural Brain Research, 341, 129-134. https://doi.org/10.1016/j.bbr.2017.12.029 Galvan, A., Hare, T. A., Parra, C. E., Penn, J., Voss, H., Glover, G., & Casey, B. J. (2006). Earlier development of the Accumbens relative to Orbitofrontal cortex might underlie risk-taking behavior in adolescents. Journal of Neuroscience, 26(25), 6885-6892. https://doi.org/10.1523/jneurosci.1062-06.2006 Gao, Z., Wang, H., Lu, C., Lu, T., Froudist-Walsh, S., Chen, M., Wang, X., Hu, J., & Sun, W. (2021). The neural basis of delayed gratification. Science Advances, 7(49). https://doi.org/10.1126/sciadv.abg6611 Gbadeyan, O., McMahon, K., Steinhauser, M., & Meinzer, M. (2016). Stimulation of Dorsolateral prefrontal cortex enhances adaptive cognitive control: A high-definition Transcranial direct current stimulation study. The Journal of Neuroscience, 36(50), 12530-12536. https://doi.org/10.1523/jneurosci.2450-16.2016 Gong, S., & Li, Q. (2016). Rebuilding self-control after ego depletion: The role of positive emotions. Advances in Intelligent Systems and Computing, 1401-1411. https://doi.org/10.1007/978-981-10-1837-4_113 Hare, T., Camerer, C., & Rangel, A. (2009). Self-control in decision-making involves modulation of the vmPFC valuation system. NeuroImage, 47, S95. https://doi.org/10.1016/s1053-8119(09)70776-1 Konrad, K., Firk, C., & Uhlhaas, P. J. (2013). Brain development during adolescence. Deutsches Ärzteblatt international. https://doi.org/10.3238/arztebl.2013.0425 Lamichhane, B., Di Rosa, E., & Braver, T. S. (2022). Delay of gratification dissociates cognitive control and valuation brain regions in healthy young adults. Neuropsychologia, 173, 108303. https://doi.org/10.1016/j.neuropsychologia.2022.108303 Lebreton, M., Bertoux, M., Boutet, C., Lehericy, S., Dubois, B., Fossati, P., & Pessiglione, M. (2013). A critical role for the hippocampus in the valuation of imagined outcomes. PLoS Biology, 11(10), e1001684. https://doi.org/10.1371/journal.pbio.1001684 Plichta, M. M., & Scheres, A. (2014). Ventral–striatal responsiveness during reward anticipation in ADHD and its relation to trait impulsivity in the healthy population: A meta-analytic review of the fMRI literature. Neuroscience & Biobehavioral Reviews, 38, 125-134. https://doi.org/10.1016/j.neubiorev.2013.07.012 Reeve, J. (2018). Understanding motivation and emotion. Wiley. Sasse, L. K., Peters, J., Büchel, C., & Brassen, S. (2015). Effects of prospective thinking on intertemporal choice: The role of familiarity. Human Brain Mapping, 36(10), 4210-4221. https://doi.org/10.1002/hbm.22912 Sellitto, M., Ciaramelli, E., & Di Pellegrino, G. (2010). Myopic discounting of future rewards after medial Orbitofrontal damage in humans. The Journal of Neuroscience, 30(49), 16429-16436. https://doi.org/10.1523/jneurosci.2516-10.2010 Xu, H. (2021). Self-control in career decision-making: The roles of the hot and cool systems of gratification delay. Journal of Vocational Behavior, 131, 103652. https://doi.org/10.1016/j.jvb.2021.103652 Yanaoka, K., Michaelson, L. E., Guild, R. M., Dostart, G., Yonehiro, J., Saito, S., & Munakata, Y. (2022). Cultures crossing: The power of habit in delaying gratification. Psychological Science, 33(7), 1172-1181. https://doi.org/10.1177/09567976221074650 }} ==External links== * [https://neurosciencenews.com/delayed-gratification-hippocampus-2374/ Brain structures involved in delayed gratification identified] (neurosciencenews.com) * [https://www.simplypsychology.org/phineas-gage.html Phineas Gage: His accident and impact on psychology] (simplypsychology.org) * [https://positivepsychology.com/delayed-gratification/ What Is delayed gratification? 5 examples & definition] (positivepsychology.com) [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Delayed gratification]] [[Category:Motivation and emotion/Book/Neuroscience]] [[Category:Motivation and emotion/Book/Reward]] 0lf2e1tbhes9benkd4pdrb2fqmh9yi2 2 326765 2818489 2818467 2026-07-18T18:21:13Z Dc.samizdat 2856930 /* The 24-cell */ 2818489 wikitext text/x-wiki = Golden chords of the 120-cell = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - June 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by 5² non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). Consequently 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which contains 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). Consequently the 8-point (16-cell) compounds by 3 times 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} == The 16-cell 4-orthoplex == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=\sqrt{2}+1</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. Their 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}+1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-1</math> . Over the third <math>r_3</math> chord it also moves a distance of <math>-1</math>. Fontaine and Hurley also demonstrated the significance of <math>1/r_i</math> in Steinbach's Diagonal Product Formula, which says that every chord length is the sum of certain smaller chord lengths. The smaller chords are certain diagonals of the same regular polygon of a smaller edge length, specifically edge length <math>1/r_i</math> rather than <math>1</math>. If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{3,3,4\}</math></small>. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <math>\sqrt{2}</math> edges except opposite pairs. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Each chord is a distinct 4-vector with a length and a direction. 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, where vertices circle over the chords of an <math>r_i</math> polygon. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. The angle between two <math>r_i</math> chords is <math>180^\circ - i \times 45^\circ</math>. [[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 {8/1} 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 the 16-cell form an ''edge polygon'' {8/2}=2{4}. The 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 <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> (it is <math>6\pi</math> in this case), and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> chords, each vertex makes twelve 45° 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 revolution 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. We shall refer to this isoclinic rotation as the ''great square rotation characteristic of the 16-cell'', and note once again that it is Fontaine and Hurley's counterclockwise rotation over the <math>r_3</math> {8/3} star polygon, which constructs <math>1/r_3</math>. == The 8-cell tesseract == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. [[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract <small><math>\{4,3,3\}</math></small> performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]] The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{4,3,3\}</math></small>. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedrons]]. 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 the great square rotation characteristic of the 16-cell, with the same effect on both alternate-position 16-cells. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords. They form a circular double helix which intersects each vertex of the tesseract once. The double helix is an 8-rung ladder twisted around 3 times, and bent into a circle in the fourth dimension with its ends joined. Each rung is a <math>\sqrt{3}</math> chord. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron. The 24-cell has the same chord set as the 4-hypercube tesseract: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]] The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters. The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords: :<math>r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The <math>r_1</math> and <math>r_5</math> chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. The <math>r_1</math> chords of the 24-cell form a Petrie polygon {12/1} which zig-zags back and forth, in the left and right rotational directions, between two Clifford parallel great hexagons formed by <math>r_2</math> chords. The <math>r_2</math> chords of the 24-cell form a compound great hexagon edge polygon {24/4}=4{6}. The four great hexagons lie Clifford parallel to each other. A ''simple'' rotation of the 24-cell in ''one'' of those hexagon central planes rotates that hexagon like a wheel, while the completely orthogonal central plane does not move. The six vertices of the rotating hexagon orbit on a great circle in the plane. The <math>r_3</math> chords of the 24-cell form a compound great square edge polygon {24/6}=6{4}. The six great squares lie Clifford parallel to each other. They include one pair of completely orthogonal great squares from each of the three 16-cells. The <math>r_5</math> chords of the 16-cell form a circular helix isocline, visible as a green {12/5} dodecagram Clifford polygon in the orthogonal projection. An isoclinic rotation of the 24-cell, in two Clifford parallel invariant <math>r_2</math> hexagon planes at once by equal angles, moves the twelve vertices along the circular helix over <math>r_5</math> chords. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges of Clifford parallel great squares, also isocline chords in great square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in great hexagon rotations. The green {12/5} dodecagram is a Clifford polygon.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} shows three octagram isoclines of <small><math>\sqrt{2}</math> </small>chords in the 24-cell]] We can rotate the 24-cell isoclinically in 6 Clifford parallel invariant great square planes containing 16-cell edges, in the great square rotation characteristic of the 16-cell, with the same effect on all three 16-cells. In 720° each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The rotational curve over each 90° <small><math>\sqrt{2}</math></small> chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <small><math>\sqrt{2}</math></small> chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. The triple helix is an 8-step circular staircase that twists around 3 times, and is bent into a torus in the fourth dimension. Each staircase step is a great triangle of <small><math>\sqrt{3}</math></small> chords. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} shows 2 dodecagram isoclines of <small><math>\sqrt{3}</math></small> chords in the 24-cell]]We can rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing six <math>r_{2}</math> 24-cell edges each, over <math>r_{5}</math> isocline chords. This is the ''great hexagon rotation characteristic of the 24-cell'', also Fontaine and Hurley's counterclockwise rotation over the <math>r_5</math> {12/5} star polygon which constructs <math>1/r_5</math>. A complete 24-cell great circle edge plane revolution requires 720° like a complete 16-cell great circle edge plane revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. An isoclinic rotation by 60° in an invariant central plane containing a 24-cell edge takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all 4 invariant great hexagon planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square. All 24 vertices move at once on Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 12 <math>r_5</math> <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane 5 times in a moving invariant rotation plane. The rotational curve over each <math>r_5</math> 120° chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the course of a 720° revolution each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="6" |6 distinct 180° chord pairs make 6 distinct isoclinic rotations |- ! colspan="3" |Edge chord ! colspan="3" |Isocline chord |- style="background: gainsboro;" | | rowspan="4" |<math>t_1</math> |60° | rowspan="4" |[[File:Regular_polygon_24.svg|100px]]<br>{24/1}={24} | rowspan="4" |[[File:Regular_star_polygon_24-11.svg|100px]]<br>{24/11} |120° | rowspan="4" |<math>t_{11}</math> |- style="background: gainsboro;" | |{{radic|1}} |{{radic|3}} |- style="background: gainsboro;" | |1 |1.732~ |- style="background: gainsboro;" | |15° |165° |- style="background: palegreen;" | | rowspan="4" |<math>t_2</math> |60° | rowspan="4" |[[File:Regular_star_figure_2(12,1).svg|100px]]<br>{24/2}=2{12} | rowspan="4" |[[File:Regular_star_figure_2(12,5).svg|100px]]<br>{24/10}=2{12/5} |120° | rowspan="4" |<math>t_{10}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | |30° |150° |- style="background: seashell;" | | rowspan="4" |<math>t_3</math> |90° | rowspan="4" |[[File:Regular_star_figure_3(8,1).svg|100px]]<br>{24/3}=3{8} | rowspan="4" |[[File:Regular_star_figure_3(8,3).svg|100px]]<br>{24/9}=3{8/3} |90° | rowspan="4" |<math>t_{9}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |- style="background: seashell;" | |45° |135° |- style="background: palegreen;" | | rowspan="4" |<math>t_4</math> |60° | rowspan="4" |[[File:Regular_star_figure_4(6,1).svg|100px]]<br>{24/4}=4{6} | rowspan="4" |[[File:Regular_star_figure_8(3,1).svg|100px]]<br>{24/8}=8{3} |120° | rowspan="4" |<math>t_{8}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | |60° |120° |- style="background: gainsboro;" | | rowspan="4" |<math>t_5</math> |60° | rowspan="4" |[[File:Regular_star_polygon_24-5.svg|100px]]<br>{24/5} | rowspan="4" |[[File:Regular_star_polygon_24-7.svg|100px]]<br>{24/7} |120° | rowspan="4" |<math>t_{7}</math> |- style="background: gainsboro;" | |{{radic|1}} |{{radic|3}} |- style="background: gainsboro;" | |1 |1.732~ |- style="background: gainsboro;" | |75° |105° |- style="background: gainsboro;" | | rowspan="4" |<math>t_6</math> |90° | rowspan="4" |[[File:Regular_star_figure_6(4,1).svg|100px]]<br>{24/6}=6{4} | rowspan="4" |[[File:Regular_star_figure_6(4,1).svg|100px]]<br>{24/6}=6{4} |90° | rowspan="4" |<math>t_{6}</math> |- style="background: gainsboro;" | |{{radic|2}} |{{radic|2}} |- style="background: gainsboro;" | |1.414~ |1.414~ |- style="background: gainsboro;" | |90° |90° |} By examining the chords <math>r_i</math> of the 24-cell's Petrie {12}-gon we found two distinct isoclinic rotations, the great square rotation characteristic of the 16-cell and the great hexagon rotation characteristic of the 24-cell. If we examine the chords <math>t_i</math> of the 24-cell's {24}-gon we find these and also four other distinct isoclinic rotations. Each row of the table is a distinct isoclinic rotation of the 24-cell characterized by a pair of chords that sum to 180°. The edge chords form the rotation's edge {24}-gon, and lie in invariant planes of the rotation. The isocline chords form the rotation's Clifford {24}-gon and lie in the invariant planes completely orthogonal to the edge planes. The rotational angle between successive edge chords and the rotational angle between successive isocline chords also sum to 180°. ... {{Clear}} == The 600-cell == [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are: [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</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. {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="7" |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra |- ! colspan="3" |Short edge chord ! Section ! colspan="3" |Long isocline chord |- style="background: palegreen;" | | rowspan="4" |<math>r_0</math> |0° | rowspan="4" | | rowspan="4" | | rowspan="4" |[[File:Regular_star_figure_15(2,1).svg|100px]]<br>{30/15}=15{2} |180° | rowspan="4" |<math>r_{15}</math> |- style="background: palegreen;" | |{{radic|0}} |{{radic|4}} |- style="background: palegreen;" | |0 |2 |- style="background: palegreen;" | |0° |180° |- style="background: palegreen;" | | rowspan="4" |<math>r_1</math> |36° | rowspan="4" |[[File:Regular_polygon_30.svg|100px]]<br>{30/1} | rowspan="4" | | rowspan="4" |[[File:Regular_star_figure_2(15,7).svg|100px]]<br>{30/14}=2{15/7} |144° | rowspan="4" |<math>r_{14}</math> |- style="background: palegreen;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: palegreen;" | |0.618~ |1.902~ |- style="background: palegreen;" | |12° |168° |- style="background: gainsboro;" | | rowspan="4" |<math>r_2</math> |36° | rowspan="4" |[[File:Regular_star_figure_2(15,1).svg|100px]]<br>{30/2}=2{15} | rowspan="4" | | rowspan="4" |[[File:Regular_star_polygon_30-13.svg|100px]]<br>{30/13} |144° | rowspan="4" |<math>r_{13}</math> |- style="background: gainsboro;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: gainsboro;" | |0.618~ |1.902~ |- style="background: gainsboro;" | |24° |156° |- style="background: yellow;" | | rowspan="4" |<math>r_3</math> |36° | rowspan="4" |[[File:Regular_star_figure_3(10,1).svg|100px]]<br>{30/3}=3{10} | rowspan="4" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="4" |[[File:Regular_star_figure_6(5,2).svg|100px]]<br>{30/12}=6{5/2} |144° | rowspan="4" |<math>r_{12}</math> |- style="background: yellow;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: yellow;" | |0.618~ |1.902~ |- style="background: yellow;" | |36° |144° |- style="background: palegreen;" | | rowspan="4" |<math>r_4</math> |60° | rowspan="4" |[[File:Regular_star_figure_2(15,2).svg|100px]]<br>{30/4}=2{15/2} | rowspan="4" | | rowspan="4" |[[File:Regular_star_polygon_30-11.svg|100px]]<br>{30/11} |120° | rowspan="4" |<math>r_{11}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | |48° |132° |- style="background: palegreen;" | | rowspan="4" |<math>r_5</math> |60° | rowspan="4" |[[File:Regular_star_figure_5(6,1).svg|100px]]<br>{30/5}=5{6} | rowspan="4" |[[File:V2 dodecahedron.png|100px]]<br>Dodecahedron | rowspan="4" |[[File:Regular_star_figure_10(3,1).svg|100px]]<br>{30/10}=10{3} |120° | rowspan="4" |<math>r_{10}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | |60° |120° |- style="background: yellow;" | | rowspan="4" |<math>r_{6}</math> |72° | rowspan="4" |[[File:Regular_star_figure_6(5,1).svg|100px]]<br>{30/6}=6{5} | rowspan="4" |[[File:V3 icosahedron.png|100px]]<br>Icosahedron | rowspan="4" |[[File:Regular_star_figure_3(10,3).svg|100px]]<br>{30/9}=3{10/3} |108° | rowspan="4" |<math>r_{9}</math> |- style="background: yellow;" | |{{radic|1.382~}} |{{radic|2.618~}} |- style="background: yellow;" | |1.176~ |1.618~ |- style="background: yellow;" | |72° |108° |- style="background: seashell;" | | rowspan="4" |<math>r_{7}</math> |90° | rowspan="4" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} | rowspan="4" |[[File:V4 icosidodecahedron.png|100px]]<br>Icosidodecahedron | rowspan="4" |[[File:Regular_star_figure_2(15,4).svg|100px]]<br>{30/8}=2{15/4} |90° | rowspan="4" |<math>r_{8}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |- style="background: seashell;" | |84° |96° |} The list of 600-cell chords <math>r_{i}</math> can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. The short chord and long chord each have their characteristic {30/n}-gon. Each row identifies a discrete isoclinic rotation of the 600-cell in invariant central planes containing the edges of the short chord {30}-gon, over the isocline chords of the long chord {30}-gon, the rotation's Clifford polygon. Each distinct pair of complementary chord lengths is identified with a distinct [[w:600-cell#Polyhedral sections|polyhedral section of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance <math>\phi^{-1}</math> is a icosahedron vertex figure, and the largest section at radial distance <math>\sqrt{2}</math> is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. Because [[w:3-sphere|<math>\mathbb{S}^3</math>]] is spherical, at radial distances greater than <math>\sqrt{2}</math> the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance <math>\sqrt{2+\phi}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>\sqrt{2}</math></small>]] We can rotate the 600-cell isoclinically in the great square rotation characteristic of the 16-cell, with the same effect on 15 disjoint 16-cells. Each 90° displacement takes 15 pairs of completely orthogonal invariant great square planes to each other. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The rotational curve over each 90° chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> form a circular fibration of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once. The 600-cell has another distinct isoclinic rotation in invariant great square planes, which has period 30 and visits every vertex of a 600-cell Petrie polygon. This [''great square left rotation characteristic of the 600-cell]'' takes place over <math>r_7</math> edge chords and <math>r_8</math> isocline chords. The {30/7} edge polygon is a skew helix of circumference <math>14\pi</math> with each <math>r_7</math> edge belonging to a distinct great square. The four {30/7} polygrams contribute one edge each to 30 great squares. Each 90° displacement takes every 16-cell to another 16-cell. The vertices of the invariant great squares each make seven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 90° {30/7} edge makes seven 12° turns. Four Clifford parallel {30/7} geodesics of circumference <math>16\pi</math> form a circular quadruple helix that intersects each 600-cell vertex once. The {30/8}=2{15/4} Clifford polygon is a compound of two skew {15/4} pentadecagrams of circumference <math>16\pi</math> with each <math>r_8</math> isocline chord belonging to a distinct 16-cell. The four {30/8} polygrams contribute one edge each to 30 great squares. The rotational curve over each 90° {30/8} isocline chord makes eight 12° turns. Four Clifford parallel {30/8} geodesics of circumference <math>16\pi</math> form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>\sqrt{3}</math></small> ]] We can rotate the 600-cell isoclinically in the great hexagon rotation characteristic of the 24-cell, over <math>\sqrt{1}</math> edge chords and <math>\sqrt{3}</math> isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° revolution each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> form a circular fibration of ten twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. The 600-cell has another distinct isoclinic rotation in invariant great hexagon planes, over <math>r_{4}=\sqrt{1}</math> edge chords and <math>r_{11}=\sqrt{3}</math> isocline chords This [''invariant great hexagon left rotation characteristic of the 600-cell]'' has period 30 and visits every vertex of a 600-cell Petrie polygon. Its {30/11} Clifford polygon is a skew helix where each <math>r_{11}</math> isocline chord is the <math>\sqrt{3}</math> diagonal of a great hexagon of a distinct 24-cell. The vertices of the invariant great hexagons of this rotation each make eleven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 120° <math>r_{11}</math> isocline chord makes eleven 12° turns. Four Clifford parallel {30/11} geodesic isoclines of circumference <math>22\pi</math> over <math>r_{11}</math> chords form a circular quadruple helix that intersects each 600-cell vertex once. We can rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its 36° <math>r_{3}</math> edges, over 144° <math>r_{12}</math> isocline chords. This ''invariant great decagon rotation characteristic of the 600-cell'' has period 5 and takes disjoint 24-cells to each other. The rotational curve over each <math>r_{12}</math> chord of its {5/2} Clifford polygon makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference <math>4\pi</math> over five <math>r_{12}</math> chords form a circular fibration of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. The rotation of the 600-cell 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. The 24-cells revolve within the 600-cell, as the 16-cells revolve within the 24-cells. All 120 vertices move at once on four Clifford parallel geodesic isoclines, displaced 144° in different directions. The 600-cell has another distinct isoclinic rotation in invariant great decagon planes containing its 36° <math>r_{2}</math> edges, over 144° <math>r_{13}</math> isocline chords. This [''great decagon left rotation characteristic of the 600-cell]'' has period 30 and visits every vertex of a 600-cell Petrie polygon. The rotational curve over each 144° <math>r_{13}</math> isocline chord makes thirteen 12° turns. Four Clifford parallel {30/13} geodesic isoclines of circumference <math>26\pi</math> form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} == The 5-cell 4-simplex == {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="9" |30 chords (15 180° pairs) make 15 distinct section polyhedra |- ! colspan="3" |Short chord ! Section ! colspan="3" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px]]<br>{30/15}=15{2} |180° | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} |{{radic|4}} |- style="background: palegreen;" | |0 |2 |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° | rowspan="3" |[[File:Regular_polygon_30.svg|100px]]<br>{30/1} | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px]]<br>{30/14} |164.5~° | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} |{{radic|3.927~}} |- style="background: palegreen;" | |0.270~ |1.982~ |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px]]<br>{30/2}=2{15} | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px]]<br>{30/13} |154.8~° | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} |{{radic|3.809~}} |- style="background: gainsboro;" | |0.437~ |1.952~ |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px]]<br>{30/3}=3{10} | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px]]<br>{30/12}=6{5/2} |144° | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: yellow;" | |0.618~ |1.902~ |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |138.6~° | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} |{{radic|3.5}} |- style="background: gainsboro;" | |0.707~ |1.871~ |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px]]<br>{30/4}=2{15/2} | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px]]<br>{30/11} |135.5~° | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} |{{radic|3.427~}} |- style="background: palegreen;" | |0.757~ |1.851~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |130.9~° | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} |{{radic|3.309~}} |- style="background: gainsboro;" | |0.831~ |1.819~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° | rowspan="3" | | rowspan="3" | | rowspan="3" | |124° | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} |{{radic|3.118~}} |- style="background: gainsboro;" | |0.939~ |1.766~ |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px]]<br>{30/5}=5{6} | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px]]<br>{30/10}=10{3} |120° | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |113.9~° | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} |{{radic|2.809~}} |- style="background: gainsboro;" | |1.091~ |1.676~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |110.2~° | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} |{{radic|2.691~}} |- style="background: gainsboro;" | |1.144~ |1.640~ |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px]]<br>{30/6}=6{5} | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px]]<br>{30/9}=3{10/3} |108° | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} |{{radic|2.618~}} |- style="background: yellow;" | |1.176~ |1.618~ |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px]]<br>{30/8}=2{15/4} |104.5~° | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} |{{radic|2.5}} |- style="background: palegreen;" | |1.224~ |1.581~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |98.9~° | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} |{{radic|2.309~}} |- style="background: gainsboro;" | |1.300~ |1.520~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |95.5~° | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} |{{radic|2.191~}} |- style="background: gainsboro;" | |1.345~ |1.480~ |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} |90° | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |} The [[User:Dc.samizdat/Golden chords of the 120-cell#Thirty distinguished distances|list of thirty 120-cell chords]] <math>c_{t}</math> can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct [[w:120-cell#Concentric_hulls|polyhedral section of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance <math>c_1</math> is a tetrahedron vertex figure, and the largest section at radial distance <math>c_{15}</math> is a central section bisecting the 120-cell. Because [[w:3-sphere|<math>\mathbb{S}^3</math>]] is spherical, at radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance <math>c_{29}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. Only 8 of the 30 chords in the table occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. Since all those polytopes except the 5-cell occur in the 600-cell, and the 600-cell and the 120-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. ... {{Clear}} == Finally the 120-cell == The [[120-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{5,3,3\}</math></small>. It has 600 vertices, 1200 edges, 720 pentagon faces, and 120 dodecahedron cells. It is the four-dimensional analogue of the dodecahedron. The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. ... {{Clear}} == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.] The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} m81yi3uyxjvno2jonugzlt9mc1e1pk0 2818491 2818489 2026-07-18T21:14:14Z Dc.samizdat 2856930 /* The 24-cell */ 2818491 wikitext text/x-wiki = Golden chords of the 120-cell = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - June 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by 5² non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). Consequently 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which contains 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). Consequently the 8-point (16-cell) compounds by 3 times 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} == The 16-cell 4-orthoplex == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=\sqrt{2}+1</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. Their 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}+1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-1</math> . Over the third <math>r_3</math> chord it also moves a distance of <math>-1</math>. Fontaine and Hurley also demonstrated the significance of <math>1/r_i</math> in Steinbach's Diagonal Product Formula, which says that every chord length is the sum of certain smaller chord lengths. The smaller chords are certain diagonals of the same regular polygon of a smaller edge length, specifically edge length <math>1/r_i</math> rather than <math>1</math>. If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{3,3,4\}</math></small>. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <math>\sqrt{2}</math> edges except opposite pairs. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Each chord is a distinct 4-vector with a length and a direction. 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, where vertices circle over the chords of an <math>r_i</math> polygon. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. The angle between two <math>r_i</math> chords is <math>180^\circ - i \times 45^\circ</math>. [[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 {8/1} 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 the 16-cell form an ''edge polygon'' {8/2}=2{4}. The 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 <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> (it is <math>6\pi</math> in this case), and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> chords, each vertex makes twelve 45° 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 revolution 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. We shall refer to this isoclinic rotation as the ''great square rotation characteristic of the 16-cell'', and note once again that it is Fontaine and Hurley's counterclockwise rotation over the <math>r_3</math> {8/3} star polygon, which constructs <math>1/r_3</math>. == The 8-cell tesseract == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. [[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract <small><math>\{4,3,3\}</math></small> performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]] The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{4,3,3\}</math></small>. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedrons]]. 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 the great square rotation characteristic of the 16-cell, with the same effect on both alternate-position 16-cells. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords. They form a circular double helix which intersects each vertex of the tesseract once. The double helix is an 8-rung ladder twisted around 3 times, and bent into a circle in the fourth dimension with its ends joined. Each rung is a <math>\sqrt{3}</math> chord. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron. The 24-cell has the same chord set as the 4-hypercube tesseract: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]] The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters. The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords: :<math>r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The <math>r_1</math> and <math>r_5</math> chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. The <math>r_1</math> chords of the 24-cell form a Petrie polygon {12/1} which zig-zags back and forth, in the left and right rotational directions, between two Clifford parallel great hexagons formed by <math>r_2</math> chords. The <math>r_3</math> chords of the 24-cell form a compound great square edge polygon {24/6}=6{4}. The six great squares lie Clifford parallel to each other. They include one pair of completely orthogonal great squares from each of the three 16-cells. The <math>r_4</math> chords of the 24-cell form a compound great hexagon edge polygon {24/4}=4{6}. The four great hexagons lie Clifford parallel to each other. A ''simple'' rotation of the 24-cell in ''one'' of those hexagon central planes rotates that hexagon like a wheel, while the completely orthogonal invariant central plane does not move. The six vertices of the rotating hexagon orbit on a great circle in the plane. The <math>r_5</math> chords of the 16-cell form a circular helix isocline, visible as a green {12/5} dodecagram Clifford polygon in the orthogonal projection. An isoclinic rotation of the 24-cell, in two Clifford parallel invariant <math>r_2</math> hexagon planes at once by equal angles, moves the twelve vertices along the circular helix over <math>r_5</math> chords. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges of Clifford parallel great squares, also isocline chords in great square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in great hexagon rotations. The green {12/5} dodecagram is a Clifford polygon.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} shows three octagram isoclines of <small><math>\sqrt{2}</math> </small>chords in the 24-cell]] We can rotate the 24-cell isoclinically in 6 Clifford parallel invariant great square planes containing four <math>r_{3}</math> 16-cell edges each, in the great square rotation characteristic of the 16-cell, with the same effect on all three 16-cells. In 720° each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The rotational curve over each 90° <small><math>\sqrt{2}</math></small> chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <small><math>\sqrt{2}</math></small> chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. The triple helix is an 8-step circular staircase that twists around 3 times, and is bent into a torus in the fourth dimension. Each staircase step is a great triangle of <small><math>\sqrt{3}</math></small> chords. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} shows 2 dodecagram isoclines of <small><math>\sqrt{3}</math></small> chords in the 24-cell]]We can rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing six <math>r_{2}</math> 24-cell edges each, over <math>r_{5}</math> isocline chords. This is the ''great hexagon rotation characteristic of the 24-cell'', also Fontaine and Hurley's counterclockwise rotation over the <math>r_5</math> {12/5} star polygon which constructs <math>1/r_5</math>. A complete 24-cell great circle edge plane revolution requires 720° like a complete 16-cell great circle edge plane revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. An isoclinic rotation by 60° in an invariant central plane containing a 24-cell edge takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all 4 invariant great hexagon planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square. All 24 vertices move at once on Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 12 <math>r_5</math> <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane 5 times in a moving invariant rotation plane. The rotational curve over each <math>r_5</math> 120° chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the course of a 720° revolution each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="6" |6 distinct 180° chord pairs make 6 distinct isoclinic rotations |- ! colspan="3" |Edge chord ! colspan="3" |Isocline chord |- style="background: gainsboro;" | | rowspan="4" |<math>t_1</math> |60° | rowspan="4" |[[File:Regular_polygon_24.svg|100px]]<br>{24/1}={24} | rowspan="4" |[[File:Regular_star_polygon_24-11.svg|100px]]<br>{24/11} |120° | rowspan="4" |<math>t_{11}</math> |- style="background: gainsboro;" | |{{radic|1}} |{{radic|3}} |- style="background: gainsboro;" | |1 |1.732~ |- style="background: gainsboro;" | |15° |165° |- style="background: palegreen;" | | rowspan="4" |<math>t_2</math> |60° | rowspan="4" |[[File:Regular_star_figure_2(12,1).svg|100px]]<br>{24/2}=2{12} | rowspan="4" |[[File:Regular_star_figure_2(12,5).svg|100px]]<br>{24/10}=2{12/5} |120° | rowspan="4" |<math>t_{10}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | |30° |150° |- style="background: seashell;" | | rowspan="4" |<math>t_3</math> |90° | rowspan="4" |[[File:Regular_star_figure_3(8,1).svg|100px]]<br>{24/3}=3{8} | rowspan="4" |[[File:Regular_star_figure_3(8,3).svg|100px]]<br>{24/9}=3{8/3} |90° | rowspan="4" |<math>t_{9}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |- style="background: seashell;" | |45° |135° |- style="background: palegreen;" | | rowspan="4" |<math>t_4</math> |60° | rowspan="4" |[[File:Regular_star_figure_4(6,1).svg|100px]]<br>{24/4}=4{6} | rowspan="4" |[[File:Regular_star_figure_8(3,1).svg|100px]]<br>{24/8}=8{3} |120° | rowspan="4" |<math>t_{8}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | |60° |120° |- style="background: gainsboro;" | | rowspan="4" |<math>t_5</math> |60° | rowspan="4" |[[File:Regular_star_polygon_24-5.svg|100px]]<br>{24/5} | rowspan="4" |[[File:Regular_star_polygon_24-7.svg|100px]]<br>{24/7} |120° | rowspan="4" |<math>t_{7}</math> |- style="background: gainsboro;" | |{{radic|1}} |{{radic|3}} |- style="background: gainsboro;" | |1 |1.732~ |- style="background: gainsboro;" | |75° |105° |- style="background: gainsboro;" | | rowspan="4" |<math>t_6</math> |90° | rowspan="4" |[[File:Regular_star_figure_6(4,1).svg|100px]]<br>{24/6}=6{4} | rowspan="4" |[[File:Regular_star_figure_6(4,1).svg|100px]]<br>{24/6}=6{4} |90° | rowspan="4" |<math>t_{6}</math> |- style="background: gainsboro;" | |{{radic|2}} |{{radic|2}} |- style="background: gainsboro;" | |1.414~ |1.414~ |- style="background: gainsboro;" | |90° |90° |} By examining the chords <math>r_i</math> of the 24-cell's Petrie {12}-gon we have found two distinct isoclinic rotations, the great square rotation characteristic of the 16-cell and the great hexagon rotation characteristic of the 24-cell. If we examine the chords <math>t_i</math> of the 24-cell's {24}-gon we find these, and also four other distinct isoclinic rotations. Each row of the table is a distinct isoclinic rotation of the 24-cell characterized by a pair of chords that sum to 180°. The 24 edge chords form the rotation's edge {24}-gon, and lie in invariant planes of the rotation. The 24 isocline chords form the rotation's Clifford {24}-gon and lie in invariant planes completely orthogonal to the corresponding edge chord's invariant plane. The rotational angle between successive edge chords and the rotational angle between successive isocline chords also sum to 180°. ... {{Clear}} == The 600-cell == [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are: [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</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. {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="7" |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra |- ! colspan="3" |Short edge chord ! Section ! colspan="3" |Long isocline chord |- style="background: palegreen;" | | rowspan="4" |<math>r_0</math> |0° | rowspan="4" | | rowspan="4" | | rowspan="4" |[[File:Regular_star_figure_15(2,1).svg|100px]]<br>{30/15}=15{2} |180° | rowspan="4" |<math>r_{15}</math> |- style="background: palegreen;" | |{{radic|0}} |{{radic|4}} |- style="background: palegreen;" | |0 |2 |- style="background: palegreen;" | |0° |180° |- style="background: palegreen;" | | rowspan="4" |<math>r_1</math> |36° | rowspan="4" |[[File:Regular_polygon_30.svg|100px]]<br>{30/1} | rowspan="4" | | rowspan="4" |[[File:Regular_star_figure_2(15,7).svg|100px]]<br>{30/14}=2{15/7} |144° | rowspan="4" |<math>r_{14}</math> |- style="background: palegreen;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: palegreen;" | |0.618~ |1.902~ |- style="background: palegreen;" | |12° |168° |- style="background: gainsboro;" | | rowspan="4" |<math>r_2</math> |36° | rowspan="4" |[[File:Regular_star_figure_2(15,1).svg|100px]]<br>{30/2}=2{15} | rowspan="4" | | rowspan="4" |[[File:Regular_star_polygon_30-13.svg|100px]]<br>{30/13} |144° | rowspan="4" |<math>r_{13}</math> |- style="background: gainsboro;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: gainsboro;" | |0.618~ |1.902~ |- style="background: gainsboro;" | |24° |156° |- style="background: yellow;" | | rowspan="4" |<math>r_3</math> |36° | rowspan="4" |[[File:Regular_star_figure_3(10,1).svg|100px]]<br>{30/3}=3{10} | rowspan="4" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="4" |[[File:Regular_star_figure_6(5,2).svg|100px]]<br>{30/12}=6{5/2} |144° | rowspan="4" |<math>r_{12}</math> |- style="background: yellow;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: yellow;" | |0.618~ |1.902~ |- style="background: yellow;" | |36° |144° |- style="background: palegreen;" | | rowspan="4" |<math>r_4</math> |60° | rowspan="4" |[[File:Regular_star_figure_2(15,2).svg|100px]]<br>{30/4}=2{15/2} | rowspan="4" | | rowspan="4" |[[File:Regular_star_polygon_30-11.svg|100px]]<br>{30/11} |120° | rowspan="4" |<math>r_{11}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | |48° |132° |- style="background: palegreen;" | | rowspan="4" |<math>r_5</math> |60° | rowspan="4" |[[File:Regular_star_figure_5(6,1).svg|100px]]<br>{30/5}=5{6} | rowspan="4" |[[File:V2 dodecahedron.png|100px]]<br>Dodecahedron | rowspan="4" |[[File:Regular_star_figure_10(3,1).svg|100px]]<br>{30/10}=10{3} |120° | rowspan="4" |<math>r_{10}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | |60° |120° |- style="background: yellow;" | | rowspan="4" |<math>r_{6}</math> |72° | rowspan="4" |[[File:Regular_star_figure_6(5,1).svg|100px]]<br>{30/6}=6{5} | rowspan="4" |[[File:V3 icosahedron.png|100px]]<br>Icosahedron | rowspan="4" |[[File:Regular_star_figure_3(10,3).svg|100px]]<br>{30/9}=3{10/3} |108° | rowspan="4" |<math>r_{9}</math> |- style="background: yellow;" | |{{radic|1.382~}} |{{radic|2.618~}} |- style="background: yellow;" | |1.176~ |1.618~ |- style="background: yellow;" | |72° |108° |- style="background: seashell;" | | rowspan="4" |<math>r_{7}</math> |90° | rowspan="4" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} | rowspan="4" |[[File:V4 icosidodecahedron.png|100px]]<br>Icosidodecahedron | rowspan="4" |[[File:Regular_star_figure_2(15,4).svg|100px]]<br>{30/8}=2{15/4} |90° | rowspan="4" |<math>r_{8}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |- style="background: seashell;" | |84° |96° |} The list of 600-cell chords <math>r_{i}</math> can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. The short chord and long chord each have their characteristic {30/n}-gon. Each row identifies a discrete isoclinic rotation of the 600-cell in invariant central planes containing the edges of the short chord {30}-gon, over the isocline chords of the long chord {30}-gon, the rotation's Clifford polygon. Each distinct pair of complementary chord lengths is identified with a distinct [[w:600-cell#Polyhedral sections|polyhedral section of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance <math>\phi^{-1}</math> is a icosahedron vertex figure, and the largest section at radial distance <math>\sqrt{2}</math> is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. Because [[w:3-sphere|<math>\mathbb{S}^3</math>]] is spherical, at radial distances greater than <math>\sqrt{2}</math> the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance <math>\sqrt{2+\phi}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>\sqrt{2}</math></small>]] We can rotate the 600-cell isoclinically in the great square rotation characteristic of the 16-cell, with the same effect on 15 disjoint 16-cells. Each 90° displacement takes 15 pairs of completely orthogonal invariant great square planes to each other. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The rotational curve over each 90° chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> form a circular fibration of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once. The 600-cell has another distinct isoclinic rotation in invariant great square planes, which has period 30 and visits every vertex of a 600-cell Petrie polygon. This [''great square left rotation characteristic of the 600-cell]'' takes place over <math>r_7</math> edge chords and <math>r_8</math> isocline chords. The {30/7} edge polygon is a skew helix of circumference <math>14\pi</math> with each <math>r_7</math> edge belonging to a distinct great square. The four {30/7} polygrams contribute one edge each to 30 great squares. Each 90° displacement takes every 16-cell to another 16-cell. The vertices of the invariant great squares each make seven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 90° {30/7} edge makes seven 12° turns. Four Clifford parallel {30/7} geodesics of circumference <math>16\pi</math> form a circular quadruple helix that intersects each 600-cell vertex once. The {30/8}=2{15/4} Clifford polygon is a compound of two skew {15/4} pentadecagrams of circumference <math>16\pi</math> with each <math>r_8</math> isocline chord belonging to a distinct 16-cell. The four {30/8} polygrams contribute one edge each to 30 great squares. The rotational curve over each 90° {30/8} isocline chord makes eight 12° turns. Four Clifford parallel {30/8} geodesics of circumference <math>16\pi</math> form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>\sqrt{3}</math></small> ]] We can rotate the 600-cell isoclinically in the great hexagon rotation characteristic of the 24-cell, over <math>\sqrt{1}</math> edge chords and <math>\sqrt{3}</math> isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° revolution each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> form a circular fibration of ten twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. The 600-cell has another distinct isoclinic rotation in invariant great hexagon planes, over <math>r_{4}=\sqrt{1}</math> edge chords and <math>r_{11}=\sqrt{3}</math> isocline chords This [''invariant great hexagon left rotation characteristic of the 600-cell]'' has period 30 and visits every vertex of a 600-cell Petrie polygon. Its {30/11} Clifford polygon is a skew helix where each <math>r_{11}</math> isocline chord is the <math>\sqrt{3}</math> diagonal of a great hexagon of a distinct 24-cell. The vertices of the invariant great hexagons of this rotation each make eleven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 120° <math>r_{11}</math> isocline chord makes eleven 12° turns. Four Clifford parallel {30/11} geodesic isoclines of circumference <math>22\pi</math> over <math>r_{11}</math> chords form a circular quadruple helix that intersects each 600-cell vertex once. We can rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its 36° <math>r_{3}</math> edges, over 144° <math>r_{12}</math> isocline chords. This ''invariant great decagon rotation characteristic of the 600-cell'' has period 5 and takes disjoint 24-cells to each other. The rotational curve over each <math>r_{12}</math> chord of its {5/2} Clifford polygon makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference <math>4\pi</math> over five <math>r_{12}</math> chords form a circular fibration of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. The rotation of the 600-cell 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. The 24-cells revolve within the 600-cell, as the 16-cells revolve within the 24-cells. All 120 vertices move at once on four Clifford parallel geodesic isoclines, displaced 144° in different directions. The 600-cell has another distinct isoclinic rotation in invariant great decagon planes containing its 36° <math>r_{2}</math> edges, over 144° <math>r_{13}</math> isocline chords. This [''great decagon left rotation characteristic of the 600-cell]'' has period 30 and visits every vertex of a 600-cell Petrie polygon. The rotational curve over each 144° <math>r_{13}</math> isocline chord makes thirteen 12° turns. Four Clifford parallel {30/13} geodesic isoclines of circumference <math>26\pi</math> form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} == The 5-cell 4-simplex == {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="9" |30 chords (15 180° pairs) make 15 distinct section polyhedra |- ! colspan="3" |Short chord ! Section ! colspan="3" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px]]<br>{30/15}=15{2} |180° | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} |{{radic|4}} |- style="background: palegreen;" | |0 |2 |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° | rowspan="3" |[[File:Regular_polygon_30.svg|100px]]<br>{30/1} | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px]]<br>{30/14} |164.5~° | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} |{{radic|3.927~}} |- style="background: palegreen;" | |0.270~ |1.982~ |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px]]<br>{30/2}=2{15} | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px]]<br>{30/13} |154.8~° | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} |{{radic|3.809~}} |- style="background: gainsboro;" | |0.437~ |1.952~ |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px]]<br>{30/3}=3{10} | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px]]<br>{30/12}=6{5/2} |144° | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: yellow;" | |0.618~ |1.902~ |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |138.6~° | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} |{{radic|3.5}} |- style="background: gainsboro;" | |0.707~ |1.871~ |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px]]<br>{30/4}=2{15/2} | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px]]<br>{30/11} |135.5~° | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} |{{radic|3.427~}} |- style="background: palegreen;" | |0.757~ |1.851~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |130.9~° | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} |{{radic|3.309~}} |- style="background: gainsboro;" | |0.831~ |1.819~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° | rowspan="3" | | rowspan="3" | | rowspan="3" | |124° | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} |{{radic|3.118~}} |- style="background: gainsboro;" | |0.939~ |1.766~ |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px]]<br>{30/5}=5{6} | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px]]<br>{30/10}=10{3} |120° | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |113.9~° | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} |{{radic|2.809~}} |- style="background: gainsboro;" | |1.091~ |1.676~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |110.2~° | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} |{{radic|2.691~}} |- style="background: gainsboro;" | |1.144~ |1.640~ |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px]]<br>{30/6}=6{5} | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px]]<br>{30/9}=3{10/3} |108° | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} |{{radic|2.618~}} |- style="background: yellow;" | |1.176~ |1.618~ |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px]]<br>{30/8}=2{15/4} |104.5~° | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} |{{radic|2.5}} |- style="background: palegreen;" | |1.224~ |1.581~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |98.9~° | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} |{{radic|2.309~}} |- style="background: gainsboro;" | |1.300~ |1.520~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |95.5~° | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} |{{radic|2.191~}} |- style="background: gainsboro;" | |1.345~ |1.480~ |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} |90° | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |} The [[User:Dc.samizdat/Golden chords of the 120-cell#Thirty distinguished distances|list of thirty 120-cell chords]] <math>c_{t}</math> can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct [[w:120-cell#Concentric_hulls|polyhedral section of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance <math>c_1</math> is a tetrahedron vertex figure, and the largest section at radial distance <math>c_{15}</math> is a central section bisecting the 120-cell. Because [[w:3-sphere|<math>\mathbb{S}^3</math>]] is spherical, at radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance <math>c_{29}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. Only 8 of the 30 chords in the table occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. Since all those polytopes except the 5-cell occur in the 600-cell, and the 600-cell and the 120-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. ... {{Clear}} == Finally the 120-cell == The [[120-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{5,3,3\}</math></small>. It has 600 vertices, 1200 edges, 720 pentagon faces, and 120 dodecahedron cells. It is the four-dimensional analogue of the dodecahedron. The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. ... {{Clear}} == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.] The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} s0ccspkn9wdscgqbcoij5xuxfk4d8ix 2818496 2818491 2026-07-18T23:24:47Z Dc.samizdat 2856930 /* The 24-cell */ 2818496 wikitext text/x-wiki = Golden chords of the 120-cell = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - June 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by 5² non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). Consequently 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which contains 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). Consequently the 8-point (16-cell) compounds by 3 times 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} == The 16-cell 4-orthoplex == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=\sqrt{2}+1</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. Their 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}+1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-1</math> . Over the third <math>r_3</math> chord it also moves a distance of <math>-1</math>. Fontaine and Hurley also demonstrated the significance of <math>1/r_i</math> in Steinbach's Diagonal Product Formula, which says that every chord length is the sum of certain smaller chord lengths. The smaller chords are certain diagonals of the same regular polygon of a smaller edge length, specifically edge length <math>1/r_i</math> rather than <math>1</math>. If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{3,3,4\}</math></small>. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <math>\sqrt{2}</math> edges except opposite pairs. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Each chord is a distinct 4-vector with a length and a direction. 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, where vertices circle over the chords of an <math>r_i</math> polygon. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. The angle between two <math>r_i</math> chords is <math>180^\circ - i \times 45^\circ</math>. [[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 {8/1} 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 the 16-cell form an ''edge polygon'' {8/2}=2{4}. The 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 <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> (it is <math>6\pi</math> in this case), and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> chords, each vertex makes twelve 45° 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 revolution 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. We shall refer to this isoclinic rotation as the ''great square rotation characteristic of the 16-cell'', and note once again that it is Fontaine and Hurley's counterclockwise rotation over the <math>r_3</math> {8/3} star polygon, which constructs <math>1/r_3</math>. == The 8-cell tesseract == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. [[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract <small><math>\{4,3,3\}</math></small> performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]] The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{4,3,3\}</math></small>. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedrons]]. 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 the great square rotation characteristic of the 16-cell, with the same effect on both alternate-position 16-cells. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords. They form a circular double helix which intersects each vertex of the tesseract once. The double helix is an 8-rung ladder twisted around 3 times, and bent into a circle in the fourth dimension with its ends joined. Each rung is a <math>\sqrt{3}</math> chord. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron. The 24-cell has the same chord set as the 4-hypercube tesseract: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]] The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters. The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords: :<math>r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The <math>r_1</math> and <math>r_5</math> chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. The <math>r_1</math> <small><math>\sqrt{1}</math></small> chords form a Petrie dodecagon {12/1} which zig-zags back and forth, in the left and right rotational directions, between two Clifford parallel great hexagons formed by <math>r_2</math> chords. The <math>r_2</math> <small><math>\sqrt{1}</math></small> chords form 16 great hexagons. The 24-cell can be sliced four different ways into 4 disjoint great hexagons {24/4}=4{6} that lie Clifford parallel to each other. The <math>r_3</math> <small><math>\sqrt{2}</math></small> chords form a compound great square edge polygon {24/6}=6{4}. The six great squares lie Clifford parallel to each other. They include one pair of completely orthogonal great squares from each of the three 16-cells. The <math>r_4</math> <small><math>\sqrt{3}</math></small> chords form 32 great triangles inscribed in the 16 great hexagons. Twelve <math>r_5</math> <small><math>\sqrt{3}</math></small> chords of the 24-cell form a circular helix isocline, visible as a green {12/5} dodecagram in the orthogonal projection. An isoclinic rotation of the 24-cell in invariant <math>r_2</math> hexagon planes moves the vertices along 2 parallel circular helixes {24/2}=2{12/5} over the <math>r_5</math> chords. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges of Clifford parallel great squares, also isocline chords in great square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in great hexagon rotations. The green {12/5} dodecagram is a Clifford polygon.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} shows three octagram isoclines of <small><math>\sqrt{2}</math> </small>chords in the 24-cell]] We can rotate the 24-cell isoclinically in 6 Clifford parallel invariant great square planes containing four <math>r_{3}</math> 16-cell edges each, in the great square rotation characteristic of the 16-cell, with the same effect on all three 16-cells. In 720° each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The rotational curve over each 90° <small><math>\sqrt{2}</math></small> chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <small><math>\sqrt{2}</math></small> chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. The triple helix is an 8-step circular staircase that twists around 3 times, and is bent into a torus in the fourth dimension. Each staircase step is a great triangle of <small><math>\sqrt{3}</math></small> chords. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} shows 2 dodecagram isoclines of <small><math>\sqrt{3}</math></small> chords in the 24-cell]]We can rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing six <math>r_{2}</math> 24-cell edges each, over <math>r_{5}</math> isocline chords. This is the ''great hexagon rotation characteristic of the 24-cell'', also Fontaine and Hurley's counterclockwise rotation over the <math>r_5</math> {12/5} star polygon which constructs <math>1/r_5</math>. A complete 24-cell great circle edge plane revolution requires 720° like a complete 16-cell great circle edge plane revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. An isoclinic rotation by 60° in an invariant central plane containing a 24-cell edge takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all 4 invariant great hexagon planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square. All 24 vertices move at once on Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 12 <math>r_5</math> <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane 5 times in a moving invariant rotation plane. The rotational curve over each <math>r_5</math> 120° chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the course of a 720° revolution each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="6" |6 distinct 180° chord pairs make 6 distinct isoclinic rotations |- ! colspan="3" |Edge chord ! colspan="3" |Isocline chord |- style="background: gainsboro;" | | rowspan="4" |<math>t_1</math> |60° | rowspan="4" |[[File:Regular_polygon_24.svg|100px]]<br>{24/1}={24} | rowspan="4" |[[File:Regular_star_polygon_24-11.svg|100px]]<br>{24/11} |120° | rowspan="4" |<math>t_{11}</math> |- style="background: gainsboro;" | |{{radic|1}} |{{radic|3}} |- style="background: gainsboro;" | |1 |1.732~ |- style="background: gainsboro;" | |15° |165° |- style="background: palegreen;" | | rowspan="4" |<math>t_2</math> |60° | rowspan="4" |[[File:Regular_star_figure_2(12,1).svg|100px]]<br>{24/2}=2{12} | rowspan="4" |[[File:Regular_star_figure_2(12,5).svg|100px]]<br>{24/10}=2{12/5} |120° | rowspan="4" |<math>t_{10}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | |30° |150° |- style="background: seashell;" | | rowspan="4" |<math>t_3</math> |90° | rowspan="4" |[[File:Regular_star_figure_3(8,1).svg|100px]]<br>{24/3}=3{8} | rowspan="4" |[[File:Regular_star_figure_3(8,3).svg|100px]]<br>{24/9}=3{8/3} |90° | rowspan="4" |<math>t_{9}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |- style="background: seashell;" | |45° |135° |- style="background: palegreen;" | | rowspan="4" |<math>t_4</math> |60° | rowspan="4" |[[File:Regular_star_figure_4(6,1).svg|100px]]<br>{24/4}=4{6} | rowspan="4" |[[File:Regular_star_figure_8(3,1).svg|100px]]<br>{24/8}=8{3} |120° | rowspan="4" |<math>t_{8}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | |60° |120° |- style="background: gainsboro;" | | rowspan="4" |<math>t_5</math> |60° | rowspan="4" |[[File:Regular_star_polygon_24-5.svg|100px]]<br>{24/5} | rowspan="4" |[[File:Regular_star_polygon_24-7.svg|100px]]<br>{24/7} |120° | rowspan="4" |<math>t_{7}</math> |- style="background: gainsboro;" | |{{radic|1}} |{{radic|3}} |- style="background: gainsboro;" | |1 |1.732~ |- style="background: gainsboro;" | |75° |105° |- style="background: gainsboro;" | | rowspan="4" |<math>t_6</math> |90° | rowspan="4" |[[File:Regular_star_figure_6(4,1).svg|100px]]<br>{24/6}=6{4} | rowspan="4" |[[File:Regular_star_figure_6(4,1).svg|100px]]<br>{24/6}=6{4} |90° | rowspan="4" |<math>t_{6}</math> |- style="background: gainsboro;" | |{{radic|2}} |{{radic|2}} |- style="background: gainsboro;" | |1.414~ |1.414~ |- style="background: gainsboro;" | |90° |90° |} By examining the chords <math>r_i</math> of the 24-cell's Petrie {12}-gon we have found two distinct isoclinic rotations, the great square rotation characteristic of the 16-cell and the great hexagon rotation characteristic of the 24-cell. If we examine the chords <math>t_i</math> of the 24-cell's {24}-gon we find these, and also four other distinct isoclinic rotations. Each row of the table is a distinct isoclinic rotation of the 24-cell characterized by a pair of chords that sum to 180°. The 24 edge chords form the rotation's edge {24}-gon, and lie in invariant planes of the rotation. The 24 isocline chords form the rotation's Clifford {24}-gon and lie in invariant planes completely orthogonal to the corresponding edge chord's invariant plane. The rotational angle between successive edge chords and the rotational angle between successive isocline chords also sum to 180°. ... {{Clear}} == The 600-cell == [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are: [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</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. {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="7" |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra |- ! colspan="3" |Short edge chord ! Section ! colspan="3" |Long isocline chord |- style="background: palegreen;" | | rowspan="4" |<math>r_0</math> |0° | rowspan="4" | | rowspan="4" | | rowspan="4" |[[File:Regular_star_figure_15(2,1).svg|100px]]<br>{30/15}=15{2} |180° | rowspan="4" |<math>r_{15}</math> |- style="background: palegreen;" | |{{radic|0}} |{{radic|4}} |- style="background: palegreen;" | |0 |2 |- style="background: palegreen;" | |0° |180° |- style="background: palegreen;" | | rowspan="4" |<math>r_1</math> |36° | rowspan="4" |[[File:Regular_polygon_30.svg|100px]]<br>{30/1} | rowspan="4" | | rowspan="4" |[[File:Regular_star_figure_2(15,7).svg|100px]]<br>{30/14}=2{15/7} |144° | rowspan="4" |<math>r_{14}</math> |- style="background: palegreen;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: palegreen;" | |0.618~ |1.902~ |- style="background: palegreen;" | |12° |168° |- style="background: gainsboro;" | | rowspan="4" |<math>r_2</math> |36° | rowspan="4" |[[File:Regular_star_figure_2(15,1).svg|100px]]<br>{30/2}=2{15} | rowspan="4" | | rowspan="4" |[[File:Regular_star_polygon_30-13.svg|100px]]<br>{30/13} |144° | rowspan="4" |<math>r_{13}</math> |- style="background: gainsboro;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: gainsboro;" | |0.618~ |1.902~ |- style="background: gainsboro;" | |24° |156° |- style="background: yellow;" | | rowspan="4" |<math>r_3</math> |36° | rowspan="4" |[[File:Regular_star_figure_3(10,1).svg|100px]]<br>{30/3}=3{10} | rowspan="4" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="4" |[[File:Regular_star_figure_6(5,2).svg|100px]]<br>{30/12}=6{5/2} |144° | rowspan="4" |<math>r_{12}</math> |- style="background: yellow;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: yellow;" | |0.618~ |1.902~ |- style="background: yellow;" | |36° |144° |- style="background: palegreen;" | | rowspan="4" |<math>r_4</math> |60° | rowspan="4" |[[File:Regular_star_figure_2(15,2).svg|100px]]<br>{30/4}=2{15/2} | rowspan="4" | | rowspan="4" |[[File:Regular_star_polygon_30-11.svg|100px]]<br>{30/11} |120° | rowspan="4" |<math>r_{11}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | |48° |132° |- style="background: palegreen;" | | rowspan="4" |<math>r_5</math> |60° | rowspan="4" |[[File:Regular_star_figure_5(6,1).svg|100px]]<br>{30/5}=5{6} | rowspan="4" |[[File:V2 dodecahedron.png|100px]]<br>Dodecahedron | rowspan="4" |[[File:Regular_star_figure_10(3,1).svg|100px]]<br>{30/10}=10{3} |120° | rowspan="4" |<math>r_{10}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | |60° |120° |- style="background: yellow;" | | rowspan="4" |<math>r_{6}</math> |72° | rowspan="4" |[[File:Regular_star_figure_6(5,1).svg|100px]]<br>{30/6}=6{5} | rowspan="4" |[[File:V3 icosahedron.png|100px]]<br>Icosahedron | rowspan="4" |[[File:Regular_star_figure_3(10,3).svg|100px]]<br>{30/9}=3{10/3} |108° | rowspan="4" |<math>r_{9}</math> |- style="background: yellow;" | |{{radic|1.382~}} |{{radic|2.618~}} |- style="background: yellow;" | |1.176~ |1.618~ |- style="background: yellow;" | |72° |108° |- style="background: seashell;" | | rowspan="4" |<math>r_{7}</math> |90° | rowspan="4" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} | rowspan="4" |[[File:V4 icosidodecahedron.png|100px]]<br>Icosidodecahedron | rowspan="4" |[[File:Regular_star_figure_2(15,4).svg|100px]]<br>{30/8}=2{15/4} |90° | rowspan="4" |<math>r_{8}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |- style="background: seashell;" | |84° |96° |} The list of 600-cell chords <math>r_{i}</math> can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. The short chord and long chord each have their characteristic {30/n}-gon. Each row identifies a discrete isoclinic rotation of the 600-cell in invariant central planes containing the edges of the short chord {30}-gon, over the isocline chords of the long chord {30}-gon, the rotation's Clifford polygon. Each distinct pair of complementary chord lengths is identified with a distinct [[w:600-cell#Polyhedral sections|polyhedral section of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance <math>\phi^{-1}</math> is a icosahedron vertex figure, and the largest section at radial distance <math>\sqrt{2}</math> is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. Because [[w:3-sphere|<math>\mathbb{S}^3</math>]] is spherical, at radial distances greater than <math>\sqrt{2}</math> the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance <math>\sqrt{2+\phi}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>\sqrt{2}</math></small>]] We can rotate the 600-cell isoclinically in the great square rotation characteristic of the 16-cell, with the same effect on 15 disjoint 16-cells. Each 90° displacement takes 15 pairs of completely orthogonal invariant great square planes to each other. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The rotational curve over each 90° chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> form a circular fibration of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once. The 600-cell has another distinct isoclinic rotation in invariant great square planes, which has period 30 and visits every vertex of a 600-cell Petrie polygon. This [''great square left rotation characteristic of the 600-cell]'' takes place over <math>r_7</math> edge chords and <math>r_8</math> isocline chords. The {30/7} edge polygon is a skew helix of circumference <math>14\pi</math> with each <math>r_7</math> edge belonging to a distinct great square. The four {30/7} polygrams contribute one edge each to 30 great squares. Each 90° displacement takes every 16-cell to another 16-cell. The vertices of the invariant great squares each make seven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 90° {30/7} edge makes seven 12° turns. Four Clifford parallel {30/7} geodesics of circumference <math>16\pi</math> form a circular quadruple helix that intersects each 600-cell vertex once. The {30/8}=2{15/4} Clifford polygon is a compound of two skew {15/4} pentadecagrams of circumference <math>16\pi</math> with each <math>r_8</math> isocline chord belonging to a distinct 16-cell. The four {30/8} polygrams contribute one edge each to 30 great squares. The rotational curve over each 90° {30/8} isocline chord makes eight 12° turns. Four Clifford parallel {30/8} geodesics of circumference <math>16\pi</math> form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>\sqrt{3}</math></small> ]] We can rotate the 600-cell isoclinically in the great hexagon rotation characteristic of the 24-cell, over <math>\sqrt{1}</math> edge chords and <math>\sqrt{3}</math> isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° revolution each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> form a circular fibration of ten twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. The 600-cell has another distinct isoclinic rotation in invariant great hexagon planes, over <math>r_{4}=\sqrt{1}</math> edge chords and <math>r_{11}=\sqrt{3}</math> isocline chords This [''invariant great hexagon left rotation characteristic of the 600-cell]'' has period 30 and visits every vertex of a 600-cell Petrie polygon. Its {30/11} Clifford polygon is a skew helix where each <math>r_{11}</math> isocline chord is the <math>\sqrt{3}</math> diagonal of a great hexagon of a distinct 24-cell. The vertices of the invariant great hexagons of this rotation each make eleven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 120° <math>r_{11}</math> isocline chord makes eleven 12° turns. Four Clifford parallel {30/11} geodesic isoclines of circumference <math>22\pi</math> over <math>r_{11}</math> chords form a circular quadruple helix that intersects each 600-cell vertex once. We can rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its 36° <math>r_{3}</math> edges, over 144° <math>r_{12}</math> isocline chords. This ''invariant great decagon rotation characteristic of the 600-cell'' has period 5 and takes disjoint 24-cells to each other. The rotational curve over each <math>r_{12}</math> chord of its {5/2} Clifford polygon makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference <math>4\pi</math> over five <math>r_{12}</math> chords form a circular fibration of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. The rotation of the 600-cell 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. The 24-cells revolve within the 600-cell, as the 16-cells revolve within the 24-cells. All 120 vertices move at once on four Clifford parallel geodesic isoclines, displaced 144° in different directions. The 600-cell has another distinct isoclinic rotation in invariant great decagon planes containing its 36° <math>r_{2}</math> edges, over 144° <math>r_{13}</math> isocline chords. This [''great decagon left rotation characteristic of the 600-cell]'' has period 30 and visits every vertex of a 600-cell Petrie polygon. The rotational curve over each 144° <math>r_{13}</math> isocline chord makes thirteen 12° turns. Four Clifford parallel {30/13} geodesic isoclines of circumference <math>26\pi</math> form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} == The 5-cell 4-simplex == {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="9" |30 chords (15 180° pairs) make 15 distinct section polyhedra |- ! colspan="3" |Short chord ! Section ! colspan="3" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px]]<br>{30/15}=15{2} |180° | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} |{{radic|4}} |- style="background: palegreen;" | |0 |2 |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° | rowspan="3" |[[File:Regular_polygon_30.svg|100px]]<br>{30/1} | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px]]<br>{30/14} |164.5~° | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} |{{radic|3.927~}} |- style="background: palegreen;" | |0.270~ |1.982~ |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px]]<br>{30/2}=2{15} | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px]]<br>{30/13} |154.8~° | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} |{{radic|3.809~}} |- style="background: gainsboro;" | |0.437~ |1.952~ |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px]]<br>{30/3}=3{10} | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px]]<br>{30/12}=6{5/2} |144° | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: yellow;" | |0.618~ |1.902~ |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |138.6~° | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} |{{radic|3.5}} |- style="background: gainsboro;" | |0.707~ |1.871~ |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px]]<br>{30/4}=2{15/2} | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px]]<br>{30/11} |135.5~° | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} |{{radic|3.427~}} |- style="background: palegreen;" | |0.757~ |1.851~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |130.9~° | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} |{{radic|3.309~}} |- style="background: gainsboro;" | |0.831~ |1.819~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° | rowspan="3" | | rowspan="3" | | rowspan="3" | |124° | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} |{{radic|3.118~}} |- style="background: gainsboro;" | |0.939~ |1.766~ |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px]]<br>{30/5}=5{6} | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px]]<br>{30/10}=10{3} |120° | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |113.9~° | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} |{{radic|2.809~}} |- style="background: gainsboro;" | |1.091~ |1.676~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |110.2~° | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} |{{radic|2.691~}} |- style="background: gainsboro;" | |1.144~ |1.640~ |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px]]<br>{30/6}=6{5} | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px]]<br>{30/9}=3{10/3} |108° | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} |{{radic|2.618~}} |- style="background: yellow;" | |1.176~ |1.618~ |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px]]<br>{30/8}=2{15/4} |104.5~° | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} |{{radic|2.5}} |- style="background: palegreen;" | |1.224~ |1.581~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |98.9~° | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} |{{radic|2.309~}} |- style="background: gainsboro;" | |1.300~ |1.520~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |95.5~° | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} |{{radic|2.191~}} |- style="background: gainsboro;" | |1.345~ |1.480~ |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} |90° | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |} The [[User:Dc.samizdat/Golden chords of the 120-cell#Thirty distinguished distances|list of thirty 120-cell chords]] <math>c_{t}</math> can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct [[w:120-cell#Concentric_hulls|polyhedral section of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance <math>c_1</math> is a tetrahedron vertex figure, and the largest section at radial distance <math>c_{15}</math> is a central section bisecting the 120-cell. Because [[w:3-sphere|<math>\mathbb{S}^3</math>]] is spherical, at radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance <math>c_{29}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. Only 8 of the 30 chords in the table occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. Since all those polytopes except the 5-cell occur in the 600-cell, and the 600-cell and the 120-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. ... {{Clear}} == Finally the 120-cell == The [[120-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{5,3,3\}</math></small>. It has 600 vertices, 1200 edges, 720 pentagon faces, and 120 dodecahedron cells. It is the four-dimensional analogue of the dodecahedron. The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. ... {{Clear}} == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.] The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} og3zsvbxpjhbnv4uewm8k05whih2f1e 2818497 2818496 2026-07-19T00:07:01Z Dc.samizdat 2856930 /* The 24-cell */ 2818497 wikitext text/x-wiki = Golden chords of the 120-cell = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - June 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by 5² non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). Consequently 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which contains 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). Consequently the 8-point (16-cell) compounds by 3 times 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} == The 16-cell 4-orthoplex == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=\sqrt{2}+1</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. Their 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}+1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-1</math> . Over the third <math>r_3</math> chord it also moves a distance of <math>-1</math>. Fontaine and Hurley also demonstrated the significance of <math>1/r_i</math> in Steinbach's Diagonal Product Formula, which says that every chord length is the sum of certain smaller chord lengths. The smaller chords are certain diagonals of the same regular polygon of a smaller edge length, specifically edge length <math>1/r_i</math> rather than <math>1</math>. If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{3,3,4\}</math></small>. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <math>\sqrt{2}</math> edges except opposite pairs. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Each chord is a distinct 4-vector with a length and a direction. 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, where vertices circle over the chords of an <math>r_i</math> polygon. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. The angle between two <math>r_i</math> chords is <math>180^\circ - i \times 45^\circ</math>. [[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 {8/1} 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 the 16-cell form an ''edge polygon'' {8/2}=2{4}. The 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 <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> (it is <math>6\pi</math> in this case), and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> chords, each vertex makes twelve 45° 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 revolution 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. We shall refer to this isoclinic rotation as the ''great square rotation characteristic of the 16-cell'', and note once again that it is Fontaine and Hurley's counterclockwise rotation over the <math>r_3</math> {8/3} star polygon, which constructs <math>1/r_3</math>. == The 8-cell tesseract == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. [[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract <small><math>\{4,3,3\}</math></small> performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]] The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{4,3,3\}</math></small>. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedrons]]. 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 the great square rotation characteristic of the 16-cell, with the same effect on both alternate-position 16-cells. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords. They form a circular double helix which intersects each vertex of the tesseract once. The double helix is an 8-rung ladder twisted around 3 times, and bent into a circle in the fourth dimension with its ends joined. Each rung is a <math>\sqrt{3}</math> chord. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron. The 24-cell has the same chord set as the 4-hypercube tesseract: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]] The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters. The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords: :<math>r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The <math>r_1</math> and <math>r_5</math> chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. The <math>r_1</math> <small><math>\sqrt{1}</math></small> chords form a 8 Petrie dodecagons {12/1} which zig-zag back and forth, in the left and right rotational directions, between two Clifford parallel great hexagons formed by <math>r_2</math> chords. The <math>r_2</math> <small><math>\sqrt{1}</math></small> chords form 16 great hexagons, which can be divided four different ways into 4 disjoint great hexagons {24/4}=4{6} that lie Clifford parallel to each other. The <math>r_3</math> <small><math>\sqrt{2}</math></small> chords form 18 great squares, which can be divided three different ways into 6 disjoint great squares {24/6}=6{4} that lie Clifford parallel to each other. They include one pair of completely orthogonal great squares from each of the three 16-cells. The <math>r_4</math> <small><math>\sqrt{3}</math></small> chords form 32 great triangles, which can be divided four different ways into 8 disjoint great triangles {24/8}=8{3} inscribed in 4 disjoint great hexagons. The <math>r_5</math> <small><math>\sqrt{3}</math></small> chords form 8 circular helix Clifford polygons, visible as a green {12/5} dodecagram in the orthogonal projection. An isoclinic rotation of the 24-cell in 4 invariant <math>r_2</math> hexagon planes moves the vertices along 2 parallel circular isoclines {24/2}=2{12/5} over <math>r_5</math> chords. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges of Clifford parallel great squares, also isocline chords in great square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in great hexagon rotations. The green {12/5} dodecagram is a Clifford polygon.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} shows three octagram isoclines of <small><math>\sqrt{2}</math> </small>chords in the 24-cell]] We can rotate the 24-cell isoclinically in 6 Clifford parallel invariant great square planes containing four <math>r_{3}</math> 16-cell edges each, in the great square rotation characteristic of the 16-cell, with the same effect on all three 16-cells. In 720° each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The rotational curve over each 90° <small><math>\sqrt{2}</math></small> chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <small><math>\sqrt{2}</math></small> chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. The triple helix is an 8-step circular staircase that twists around 3 times, and is bent into a torus in the fourth dimension. Each staircase step is a great triangle of <small><math>\sqrt{3}</math></small> chords. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} shows 2 dodecagram isoclines of <small><math>\sqrt{3}</math></small> chords in the 24-cell]]We can rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing six <math>r_{2}</math> 24-cell edges each, over <math>r_{5}</math> isocline chords. This is the ''great hexagon rotation characteristic of the 24-cell'', also Fontaine and Hurley's counterclockwise rotation over the <math>r_5</math> {12/5} star polygon which constructs <math>1/r_5</math>. A complete 24-cell great circle edge plane revolution requires 720° like a complete 16-cell great circle edge plane revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. An isoclinic rotation by 60° in an invariant central plane containing a 24-cell edge takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all 4 invariant great hexagon planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square. All 24 vertices move at once on Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 12 <math>r_5</math> <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane 5 times in a moving invariant rotation plane. The rotational curve over each <math>r_5</math> 120° chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the course of a 720° revolution each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="6" |6 distinct 180° chord pairs make 6 distinct isoclinic rotations |- ! colspan="3" |Edge chord ! colspan="3" |Isocline chord |- style="background: gainsboro;" | | rowspan="4" |<math>t_1</math> |60° | rowspan="4" |[[File:Regular_polygon_24.svg|100px]]<br>{24/1}={24} | rowspan="4" |[[File:Regular_star_polygon_24-11.svg|100px]]<br>{24/11} |120° | rowspan="4" |<math>t_{11}</math> |- style="background: gainsboro;" | |{{radic|1}} |{{radic|3}} |- style="background: gainsboro;" | |1 |1.732~ |- style="background: gainsboro;" | |15° |165° |- style="background: palegreen;" | | rowspan="4" |<math>t_2</math> |60° | rowspan="4" |[[File:Regular_star_figure_2(12,1).svg|100px]]<br>{24/2}=2{12} | rowspan="4" |[[File:Regular_star_figure_2(12,5).svg|100px]]<br>{24/10}=2{12/5} |120° | rowspan="4" |<math>t_{10}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | |30° |150° |- style="background: seashell;" | | rowspan="4" |<math>t_3</math> |90° | rowspan="4" |[[File:Regular_star_figure_3(8,1).svg|100px]]<br>{24/3}=3{8} | rowspan="4" |[[File:Regular_star_figure_3(8,3).svg|100px]]<br>{24/9}=3{8/3} |90° | rowspan="4" |<math>t_{9}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |- style="background: seashell;" | |45° |135° |- style="background: palegreen;" | | rowspan="4" |<math>t_4</math> |60° | rowspan="4" |[[File:Regular_star_figure_4(6,1).svg|100px]]<br>{24/4}=4{6} | rowspan="4" |[[File:Regular_star_figure_8(3,1).svg|100px]]<br>{24/8}=8{3} |120° | rowspan="4" |<math>t_{8}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | |60° |120° |- style="background: gainsboro;" | | rowspan="4" |<math>t_5</math> |60° | rowspan="4" |[[File:Regular_star_polygon_24-5.svg|100px]]<br>{24/5} | rowspan="4" |[[File:Regular_star_polygon_24-7.svg|100px]]<br>{24/7} |120° | rowspan="4" |<math>t_{7}</math> |- style="background: gainsboro;" | |{{radic|1}} |{{radic|3}} |- style="background: gainsboro;" | |1 |1.732~ |- style="background: gainsboro;" | |75° |105° |- style="background: gainsboro;" | | rowspan="4" |<math>t_6</math> |90° | rowspan="4" |[[File:Regular_star_figure_6(4,1).svg|100px]]<br>{24/6}=6{4} | rowspan="4" |[[File:Regular_star_figure_6(4,1).svg|100px]]<br>{24/6}=6{4} |90° | rowspan="4" |<math>t_{6}</math> |- style="background: gainsboro;" | |{{radic|2}} |{{radic|2}} |- style="background: gainsboro;" | |1.414~ |1.414~ |- style="background: gainsboro;" | |90° |90° |} By examining the chords <math>r_i</math> of the 24-cell's Petrie {12}-gon we have found two distinct isoclinic rotations, the great square rotation characteristic of the 16-cell and the great hexagon rotation characteristic of the 24-cell. If we examine the chords <math>t_i</math> of the 24-cell's {24}-gon we find these, and also four other distinct isoclinic rotations. Each row of the table is a distinct isoclinic rotation of the 24-cell characterized by a pair of chords that sum to 180°. The 24 edge chords form the rotation's edge {24}-gon, and lie in invariant planes of the rotation. The 24 isocline chords form the rotation's Clifford {24}-gon and lie in invariant planes completely orthogonal to the corresponding edge chord's invariant plane. The rotational angle between successive edge chords and the rotational angle between successive isocline chords also sum to 180°. ... {{Clear}} == The 600-cell == [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are: [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</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. {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="7" |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra |- ! colspan="3" |Short edge chord ! Section ! colspan="3" |Long isocline chord |- style="background: palegreen;" | | rowspan="4" |<math>r_0</math> |0° | rowspan="4" | | rowspan="4" | | rowspan="4" |[[File:Regular_star_figure_15(2,1).svg|100px]]<br>{30/15}=15{2} |180° | rowspan="4" |<math>r_{15}</math> |- style="background: palegreen;" | |{{radic|0}} |{{radic|4}} |- style="background: palegreen;" | |0 |2 |- style="background: palegreen;" | |0° |180° |- style="background: palegreen;" | | rowspan="4" |<math>r_1</math> |36° | rowspan="4" |[[File:Regular_polygon_30.svg|100px]]<br>{30/1} | rowspan="4" | | rowspan="4" |[[File:Regular_star_figure_2(15,7).svg|100px]]<br>{30/14}=2{15/7} |144° | rowspan="4" |<math>r_{14}</math> |- style="background: palegreen;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: palegreen;" | |0.618~ |1.902~ |- style="background: palegreen;" | |12° |168° |- style="background: gainsboro;" | | rowspan="4" |<math>r_2</math> |36° | rowspan="4" |[[File:Regular_star_figure_2(15,1).svg|100px]]<br>{30/2}=2{15} | rowspan="4" | | rowspan="4" |[[File:Regular_star_polygon_30-13.svg|100px]]<br>{30/13} |144° | rowspan="4" |<math>r_{13}</math> |- style="background: gainsboro;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: gainsboro;" | |0.618~ |1.902~ |- style="background: gainsboro;" | |24° |156° |- style="background: yellow;" | | rowspan="4" |<math>r_3</math> |36° | rowspan="4" |[[File:Regular_star_figure_3(10,1).svg|100px]]<br>{30/3}=3{10} | rowspan="4" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="4" |[[File:Regular_star_figure_6(5,2).svg|100px]]<br>{30/12}=6{5/2} |144° | rowspan="4" |<math>r_{12}</math> |- style="background: yellow;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: yellow;" | |0.618~ |1.902~ |- style="background: yellow;" | |36° |144° |- style="background: palegreen;" | | rowspan="4" |<math>r_4</math> |60° | rowspan="4" |[[File:Regular_star_figure_2(15,2).svg|100px]]<br>{30/4}=2{15/2} | rowspan="4" | | rowspan="4" |[[File:Regular_star_polygon_30-11.svg|100px]]<br>{30/11} |120° | rowspan="4" |<math>r_{11}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | |48° |132° |- style="background: palegreen;" | | rowspan="4" |<math>r_5</math> |60° | rowspan="4" |[[File:Regular_star_figure_5(6,1).svg|100px]]<br>{30/5}=5{6} | rowspan="4" |[[File:V2 dodecahedron.png|100px]]<br>Dodecahedron | rowspan="4" |[[File:Regular_star_figure_10(3,1).svg|100px]]<br>{30/10}=10{3} |120° | rowspan="4" |<math>r_{10}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | |60° |120° |- style="background: yellow;" | | rowspan="4" |<math>r_{6}</math> |72° | rowspan="4" |[[File:Regular_star_figure_6(5,1).svg|100px]]<br>{30/6}=6{5} | rowspan="4" |[[File:V3 icosahedron.png|100px]]<br>Icosahedron | rowspan="4" |[[File:Regular_star_figure_3(10,3).svg|100px]]<br>{30/9}=3{10/3} |108° | rowspan="4" |<math>r_{9}</math> |- style="background: yellow;" | |{{radic|1.382~}} |{{radic|2.618~}} |- style="background: yellow;" | |1.176~ |1.618~ |- style="background: yellow;" | |72° |108° |- style="background: seashell;" | | rowspan="4" |<math>r_{7}</math> |90° | rowspan="4" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} | rowspan="4" |[[File:V4 icosidodecahedron.png|100px]]<br>Icosidodecahedron | rowspan="4" |[[File:Regular_star_figure_2(15,4).svg|100px]]<br>{30/8}=2{15/4} |90° | rowspan="4" |<math>r_{8}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |- style="background: seashell;" | |84° |96° |} The list of 600-cell chords <math>r_{i}</math> can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. The short chord and long chord each have their characteristic {30/n}-gon. Each row identifies a discrete isoclinic rotation of the 600-cell in invariant central planes containing the edges of the short chord {30}-gon, over the isocline chords of the long chord {30}-gon, the rotation's Clifford polygon. Each distinct pair of complementary chord lengths is identified with a distinct [[w:600-cell#Polyhedral sections|polyhedral section of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance <math>\phi^{-1}</math> is a icosahedron vertex figure, and the largest section at radial distance <math>\sqrt{2}</math> is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. Because [[w:3-sphere|<math>\mathbb{S}^3</math>]] is spherical, at radial distances greater than <math>\sqrt{2}</math> the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance <math>\sqrt{2+\phi}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>\sqrt{2}</math></small>]] We can rotate the 600-cell isoclinically in the great square rotation characteristic of the 16-cell, with the same effect on 15 disjoint 16-cells. Each 90° displacement takes 15 pairs of completely orthogonal invariant great square planes to each other. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The rotational curve over each 90° chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> form a circular fibration of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once. The 600-cell has another distinct isoclinic rotation in invariant great square planes, which has period 30 and visits every vertex of a 600-cell Petrie polygon. This [''great square left rotation characteristic of the 600-cell]'' takes place over <math>r_7</math> edge chords and <math>r_8</math> isocline chords. The {30/7} edge polygon is a skew helix of circumference <math>14\pi</math> with each <math>r_7</math> edge belonging to a distinct great square. The four {30/7} polygrams contribute one edge each to 30 great squares. Each 90° displacement takes every 16-cell to another 16-cell. The vertices of the invariant great squares each make seven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 90° {30/7} edge makes seven 12° turns. Four Clifford parallel {30/7} geodesics of circumference <math>16\pi</math> form a circular quadruple helix that intersects each 600-cell vertex once. The {30/8}=2{15/4} Clifford polygon is a compound of two skew {15/4} pentadecagrams of circumference <math>16\pi</math> with each <math>r_8</math> isocline chord belonging to a distinct 16-cell. The four {30/8} polygrams contribute one edge each to 30 great squares. The rotational curve over each 90° {30/8} isocline chord makes eight 12° turns. Four Clifford parallel {30/8} geodesics of circumference <math>16\pi</math> form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>\sqrt{3}</math></small> ]] We can rotate the 600-cell isoclinically in the great hexagon rotation characteristic of the 24-cell, over <math>\sqrt{1}</math> edge chords and <math>\sqrt{3}</math> isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° revolution each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> form a circular fibration of ten twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. The 600-cell has another distinct isoclinic rotation in invariant great hexagon planes, over <math>r_{4}=\sqrt{1}</math> edge chords and <math>r_{11}=\sqrt{3}</math> isocline chords This [''invariant great hexagon left rotation characteristic of the 600-cell]'' has period 30 and visits every vertex of a 600-cell Petrie polygon. Its {30/11} Clifford polygon is a skew helix where each <math>r_{11}</math> isocline chord is the <math>\sqrt{3}</math> diagonal of a great hexagon of a distinct 24-cell. The vertices of the invariant great hexagons of this rotation each make eleven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 120° <math>r_{11}</math> isocline chord makes eleven 12° turns. Four Clifford parallel {30/11} geodesic isoclines of circumference <math>22\pi</math> over <math>r_{11}</math> chords form a circular quadruple helix that intersects each 600-cell vertex once. We can rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its 36° <math>r_{3}</math> edges, over 144° <math>r_{12}</math> isocline chords. This ''invariant great decagon rotation characteristic of the 600-cell'' has period 5 and takes disjoint 24-cells to each other. The rotational curve over each <math>r_{12}</math> chord of its {5/2} Clifford polygon makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference <math>4\pi</math> over five <math>r_{12}</math> chords form a circular fibration of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. The rotation of the 600-cell 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. The 24-cells revolve within the 600-cell, as the 16-cells revolve within the 24-cells. All 120 vertices move at once on four Clifford parallel geodesic isoclines, displaced 144° in different directions. The 600-cell has another distinct isoclinic rotation in invariant great decagon planes containing its 36° <math>r_{2}</math> edges, over 144° <math>r_{13}</math> isocline chords. This [''great decagon left rotation characteristic of the 600-cell]'' has period 30 and visits every vertex of a 600-cell Petrie polygon. The rotational curve over each 144° <math>r_{13}</math> isocline chord makes thirteen 12° turns. Four Clifford parallel {30/13} geodesic isoclines of circumference <math>26\pi</math> form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} == The 5-cell 4-simplex == {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="9" |30 chords (15 180° pairs) make 15 distinct section polyhedra |- ! colspan="3" |Short chord ! Section ! colspan="3" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px]]<br>{30/15}=15{2} |180° | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} |{{radic|4}} |- style="background: palegreen;" | |0 |2 |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° | rowspan="3" |[[File:Regular_polygon_30.svg|100px]]<br>{30/1} | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px]]<br>{30/14} |164.5~° | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} |{{radic|3.927~}} |- style="background: palegreen;" | |0.270~ |1.982~ |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px]]<br>{30/2}=2{15} | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px]]<br>{30/13} |154.8~° | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} |{{radic|3.809~}} |- style="background: gainsboro;" | |0.437~ |1.952~ |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px]]<br>{30/3}=3{10} | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px]]<br>{30/12}=6{5/2} |144° | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: yellow;" | |0.618~ |1.902~ |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |138.6~° | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} |{{radic|3.5}} |- style="background: gainsboro;" | |0.707~ |1.871~ |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px]]<br>{30/4}=2{15/2} | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px]]<br>{30/11} |135.5~° | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} |{{radic|3.427~}} |- style="background: palegreen;" | |0.757~ |1.851~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |130.9~° | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} |{{radic|3.309~}} |- style="background: gainsboro;" | |0.831~ |1.819~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° | rowspan="3" | | rowspan="3" | | rowspan="3" | |124° | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} |{{radic|3.118~}} |- style="background: gainsboro;" | |0.939~ |1.766~ |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px]]<br>{30/5}=5{6} | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px]]<br>{30/10}=10{3} |120° | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |113.9~° | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} |{{radic|2.809~}} |- style="background: gainsboro;" | |1.091~ |1.676~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |110.2~° | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} |{{radic|2.691~}} |- style="background: gainsboro;" | |1.144~ |1.640~ |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px]]<br>{30/6}=6{5} | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px]]<br>{30/9}=3{10/3} |108° | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} |{{radic|2.618~}} |- style="background: yellow;" | |1.176~ |1.618~ |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px]]<br>{30/8}=2{15/4} |104.5~° | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} |{{radic|2.5}} |- style="background: palegreen;" | |1.224~ |1.581~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |98.9~° | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} |{{radic|2.309~}} |- style="background: gainsboro;" | |1.300~ |1.520~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |95.5~° | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} |{{radic|2.191~}} |- style="background: gainsboro;" | |1.345~ |1.480~ |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} |90° | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |} The [[User:Dc.samizdat/Golden chords of the 120-cell#Thirty distinguished distances|list of thirty 120-cell chords]] <math>c_{t}</math> can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct [[w:120-cell#Concentric_hulls|polyhedral section of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance <math>c_1</math> is a tetrahedron vertex figure, and the largest section at radial distance <math>c_{15}</math> is a central section bisecting the 120-cell. Because [[w:3-sphere|<math>\mathbb{S}^3</math>]] is spherical, at radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance <math>c_{29}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. Only 8 of the 30 chords in the table occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. Since all those polytopes except the 5-cell occur in the 600-cell, and the 600-cell and the 120-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. ... {{Clear}} == Finally the 120-cell == The [[120-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{5,3,3\}</math></small>. It has 600 vertices, 1200 edges, 720 pentagon faces, and 120 dodecahedron cells. It is the four-dimensional analogue of the dodecahedron. The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. ... {{Clear}} == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.] The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} ni1ve6l28tt9slf371dew93bnpl96y5 2818499 2818497 2026-07-19T00:12:11Z Dc.samizdat 2856930 /* The 24-cell */ 2818499 wikitext text/x-wiki = Golden chords of the 120-cell = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - June 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by 5² non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). Consequently 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which contains 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). Consequently the 8-point (16-cell) compounds by 3 times 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} == The 16-cell 4-orthoplex == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=\sqrt{2}+1</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. Their 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}+1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-1</math> . Over the third <math>r_3</math> chord it also moves a distance of <math>-1</math>. Fontaine and Hurley also demonstrated the significance of <math>1/r_i</math> in Steinbach's Diagonal Product Formula, which says that every chord length is the sum of certain smaller chord lengths. The smaller chords are certain diagonals of the same regular polygon of a smaller edge length, specifically edge length <math>1/r_i</math> rather than <math>1</math>. If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{3,3,4\}</math></small>. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <math>\sqrt{2}</math> edges except opposite pairs. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Each chord is a distinct 4-vector with a length and a direction. 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, where vertices circle over the chords of an <math>r_i</math> polygon. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. The angle between two <math>r_i</math> chords is <math>180^\circ - i \times 45^\circ</math>. [[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 {8/1} 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 the 16-cell form an ''edge polygon'' {8/2}=2{4}. The 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 <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> (it is <math>6\pi</math> in this case), and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> chords, each vertex makes twelve 45° 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 revolution 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. We shall refer to this isoclinic rotation as the ''great square rotation characteristic of the 16-cell'', and note once again that it is Fontaine and Hurley's counterclockwise rotation over the <math>r_3</math> {8/3} star polygon, which constructs <math>1/r_3</math>. == The 8-cell tesseract == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. [[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract <small><math>\{4,3,3\}</math></small> performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]] The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{4,3,3\}</math></small>. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedrons]]. 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 the great square rotation characteristic of the 16-cell, with the same effect on both alternate-position 16-cells. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords. They form a circular double helix which intersects each vertex of the tesseract once. The double helix is an 8-rung ladder twisted around 3 times, and bent into a circle in the fourth dimension with its ends joined. Each rung is a <math>\sqrt{3}</math> chord. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron. The 24-cell has the same chord set as the 4-hypercube tesseract: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]] The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters. The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords: :<math>r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The <math>r_1</math> and <math>r_5</math> chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. The <math>r_1</math> <small><math>\sqrt{1}</math></small> chords form a 8 Petrie dodecagons {12/1} which zig-zag back and forth, in the left and right rotational directions, between two Clifford parallel great hexagons formed by <math>r_2</math> chords. The <math>r_2</math> <small><math>\sqrt{1}</math></small> chords form 16 great hexagons, which can be divided four different ways into 4 disjoint great hexagons {24/4}=4{6} that lie Clifford parallel to each other. The <math>r_3</math> <small><math>\sqrt{2}</math></small> chords form 18 great squares, which can be divided three different ways into 6 disjoint great squares {24/6}=6{4} that lie Clifford parallel to each other. They include one pair of completely orthogonal great squares from each of the three 16-cells. The <math>r_4</math> <small><math>\sqrt{3}</math></small> chords form 32 great triangles, which can be divided four different ways into 8 disjoint great triangles {24/8}=8{3} inscribed in 4 disjoint great hexagons. The <math>r_5</math> <small><math>\sqrt{3}</math></small> chords form 8 circular helix Clifford polygons, visible as a green {12/5} dodecagram in the orthogonal projection. An isoclinic rotation of the 24-cell in 4 invariant <math>r_2</math> hexagon planes moves the vertices along 2 parallel circular isoclines {24/2}=2{12/5} over <math>r_5</math> chords. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges of Clifford parallel great squares, also isocline chords in great square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in great hexagon rotations. The green {12/5} dodecagram is a Clifford polygon.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} shows three octagram isoclines of <small><math>\sqrt{2}</math> </small>chords in the 24-cell]] We can rotate the 24-cell isoclinically in 6 Clifford parallel invariant great square planes containing four <math>r_{3}</math> 16-cell edges each, in the great square rotation characteristic of the 16-cell, with the same effect on all three 16-cells. In 720° each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The rotational curve over each 90° <small><math>\sqrt{2}</math></small> chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <small><math>\sqrt{2}</math></small> chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. The triple helix is an 8-step circular staircase that twists around 3 times, and is bent into a torus in the fourth dimension. Each staircase step is a great triangle of <small><math>\sqrt{3}</math></small> chords. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} shows 2 dodecagram isoclines of <small><math>\sqrt{3}</math></small> chords in the 24-cell]]We can rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing six <math>r_{2}</math> 24-cell edges each, over <math>r_{5}</math> isocline chords. This is the ''great hexagon rotation characteristic of the 24-cell'', also Fontaine and Hurley's counterclockwise rotation over the <math>r_5</math> {12/5} star polygon which constructs <math>1/r_5</math>. A complete 24-cell great circle edge plane revolution requires 720° like a complete 16-cell great circle edge plane revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. An isoclinic rotation by 60° in an invariant central plane containing a 24-cell edge takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all 4 invariant great hexagon planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square. All 24 vertices move at once on Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 12 <math>r_5</math> <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane 5 times in a moving invariant rotation plane. The rotational curve over each <math>r_5</math> 120° chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the course of a 720° revolution each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="6" |6 distinct 180° chord pairs make 6 distinct isoclinic rotations |- ! colspan="3" |Edge chord ! colspan="3" |Isocline chord |- style="background: gainsboro;" | | rowspan="4" |<math>t_1</math> |60° | rowspan="4" |[[File:Regular_polygon_24.svg|100px]]<br>{24/1}={24} | rowspan="4" |[[File:Regular_star_polygon_24-11.svg|100px]]<br>{24/11} |120° | rowspan="4" |<math>t_{11}</math> |- style="background: gainsboro;" | |{{radic|1}} |{{radic|3}} |- style="background: gainsboro;" | |1 |1.732~ |- style="background: gainsboro;" | |15° |165° |- style="background: palegreen;" | | rowspan="4" |<math>t_2</math> |60° | rowspan="4" |[[File:Regular_star_figure_2(12,1).svg|100px]]<br>{24/2}=2{12} | rowspan="4" |[[File:Regular_star_figure_2(12,5).svg|100px]]<br>{24/10}=2{12/5} |120° | rowspan="4" |<math>t_{10}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | |30° |150° |- style="background: seashell;" | | rowspan="4" |<math>t_3</math> |90° | rowspan="4" |[[File:Regular_star_figure_3(8,1).svg|100px]]<br>{24/3}=3{8} | rowspan="4" |[[File:Regular_star_figure_3(8,3).svg|100px]]<br>{24/9}=3{8/3} |90° | rowspan="4" |<math>t_{9}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |- style="background: seashell;" | |45° |135° |- style="background: palegreen;" | | rowspan="4" |<math>t_4</math> |60° | rowspan="4" |[[File:Regular_star_figure_4(6,1).svg|100px]]<br>{24/4}=4{6} | rowspan="4" |[[File:Regular_star_figure_8(3,1).svg|100px]]<br>{24/8}=8{3} |120° | rowspan="4" |<math>t_{8}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | |60° |120° |- style="background: gainsboro;" | | rowspan="4" |<math>t_5</math> |60° | rowspan="4" |[[File:Regular_star_polygon_24-5.svg|100px]]<br>{24/5} | rowspan="4" |[[File:Regular_star_polygon_24-7.svg|100px]]<br>{24/7} |120° | rowspan="4" |<math>t_{7}</math> |- style="background: gainsboro;" | |{{radic|1}} |{{radic|3}} |- style="background: gainsboro;" | |1 |1.732~ |- style="background: gainsboro;" | |75° |105° |- style="background: gainsboro;" | | rowspan="4" |<math>t_6</math> |90° | rowspan="4" |[[File:Regular_star_figure_6(4,1).svg|100px]]<br>{24/6}=6{4} | rowspan="4" |[[File:Regular_star_figure_6(4,1).svg|100px]]<br>{24/6}=6{4} |90° | rowspan="4" |<math>t_{6}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |- style="background: seashell;" | |90° |90° |} By examining the chords <math>r_i</math> of the 24-cell's Petrie {12}-gon we have found two distinct isoclinic rotations, the great square rotation characteristic of the 16-cell and the great hexagon rotation characteristic of the 24-cell. If we examine the chords <math>t_i</math> of the 24-cell's {24}-gon we find these, and also four other distinct isoclinic rotations. Each row of the table is a distinct isoclinic rotation of the 24-cell characterized by a pair of chords that sum to 180°. The 24 edge chords form the rotation's edge {24}-gon, and lie in invariant planes of the rotation. The 24 isocline chords form the rotation's Clifford {24}-gon and lie in invariant planes completely orthogonal to the corresponding edge chord's invariant plane. The rotational angle between successive edge chords and the rotational angle between successive isocline chords also sum to 180°. ... {{Clear}} == The 600-cell == [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are: [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</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. {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="7" |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra |- ! colspan="3" |Short edge chord ! Section ! colspan="3" |Long isocline chord |- style="background: palegreen;" | | rowspan="4" |<math>r_0</math> |0° | rowspan="4" | | rowspan="4" | | rowspan="4" |[[File:Regular_star_figure_15(2,1).svg|100px]]<br>{30/15}=15{2} |180° | rowspan="4" |<math>r_{15}</math> |- style="background: palegreen;" | |{{radic|0}} |{{radic|4}} |- style="background: palegreen;" | |0 |2 |- style="background: palegreen;" | |0° |180° |- style="background: palegreen;" | | rowspan="4" |<math>r_1</math> |36° | rowspan="4" |[[File:Regular_polygon_30.svg|100px]]<br>{30/1} | rowspan="4" | | rowspan="4" |[[File:Regular_star_figure_2(15,7).svg|100px]]<br>{30/14}=2{15/7} |144° | rowspan="4" |<math>r_{14}</math> |- style="background: palegreen;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: palegreen;" | |0.618~ |1.902~ |- style="background: palegreen;" | |12° |168° |- style="background: gainsboro;" | | rowspan="4" |<math>r_2</math> |36° | rowspan="4" |[[File:Regular_star_figure_2(15,1).svg|100px]]<br>{30/2}=2{15} | rowspan="4" | | rowspan="4" |[[File:Regular_star_polygon_30-13.svg|100px]]<br>{30/13} |144° | rowspan="4" |<math>r_{13}</math> |- style="background: gainsboro;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: gainsboro;" | |0.618~ |1.902~ |- style="background: gainsboro;" | |24° |156° |- style="background: yellow;" | | rowspan="4" |<math>r_3</math> |36° | rowspan="4" |[[File:Regular_star_figure_3(10,1).svg|100px]]<br>{30/3}=3{10} | rowspan="4" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="4" |[[File:Regular_star_figure_6(5,2).svg|100px]]<br>{30/12}=6{5/2} |144° | rowspan="4" |<math>r_{12}</math> |- style="background: yellow;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: yellow;" | |0.618~ |1.902~ |- style="background: yellow;" | |36° |144° |- style="background: palegreen;" | | rowspan="4" |<math>r_4</math> |60° | rowspan="4" |[[File:Regular_star_figure_2(15,2).svg|100px]]<br>{30/4}=2{15/2} | rowspan="4" | | rowspan="4" |[[File:Regular_star_polygon_30-11.svg|100px]]<br>{30/11} |120° | rowspan="4" |<math>r_{11}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | |48° |132° |- style="background: palegreen;" | | rowspan="4" |<math>r_5</math> |60° | rowspan="4" |[[File:Regular_star_figure_5(6,1).svg|100px]]<br>{30/5}=5{6} | rowspan="4" |[[File:V2 dodecahedron.png|100px]]<br>Dodecahedron | rowspan="4" |[[File:Regular_star_figure_10(3,1).svg|100px]]<br>{30/10}=10{3} |120° | rowspan="4" |<math>r_{10}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | |60° |120° |- style="background: yellow;" | | rowspan="4" |<math>r_{6}</math> |72° | rowspan="4" |[[File:Regular_star_figure_6(5,1).svg|100px]]<br>{30/6}=6{5} | rowspan="4" |[[File:V3 icosahedron.png|100px]]<br>Icosahedron | rowspan="4" |[[File:Regular_star_figure_3(10,3).svg|100px]]<br>{30/9}=3{10/3} |108° | rowspan="4" |<math>r_{9}</math> |- style="background: yellow;" | |{{radic|1.382~}} |{{radic|2.618~}} |- style="background: yellow;" | |1.176~ |1.618~ |- style="background: yellow;" | |72° |108° |- style="background: seashell;" | | rowspan="4" |<math>r_{7}</math> |90° | rowspan="4" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} | rowspan="4" |[[File:V4 icosidodecahedron.png|100px]]<br>Icosidodecahedron | rowspan="4" |[[File:Regular_star_figure_2(15,4).svg|100px]]<br>{30/8}=2{15/4} |90° | rowspan="4" |<math>r_{8}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |- style="background: seashell;" | |84° |96° |} The list of 600-cell chords <math>r_{i}</math> can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. The short chord and long chord each have their characteristic {30/n}-gon. Each row identifies a discrete isoclinic rotation of the 600-cell in invariant central planes containing the edges of the short chord {30}-gon, over the isocline chords of the long chord {30}-gon, the rotation's Clifford polygon. Each distinct pair of complementary chord lengths is identified with a distinct [[w:600-cell#Polyhedral sections|polyhedral section of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance <math>\phi^{-1}</math> is a icosahedron vertex figure, and the largest section at radial distance <math>\sqrt{2}</math> is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. Because [[w:3-sphere|<math>\mathbb{S}^3</math>]] is spherical, at radial distances greater than <math>\sqrt{2}</math> the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance <math>\sqrt{2+\phi}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>\sqrt{2}</math></small>]] We can rotate the 600-cell isoclinically in the great square rotation characteristic of the 16-cell, with the same effect on 15 disjoint 16-cells. Each 90° displacement takes 15 pairs of completely orthogonal invariant great square planes to each other. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The rotational curve over each 90° chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> form a circular fibration of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once. The 600-cell has another distinct isoclinic rotation in invariant great square planes, which has period 30 and visits every vertex of a 600-cell Petrie polygon. This [''great square left rotation characteristic of the 600-cell]'' takes place over <math>r_7</math> edge chords and <math>r_8</math> isocline chords. The {30/7} edge polygon is a skew helix of circumference <math>14\pi</math> with each <math>r_7</math> edge belonging to a distinct great square. The four {30/7} polygrams contribute one edge each to 30 great squares. Each 90° displacement takes every 16-cell to another 16-cell. The vertices of the invariant great squares each make seven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 90° {30/7} edge makes seven 12° turns. Four Clifford parallel {30/7} geodesics of circumference <math>16\pi</math> form a circular quadruple helix that intersects each 600-cell vertex once. The {30/8}=2{15/4} Clifford polygon is a compound of two skew {15/4} pentadecagrams of circumference <math>16\pi</math> with each <math>r_8</math> isocline chord belonging to a distinct 16-cell. The four {30/8} polygrams contribute one edge each to 30 great squares. The rotational curve over each 90° {30/8} isocline chord makes eight 12° turns. Four Clifford parallel {30/8} geodesics of circumference <math>16\pi</math> form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>\sqrt{3}</math></small> ]] We can rotate the 600-cell isoclinically in the great hexagon rotation characteristic of the 24-cell, over <math>\sqrt{1}</math> edge chords and <math>\sqrt{3}</math> isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° revolution each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> form a circular fibration of ten twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. The 600-cell has another distinct isoclinic rotation in invariant great hexagon planes, over <math>r_{4}=\sqrt{1}</math> edge chords and <math>r_{11}=\sqrt{3}</math> isocline chords This [''invariant great hexagon left rotation characteristic of the 600-cell]'' has period 30 and visits every vertex of a 600-cell Petrie polygon. Its {30/11} Clifford polygon is a skew helix where each <math>r_{11}</math> isocline chord is the <math>\sqrt{3}</math> diagonal of a great hexagon of a distinct 24-cell. The vertices of the invariant great hexagons of this rotation each make eleven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 120° <math>r_{11}</math> isocline chord makes eleven 12° turns. Four Clifford parallel {30/11} geodesic isoclines of circumference <math>22\pi</math> over <math>r_{11}</math> chords form a circular quadruple helix that intersects each 600-cell vertex once. We can rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its 36° <math>r_{3}</math> edges, over 144° <math>r_{12}</math> isocline chords. This ''invariant great decagon rotation characteristic of the 600-cell'' has period 5 and takes disjoint 24-cells to each other. The rotational curve over each <math>r_{12}</math> chord of its {5/2} Clifford polygon makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference <math>4\pi</math> over five <math>r_{12}</math> chords form a circular fibration of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. The rotation of the 600-cell 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. The 24-cells revolve within the 600-cell, as the 16-cells revolve within the 24-cells. All 120 vertices move at once on four Clifford parallel geodesic isoclines, displaced 144° in different directions. The 600-cell has another distinct isoclinic rotation in invariant great decagon planes containing its 36° <math>r_{2}</math> edges, over 144° <math>r_{13}</math> isocline chords. This [''great decagon left rotation characteristic of the 600-cell]'' has period 30 and visits every vertex of a 600-cell Petrie polygon. The rotational curve over each 144° <math>r_{13}</math> isocline chord makes thirteen 12° turns. Four Clifford parallel {30/13} geodesic isoclines of circumference <math>26\pi</math> form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} == The 5-cell 4-simplex == {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="9" |30 chords (15 180° pairs) make 15 distinct section polyhedra |- ! colspan="3" |Short chord ! Section ! colspan="3" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px]]<br>{30/15}=15{2} |180° | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} |{{radic|4}} |- style="background: palegreen;" | |0 |2 |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° | rowspan="3" |[[File:Regular_polygon_30.svg|100px]]<br>{30/1} | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px]]<br>{30/14} |164.5~° | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} |{{radic|3.927~}} |- style="background: palegreen;" | |0.270~ |1.982~ |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px]]<br>{30/2}=2{15} | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px]]<br>{30/13} |154.8~° | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} |{{radic|3.809~}} |- style="background: gainsboro;" | |0.437~ |1.952~ |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px]]<br>{30/3}=3{10} | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px]]<br>{30/12}=6{5/2} |144° | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: yellow;" | |0.618~ |1.902~ |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |138.6~° | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} |{{radic|3.5}} |- style="background: gainsboro;" | |0.707~ |1.871~ |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px]]<br>{30/4}=2{15/2} | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px]]<br>{30/11} |135.5~° | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} |{{radic|3.427~}} |- style="background: palegreen;" | |0.757~ |1.851~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |130.9~° | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} |{{radic|3.309~}} |- style="background: gainsboro;" | |0.831~ |1.819~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° | rowspan="3" | | rowspan="3" | | rowspan="3" | |124° | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} |{{radic|3.118~}} |- style="background: gainsboro;" | |0.939~ |1.766~ |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px]]<br>{30/5}=5{6} | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px]]<br>{30/10}=10{3} |120° | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |113.9~° | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} |{{radic|2.809~}} |- style="background: gainsboro;" | |1.091~ |1.676~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |110.2~° | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} |{{radic|2.691~}} |- style="background: gainsboro;" | |1.144~ |1.640~ |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px]]<br>{30/6}=6{5} | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px]]<br>{30/9}=3{10/3} |108° | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} |{{radic|2.618~}} |- style="background: yellow;" | |1.176~ |1.618~ |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px]]<br>{30/8}=2{15/4} |104.5~° | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} |{{radic|2.5}} |- style="background: palegreen;" | |1.224~ |1.581~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |98.9~° | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} |{{radic|2.309~}} |- style="background: gainsboro;" | |1.300~ |1.520~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |95.5~° | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} |{{radic|2.191~}} |- style="background: gainsboro;" | |1.345~ |1.480~ |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} |90° | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |} The [[User:Dc.samizdat/Golden chords of the 120-cell#Thirty distinguished distances|list of thirty 120-cell chords]] <math>c_{t}</math> can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct [[w:120-cell#Concentric_hulls|polyhedral section of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance <math>c_1</math> is a tetrahedron vertex figure, and the largest section at radial distance <math>c_{15}</math> is a central section bisecting the 120-cell. Because [[w:3-sphere|<math>\mathbb{S}^3</math>]] is spherical, at radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance <math>c_{29}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. Only 8 of the 30 chords in the table occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. Since all those polytopes except the 5-cell occur in the 600-cell, and the 600-cell and the 120-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. ... {{Clear}} == Finally the 120-cell == The [[120-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{5,3,3\}</math></small>. It has 600 vertices, 1200 edges, 720 pentagon faces, and 120 dodecahedron cells. It is the four-dimensional analogue of the dodecahedron. The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. ... {{Clear}} == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.] The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} 3i9i3jzbazizm4btjku9ge3tw1mxm2q 2818500 2818499 2026-07-19T00:13:51Z Dc.samizdat 2856930 /* The 24-cell */ 2818500 wikitext text/x-wiki = Golden chords of the 120-cell = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - June 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by 5² non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). Consequently 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which contains 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). Consequently the 8-point (16-cell) compounds by 3 times 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} == The 16-cell 4-orthoplex == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=\sqrt{2}+1</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. Their 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}+1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-1</math> . Over the third <math>r_3</math> chord it also moves a distance of <math>-1</math>. Fontaine and Hurley also demonstrated the significance of <math>1/r_i</math> in Steinbach's Diagonal Product Formula, which says that every chord length is the sum of certain smaller chord lengths. The smaller chords are certain diagonals of the same regular polygon of a smaller edge length, specifically edge length <math>1/r_i</math> rather than <math>1</math>. If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{3,3,4\}</math></small>. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <math>\sqrt{2}</math> edges except opposite pairs. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Each chord is a distinct 4-vector with a length and a direction. 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, where vertices circle over the chords of an <math>r_i</math> polygon. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. The angle between two <math>r_i</math> chords is <math>180^\circ - i \times 45^\circ</math>. [[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 {8/1} 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 the 16-cell form an ''edge polygon'' {8/2}=2{4}. The 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 <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> (it is <math>6\pi</math> in this case), and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> chords, each vertex makes twelve 45° 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 revolution 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. We shall refer to this isoclinic rotation as the ''great square rotation characteristic of the 16-cell'', and note once again that it is Fontaine and Hurley's counterclockwise rotation over the <math>r_3</math> {8/3} star polygon, which constructs <math>1/r_3</math>. == The 8-cell tesseract == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. [[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract <small><math>\{4,3,3\}</math></small> performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]] The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{4,3,3\}</math></small>. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedrons]]. 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 the great square rotation characteristic of the 16-cell, with the same effect on both alternate-position 16-cells. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords. They form a circular double helix which intersects each vertex of the tesseract once. The double helix is an 8-rung ladder twisted around 3 times, and bent into a circle in the fourth dimension with its ends joined. Each rung is a <math>\sqrt{3}</math> chord. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron. The 24-cell has the same chord set as the 4-hypercube tesseract: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]] The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters. The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords: :<math>r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The <math>r_1</math> and <math>r_5</math> chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. The <math>r_1</math> <small><math>\sqrt{1}</math></small> chords form a 8 Petrie dodecagons {12/1} which zig-zag back and forth, in the left and right rotational directions, between two Clifford parallel great hexagons formed by <math>r_2</math> chords. The <math>r_2</math> <small><math>\sqrt{1}</math></small> chords form 16 great hexagons, which can be divided four different ways into 4 disjoint great hexagons {24/4}=4{6} that lie Clifford parallel to each other. The <math>r_3</math> <small><math>\sqrt{2}</math></small> chords form 18 great squares, which can be divided three different ways into 6 disjoint great squares {24/6}=6{4} that lie Clifford parallel to each other. They include one pair of completely orthogonal great squares from each of the three 16-cells. The <math>r_4</math> <small><math>\sqrt{3}</math></small> chords form 32 great triangles, which can be divided four different ways into 8 disjoint great triangles {24/8}=8{3} inscribed in 4 disjoint great hexagons. The <math>r_5</math> <small><math>\sqrt{3}</math></small> chords form 8 circular helix Clifford polygons, visible as a green {12/5} dodecagram in the orthogonal projection. An isoclinic rotation of the 24-cell in 4 invariant <math>r_2</math> hexagon planes moves the vertices along 2 parallel circular isoclines {24/2}=2{12/5} over <math>r_5</math> chords. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges of Clifford parallel great squares, also isocline chords in great square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in great hexagon rotations. The green {12/5} dodecagram is a Clifford polygon.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} shows three octagram isoclines of <small><math>\sqrt{2}</math> </small>chords in the 24-cell]] We can rotate the 24-cell isoclinically in 6 Clifford parallel invariant great square planes containing four <math>r_{3}</math> 16-cell edges each, in the great square rotation characteristic of the 16-cell, with the same effect on all three 16-cells. In 720° each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The rotational curve over each 90° <small><math>\sqrt{2}</math></small> chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <small><math>\sqrt{2}</math></small> chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. The triple helix is an 8-step circular staircase that twists around 3 times, and is bent into a torus in the fourth dimension. Each staircase step is a great triangle of <small><math>\sqrt{3}</math></small> chords. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} shows 2 dodecagram isoclines of <small><math>\sqrt{3}</math></small> chords in the 24-cell]]We can rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing six <math>r_{2}</math> 24-cell edges each, over <math>r_{5}</math> isocline chords. This is the ''great hexagon rotation characteristic of the 24-cell'', also Fontaine and Hurley's counterclockwise rotation over the <math>r_5</math> {12/5} star polygon which constructs <math>1/r_5</math>. A complete 24-cell great circle edge plane revolution requires 720° like a complete 16-cell great circle edge plane revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. An isoclinic rotation by 60° in an invariant central plane containing a 24-cell edge takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all 4 invariant great hexagon planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square. All 24 vertices move at once on Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 12 <math>r_5</math> <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane 5 times in a moving invariant rotation plane. The rotational curve over each <math>r_5</math> 120° chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the course of a 720° revolution each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="6" |6 distinct 180° chord pairs make 6 distinct isoclinic rotations |- ! colspan="3" |Edge chord ! colspan="3" |Isocline chord |- style="background: gainsboro;" | | rowspan="4" |<math>t_1</math> |60° | rowspan="4" |[[File:Regular_polygon_24.svg|100px]]<br>{24/1}={24} | rowspan="4" |[[File:Regular_star_polygon_24-11.svg|100px]]<br>{24/11} |120° | rowspan="4" |<math>t_{11}</math> |- style="background: gainsboro;" | |{{radic|1}} |{{radic|3}} |- style="background: gainsboro;" | |1 |1.732~ |- style="background: gainsboro;" | |15° |165° |- style="background: palegreen;" | | rowspan="4" |<math>t_2</math> |60° | rowspan="4" |[[File:Regular_star_figure_2(12,1).svg|100px]]<br>{24/2}=2{12} | rowspan="4" |[[File:Regular_star_figure_2(12,5).svg|100px]]<br>{24/10}=2{12/5} |120° | rowspan="4" |<math>t_{10}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | |30° |150° |- style="background: seashell;" | | rowspan="4" |<math>t_3</math> |90° | rowspan="4" |[[File:Regular_star_figure_3(8,1).svg|100px]]<br>{24/3}=3{8} | rowspan="4" |[[File:Regular_star_figure_3(8,3).svg|100px]]<br>{24/9}=3{8/3} |90° | rowspan="4" |<math>t_{9}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |- style="background: seashell;" | |45° |135° |- style="background: palegreen;" | | rowspan="4" |<math>t_4</math> |60° | rowspan="4" |[[File:Regular_star_figure_4(6,1).svg|100px]]<br>{24/4}=4{6} | rowspan="4" |[[File:Regular_star_figure_8(3,1).svg|100px]]<br>{24/8}=8{3} |120° | rowspan="4" |<math>t_{8}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | |60° |120° |- style="background: gainsboro;" | | rowspan="4" |<math>t_5</math> |60° | rowspan="4" |[[File:Regular_star_polygon_24-5.svg|100px]]<br>{24/5} | rowspan="4" |[[File:Regular_star_polygon_24-7.svg|100px]]<br>{24/7} |120° | rowspan="4" |<math>t_{7}</math> |- style="background: gainsboro;" | |{{radic|1}} |{{radic|3}} |- style="background: gainsboro;" | |1 |1.732~ |- style="background: gainsboro;" | |75° |105° |- style="background: seashell;" | | rowspan="4" |<math>t_6</math> |90° | rowspan="4" |[[File:Regular_star_figure_6(4,1).svg|100px]]<br>{24/6}=6{4} | rowspan="4" |[[File:Regular_star_figure_6(4,1).svg|100px]]<br>{24/6}=6{4} |90° | rowspan="4" |<math>t_{6}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |- style="background: seashell;" | |90° |90° |} By examining the chords <math>r_i</math> of the 24-cell's Petrie {12}-gon we have found two distinct isoclinic rotations, the great square rotation characteristic of the 16-cell and the great hexagon rotation characteristic of the 24-cell. If we examine the chords <math>t_i</math> of the 24-cell's {24}-gon we find these, and also four other distinct isoclinic rotations. Each row of the table is a distinct isoclinic rotation of the 24-cell characterized by a pair of chords that sum to 180°. The 24 edge chords form the rotation's edge {24}-gon, and lie in invariant planes of the rotation. The 24 isocline chords form the rotation's Clifford {24}-gon and lie in invariant planes completely orthogonal to the corresponding edge chord's invariant plane. The rotational angle between successive edge chords and the rotational angle between successive isocline chords also sum to 180°. ... {{Clear}} == The 600-cell == [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are: [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</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. {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="7" |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra |- ! colspan="3" |Short edge chord ! Section ! colspan="3" |Long isocline chord |- style="background: palegreen;" | | rowspan="4" |<math>r_0</math> |0° | rowspan="4" | | rowspan="4" | | rowspan="4" |[[File:Regular_star_figure_15(2,1).svg|100px]]<br>{30/15}=15{2} |180° | rowspan="4" |<math>r_{15}</math> |- style="background: palegreen;" | |{{radic|0}} |{{radic|4}} |- style="background: palegreen;" | |0 |2 |- style="background: palegreen;" | |0° |180° |- style="background: palegreen;" | | rowspan="4" |<math>r_1</math> |36° | rowspan="4" |[[File:Regular_polygon_30.svg|100px]]<br>{30/1} | rowspan="4" | | rowspan="4" |[[File:Regular_star_figure_2(15,7).svg|100px]]<br>{30/14}=2{15/7} |144° | rowspan="4" |<math>r_{14}</math> |- style="background: palegreen;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: palegreen;" | |0.618~ |1.902~ |- style="background: palegreen;" | |12° |168° |- style="background: gainsboro;" | | rowspan="4" |<math>r_2</math> |36° | rowspan="4" |[[File:Regular_star_figure_2(15,1).svg|100px]]<br>{30/2}=2{15} | rowspan="4" | | rowspan="4" |[[File:Regular_star_polygon_30-13.svg|100px]]<br>{30/13} |144° | rowspan="4" |<math>r_{13}</math> |- style="background: gainsboro;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: gainsboro;" | |0.618~ |1.902~ |- style="background: gainsboro;" | |24° |156° |- style="background: yellow;" | | rowspan="4" |<math>r_3</math> |36° | rowspan="4" |[[File:Regular_star_figure_3(10,1).svg|100px]]<br>{30/3}=3{10} | rowspan="4" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="4" |[[File:Regular_star_figure_6(5,2).svg|100px]]<br>{30/12}=6{5/2} |144° | rowspan="4" |<math>r_{12}</math> |- style="background: yellow;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: yellow;" | |0.618~ |1.902~ |- style="background: yellow;" | |36° |144° |- style="background: palegreen;" | | rowspan="4" |<math>r_4</math> |60° | rowspan="4" |[[File:Regular_star_figure_2(15,2).svg|100px]]<br>{30/4}=2{15/2} | rowspan="4" | | rowspan="4" |[[File:Regular_star_polygon_30-11.svg|100px]]<br>{30/11} |120° | rowspan="4" |<math>r_{11}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | |48° |132° |- style="background: palegreen;" | | rowspan="4" |<math>r_5</math> |60° | rowspan="4" |[[File:Regular_star_figure_5(6,1).svg|100px]]<br>{30/5}=5{6} | rowspan="4" |[[File:V2 dodecahedron.png|100px]]<br>Dodecahedron | rowspan="4" |[[File:Regular_star_figure_10(3,1).svg|100px]]<br>{30/10}=10{3} |120° | rowspan="4" |<math>r_{10}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | |60° |120° |- style="background: yellow;" | | rowspan="4" |<math>r_{6}</math> |72° | rowspan="4" |[[File:Regular_star_figure_6(5,1).svg|100px]]<br>{30/6}=6{5} | rowspan="4" |[[File:V3 icosahedron.png|100px]]<br>Icosahedron | rowspan="4" |[[File:Regular_star_figure_3(10,3).svg|100px]]<br>{30/9}=3{10/3} |108° | rowspan="4" |<math>r_{9}</math> |- style="background: yellow;" | |{{radic|1.382~}} |{{radic|2.618~}} |- style="background: yellow;" | |1.176~ |1.618~ |- style="background: yellow;" | |72° |108° |- style="background: seashell;" | | rowspan="4" |<math>r_{7}</math> |90° | rowspan="4" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} | rowspan="4" |[[File:V4 icosidodecahedron.png|100px]]<br>Icosidodecahedron | rowspan="4" |[[File:Regular_star_figure_2(15,4).svg|100px]]<br>{30/8}=2{15/4} |90° | rowspan="4" |<math>r_{8}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |- style="background: seashell;" | |84° |96° |} The list of 600-cell chords <math>r_{i}</math> can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. The short chord and long chord each have their characteristic {30/n}-gon. Each row identifies a discrete isoclinic rotation of the 600-cell in invariant central planes containing the edges of the short chord {30}-gon, over the isocline chords of the long chord {30}-gon, the rotation's Clifford polygon. Each distinct pair of complementary chord lengths is identified with a distinct [[w:600-cell#Polyhedral sections|polyhedral section of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance <math>\phi^{-1}</math> is a icosahedron vertex figure, and the largest section at radial distance <math>\sqrt{2}</math> is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. Because [[w:3-sphere|<math>\mathbb{S}^3</math>]] is spherical, at radial distances greater than <math>\sqrt{2}</math> the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance <math>\sqrt{2+\phi}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>\sqrt{2}</math></small>]] We can rotate the 600-cell isoclinically in the great square rotation characteristic of the 16-cell, with the same effect on 15 disjoint 16-cells. Each 90° displacement takes 15 pairs of completely orthogonal invariant great square planes to each other. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The rotational curve over each 90° chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> form a circular fibration of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once. The 600-cell has another distinct isoclinic rotation in invariant great square planes, which has period 30 and visits every vertex of a 600-cell Petrie polygon. This [''great square left rotation characteristic of the 600-cell]'' takes place over <math>r_7</math> edge chords and <math>r_8</math> isocline chords. The {30/7} edge polygon is a skew helix of circumference <math>14\pi</math> with each <math>r_7</math> edge belonging to a distinct great square. The four {30/7} polygrams contribute one edge each to 30 great squares. Each 90° displacement takes every 16-cell to another 16-cell. The vertices of the invariant great squares each make seven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 90° {30/7} edge makes seven 12° turns. Four Clifford parallel {30/7} geodesics of circumference <math>16\pi</math> form a circular quadruple helix that intersects each 600-cell vertex once. The {30/8}=2{15/4} Clifford polygon is a compound of two skew {15/4} pentadecagrams of circumference <math>16\pi</math> with each <math>r_8</math> isocline chord belonging to a distinct 16-cell. The four {30/8} polygrams contribute one edge each to 30 great squares. The rotational curve over each 90° {30/8} isocline chord makes eight 12° turns. Four Clifford parallel {30/8} geodesics of circumference <math>16\pi</math> form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>\sqrt{3}</math></small> ]] We can rotate the 600-cell isoclinically in the great hexagon rotation characteristic of the 24-cell, over <math>\sqrt{1}</math> edge chords and <math>\sqrt{3}</math> isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° revolution each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> form a circular fibration of ten twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. The 600-cell has another distinct isoclinic rotation in invariant great hexagon planes, over <math>r_{4}=\sqrt{1}</math> edge chords and <math>r_{11}=\sqrt{3}</math> isocline chords This [''invariant great hexagon left rotation characteristic of the 600-cell]'' has period 30 and visits every vertex of a 600-cell Petrie polygon. Its {30/11} Clifford polygon is a skew helix where each <math>r_{11}</math> isocline chord is the <math>\sqrt{3}</math> diagonal of a great hexagon of a distinct 24-cell. The vertices of the invariant great hexagons of this rotation each make eleven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 120° <math>r_{11}</math> isocline chord makes eleven 12° turns. Four Clifford parallel {30/11} geodesic isoclines of circumference <math>22\pi</math> over <math>r_{11}</math> chords form a circular quadruple helix that intersects each 600-cell vertex once. We can rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its 36° <math>r_{3}</math> edges, over 144° <math>r_{12}</math> isocline chords. This ''invariant great decagon rotation characteristic of the 600-cell'' has period 5 and takes disjoint 24-cells to each other. The rotational curve over each <math>r_{12}</math> chord of its {5/2} Clifford polygon makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference <math>4\pi</math> over five <math>r_{12}</math> chords form a circular fibration of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. The rotation of the 600-cell 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. The 24-cells revolve within the 600-cell, as the 16-cells revolve within the 24-cells. All 120 vertices move at once on four Clifford parallel geodesic isoclines, displaced 144° in different directions. The 600-cell has another distinct isoclinic rotation in invariant great decagon planes containing its 36° <math>r_{2}</math> edges, over 144° <math>r_{13}</math> isocline chords. This [''great decagon left rotation characteristic of the 600-cell]'' has period 30 and visits every vertex of a 600-cell Petrie polygon. The rotational curve over each 144° <math>r_{13}</math> isocline chord makes thirteen 12° turns. Four Clifford parallel {30/13} geodesic isoclines of circumference <math>26\pi</math> form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} == The 5-cell 4-simplex == {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="9" |30 chords (15 180° pairs) make 15 distinct section polyhedra |- ! colspan="3" |Short chord ! Section ! colspan="3" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px]]<br>{30/15}=15{2} |180° | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} |{{radic|4}} |- style="background: palegreen;" | |0 |2 |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° | rowspan="3" |[[File:Regular_polygon_30.svg|100px]]<br>{30/1} | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px]]<br>{30/14} |164.5~° | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} |{{radic|3.927~}} |- style="background: palegreen;" | |0.270~ |1.982~ |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px]]<br>{30/2}=2{15} | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px]]<br>{30/13} |154.8~° | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} |{{radic|3.809~}} |- style="background: gainsboro;" | |0.437~ |1.952~ |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px]]<br>{30/3}=3{10} | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px]]<br>{30/12}=6{5/2} |144° | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: yellow;" | |0.618~ |1.902~ |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |138.6~° | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} |{{radic|3.5}} |- style="background: gainsboro;" | |0.707~ |1.871~ |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px]]<br>{30/4}=2{15/2} | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px]]<br>{30/11} |135.5~° | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} |{{radic|3.427~}} |- style="background: palegreen;" | |0.757~ |1.851~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |130.9~° | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} |{{radic|3.309~}} |- style="background: gainsboro;" | |0.831~ |1.819~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° | rowspan="3" | | rowspan="3" | | rowspan="3" | |124° | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} |{{radic|3.118~}} |- style="background: gainsboro;" | |0.939~ |1.766~ |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px]]<br>{30/5}=5{6} | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px]]<br>{30/10}=10{3} |120° | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |113.9~° | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} |{{radic|2.809~}} |- style="background: gainsboro;" | |1.091~ |1.676~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |110.2~° | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} |{{radic|2.691~}} |- style="background: gainsboro;" | |1.144~ |1.640~ |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px]]<br>{30/6}=6{5} | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px]]<br>{30/9}=3{10/3} |108° | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} |{{radic|2.618~}} |- style="background: yellow;" | |1.176~ |1.618~ |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px]]<br>{30/8}=2{15/4} |104.5~° | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} |{{radic|2.5}} |- style="background: palegreen;" | |1.224~ |1.581~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |98.9~° | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} |{{radic|2.309~}} |- style="background: gainsboro;" | |1.300~ |1.520~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |95.5~° | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} |{{radic|2.191~}} |- style="background: gainsboro;" | |1.345~ |1.480~ |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} |90° | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |} The [[User:Dc.samizdat/Golden chords of the 120-cell#Thirty distinguished distances|list of thirty 120-cell chords]] <math>c_{t}</math> can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct [[w:120-cell#Concentric_hulls|polyhedral section of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance <math>c_1</math> is a tetrahedron vertex figure, and the largest section at radial distance <math>c_{15}</math> is a central section bisecting the 120-cell. Because [[w:3-sphere|<math>\mathbb{S}^3</math>]] is spherical, at radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance <math>c_{29}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. Only 8 of the 30 chords in the table occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. Since all those polytopes except the 5-cell occur in the 600-cell, and the 600-cell and the 120-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. ... {{Clear}} == Finally the 120-cell == The [[120-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{5,3,3\}</math></small>. It has 600 vertices, 1200 edges, 720 pentagon faces, and 120 dodecahedron cells. It is the four-dimensional analogue of the dodecahedron. The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. ... {{Clear}} == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.] The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} ks3mqslhrsox8ncv5p2qopt4fkf3pey 2818501 2818500 2026-07-19T00:30:18Z Dc.samizdat 2856930 /* The 24-cell */ 2818501 wikitext text/x-wiki = Golden chords of the 120-cell = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - June 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by 5² non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). Consequently 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which contains 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). Consequently the 8-point (16-cell) compounds by 3 times 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} == The 16-cell 4-orthoplex == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=\sqrt{2}+1</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. Their 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}+1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-1</math> . Over the third <math>r_3</math> chord it also moves a distance of <math>-1</math>. Fontaine and Hurley also demonstrated the significance of <math>1/r_i</math> in Steinbach's Diagonal Product Formula, which says that every chord length is the sum of certain smaller chord lengths. The smaller chords are certain diagonals of the same regular polygon of a smaller edge length, specifically edge length <math>1/r_i</math> rather than <math>1</math>. If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{3,3,4\}</math></small>. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <math>\sqrt{2}</math> edges except opposite pairs. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Each chord is a distinct 4-vector with a length and a direction. 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, where vertices circle over the chords of an <math>r_i</math> polygon. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. The angle between two <math>r_i</math> chords is <math>180^\circ - i \times 45^\circ</math>. [[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 {8/1} 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 the 16-cell form an ''edge polygon'' {8/2}=2{4}. The 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 <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> (it is <math>6\pi</math> in this case), and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> chords, each vertex makes twelve 45° 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 revolution 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. We shall refer to this isoclinic rotation as the ''great square rotation characteristic of the 16-cell'', and note once again that it is Fontaine and Hurley's counterclockwise rotation over the <math>r_3</math> {8/3} star polygon, which constructs <math>1/r_3</math>. == The 8-cell tesseract == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. [[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract <small><math>\{4,3,3\}</math></small> performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]] The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{4,3,3\}</math></small>. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedrons]]. 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 the great square rotation characteristic of the 16-cell, with the same effect on both alternate-position 16-cells. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords. They form a circular double helix which intersects each vertex of the tesseract once. The double helix is an 8-rung ladder twisted around 3 times, and bent into a circle in the fourth dimension with its ends joined. Each rung is a <math>\sqrt{3}</math> chord. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron. The 24-cell has the same chord set as the 4-hypercube tesseract: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]] The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters. The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords: :<math>r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The <math>r_1</math> and <math>r_5</math> chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. The <math>r_1</math> <small><math>\sqrt{1}</math></small> chords form a 8 Petrie dodecagons which zig-zag back and forth, in the left and right rotational directions, between two Clifford parallel great hexagons formed by <math>r_2</math> chords. The 8 Petrie dodecagons can be divided four ways into 2 disjoint dodecagons {24/2}=2{12}. The <math>r_2</math> <small><math>\sqrt{1}</math></small> chords form 16 great hexagons, which can be divided four different ways into 4 Clifford parallel great hexagons {24/4}=4{6}. The <math>r_3</math> <small><math>\sqrt{2}</math></small> chords form 18 great squares, which can be divided three different ways into 6 Clifford parallel great squares {24/6}=6{4}, including one pair of completely orthogonal great squares from each of the three 16-cells. The <math>r_4</math> <small><math>\sqrt{3}</math></small> chords form 32 great triangles, which can be divided four different ways into 8 disjoint great triangles {24/8}=8{3} inscribed in 4 Clifford parallel great hexagons. The <math>r_5</math> <small><math>\sqrt{3}</math></small> chords form 8 circular helix Clifford polygons, visible as a green {12/5} dodecagram in the orthogonal projection. An isoclinic rotation of the 24-cell in 4 invariant <math>r_2</math> hexagon planes moves the vertices along 2 Clifford parallel circular isoclines {24/2}=2{12/5} over <math>r_5</math> chords. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges of Clifford parallel great squares, also isocline chords in great square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in great hexagon rotations. The green {12/5} dodecagram is a Clifford polygon.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} shows three octagram isoclines of <small><math>\sqrt{2}</math> </small>chords in the 24-cell]] We can rotate the 24-cell isoclinically in 6 Clifford parallel invariant great square planes containing 16-cell edges, in the great square rotation characteristic of the 16-cell, with the same effect on all three 16-cells. In 720° each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The rotational curve over each 90° <small><math>\sqrt{2}</math></small> chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <small><math>\sqrt{2}</math></small> chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. The triple helix is an 8-step circular staircase that twists around 3 times, and is bent into a torus in the fourth dimension. Each staircase step is a great triangle of <small><math>\sqrt{3}</math></small> chords. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} shows 2 dodecagram isoclines of <small><math>\sqrt{3}</math></small> chords in the 24-cell]]We can rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing 24-cell edges, over <math>r_{5}</math> isocline chords. This is the ''great hexagon rotation characteristic of the 24-cell'', also Fontaine and Hurley's counterclockwise rotation over the <math>r_5</math> {12/5} star polygon which constructs <math>1/r_5</math>. A complete 24-cell great circle edge plane revolution requires 720° like a complete 16-cell great circle edge plane revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. An isoclinic rotation by 60° in an invariant central plane containing a 24-cell edge takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as 4 invariant great hexagon planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square. All 24 vertices move at once on Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 12 <math>r_5</math> <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane 5 times in a moving invariant rotation plane. The rotational curve over each <math>r_5</math> 120° chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the course of a 720° revolution each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="6" |6 distinct 180° chord pairs make 6 distinct isoclinic rotations |- ! colspan="3" |Edge chord ! colspan="3" |Isocline chord |- style="background: gainsboro;" | | rowspan="4" |<math>t_1</math> |60° | rowspan="4" |[[File:Regular_polygon_24.svg|100px]]<br>{24/1}={24} | rowspan="4" |[[File:Regular_star_polygon_24-11.svg|100px]]<br>{24/11} |120° | rowspan="4" |<math>t_{11}</math> |- style="background: gainsboro;" | |{{radic|1}} |{{radic|3}} |- style="background: gainsboro;" | |1 |1.732~ |- style="background: gainsboro;" | |15° |165° |- style="background: palegreen;" | | rowspan="4" |<math>t_2</math> |60° | rowspan="4" |[[File:Regular_star_figure_2(12,1).svg|100px]]<br>{24/2}=2{12} | rowspan="4" |[[File:Regular_star_figure_2(12,5).svg|100px]]<br>{24/10}=2{12/5} |120° | rowspan="4" |<math>t_{10}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | |30° |150° |- style="background: seashell;" | | rowspan="4" |<math>t_3</math> |90° | rowspan="4" |[[File:Regular_star_figure_3(8,1).svg|100px]]<br>{24/3}=3{8} | rowspan="4" |[[File:Regular_star_figure_3(8,3).svg|100px]]<br>{24/9}=3{8/3} |90° | rowspan="4" |<math>t_{9}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |- style="background: seashell;" | |45° |135° |- style="background: palegreen;" | | rowspan="4" |<math>t_4</math> |60° | rowspan="4" |[[File:Regular_star_figure_4(6,1).svg|100px]]<br>{24/4}=4{6} | rowspan="4" |[[File:Regular_star_figure_8(3,1).svg|100px]]<br>{24/8}=8{3} |120° | rowspan="4" |<math>t_{8}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | |60° |120° |- style="background: gainsboro;" | | rowspan="4" |<math>t_5</math> |60° | rowspan="4" |[[File:Regular_star_polygon_24-5.svg|100px]]<br>{24/5} | rowspan="4" |[[File:Regular_star_polygon_24-7.svg|100px]]<br>{24/7} |120° | rowspan="4" |<math>t_{7}</math> |- style="background: gainsboro;" | |{{radic|1}} |{{radic|3}} |- style="background: gainsboro;" | |1 |1.732~ |- style="background: gainsboro;" | |75° |105° |- style="background: seashell;" | | rowspan="4" |<math>t_6</math> |90° | rowspan="4" |[[File:Regular_star_figure_6(4,1).svg|100px]]<br>{24/6}=6{4} | rowspan="4" |[[File:Regular_star_figure_6(4,1).svg|100px]]<br>{24/6}=6{4} |90° | rowspan="4" |<math>t_{6}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |- style="background: seashell;" | |90° |90° |} By examining the chords <math>r_i</math> of the 24-cell's Petrie {12}-gon we have found two distinct isoclinic rotations, the great square rotation characteristic of the 16-cell and the great hexagon rotation characteristic of the 24-cell. If we examine the chords <math>t_i</math> of the 24-cell's {24}-gon we find these, and also four other distinct isoclinic rotations. Each row of the table is a distinct isoclinic rotation of the 24-cell characterized by a pair of chords that sum to 180°. The 24 edge chords form the rotation's edge {24}-gon, and lie in invariant planes of the rotation. The 24 isocline chords form the rotation's Clifford {24}-gon and lie in invariant planes completely orthogonal to the corresponding edge chord's invariant plane. The rotational angle between successive edge chords and the rotational angle between successive isocline chords also sum to 180°. ... {{Clear}} == The 600-cell == [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are: [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</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. {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="7" |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra |- ! colspan="3" |Short edge chord ! Section ! colspan="3" |Long isocline chord |- style="background: palegreen;" | | rowspan="4" |<math>r_0</math> |0° | rowspan="4" | | rowspan="4" | | rowspan="4" |[[File:Regular_star_figure_15(2,1).svg|100px]]<br>{30/15}=15{2} |180° | rowspan="4" |<math>r_{15}</math> |- style="background: palegreen;" | |{{radic|0}} |{{radic|4}} |- style="background: palegreen;" | |0 |2 |- style="background: palegreen;" | |0° |180° |- style="background: palegreen;" | | rowspan="4" |<math>r_1</math> |36° | rowspan="4" |[[File:Regular_polygon_30.svg|100px]]<br>{30/1} | rowspan="4" | | rowspan="4" |[[File:Regular_star_figure_2(15,7).svg|100px]]<br>{30/14}=2{15/7} |144° | rowspan="4" |<math>r_{14}</math> |- style="background: palegreen;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: palegreen;" | |0.618~ |1.902~ |- style="background: palegreen;" | |12° |168° |- style="background: gainsboro;" | | rowspan="4" |<math>r_2</math> |36° | rowspan="4" |[[File:Regular_star_figure_2(15,1).svg|100px]]<br>{30/2}=2{15} | rowspan="4" | | rowspan="4" |[[File:Regular_star_polygon_30-13.svg|100px]]<br>{30/13} |144° | rowspan="4" |<math>r_{13}</math> |- style="background: gainsboro;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: gainsboro;" | |0.618~ |1.902~ |- style="background: gainsboro;" | |24° |156° |- style="background: yellow;" | | rowspan="4" |<math>r_3</math> |36° | rowspan="4" |[[File:Regular_star_figure_3(10,1).svg|100px]]<br>{30/3}=3{10} | rowspan="4" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="4" |[[File:Regular_star_figure_6(5,2).svg|100px]]<br>{30/12}=6{5/2} |144° | rowspan="4" |<math>r_{12}</math> |- style="background: yellow;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: yellow;" | |0.618~ |1.902~ |- style="background: yellow;" | |36° |144° |- style="background: palegreen;" | | rowspan="4" |<math>r_4</math> |60° | rowspan="4" |[[File:Regular_star_figure_2(15,2).svg|100px]]<br>{30/4}=2{15/2} | rowspan="4" | | rowspan="4" |[[File:Regular_star_polygon_30-11.svg|100px]]<br>{30/11} |120° | rowspan="4" |<math>r_{11}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | |48° |132° |- style="background: palegreen;" | | rowspan="4" |<math>r_5</math> |60° | rowspan="4" |[[File:Regular_star_figure_5(6,1).svg|100px]]<br>{30/5}=5{6} | rowspan="4" |[[File:V2 dodecahedron.png|100px]]<br>Dodecahedron | rowspan="4" |[[File:Regular_star_figure_10(3,1).svg|100px]]<br>{30/10}=10{3} |120° | rowspan="4" |<math>r_{10}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | |60° |120° |- style="background: yellow;" | | rowspan="4" |<math>r_{6}</math> |72° | rowspan="4" |[[File:Regular_star_figure_6(5,1).svg|100px]]<br>{30/6}=6{5} | rowspan="4" |[[File:V3 icosahedron.png|100px]]<br>Icosahedron | rowspan="4" |[[File:Regular_star_figure_3(10,3).svg|100px]]<br>{30/9}=3{10/3} |108° | rowspan="4" |<math>r_{9}</math> |- style="background: yellow;" | |{{radic|1.382~}} |{{radic|2.618~}} |- style="background: yellow;" | |1.176~ |1.618~ |- style="background: yellow;" | |72° |108° |- style="background: seashell;" | | rowspan="4" |<math>r_{7}</math> |90° | rowspan="4" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} | rowspan="4" |[[File:V4 icosidodecahedron.png|100px]]<br>Icosidodecahedron | rowspan="4" |[[File:Regular_star_figure_2(15,4).svg|100px]]<br>{30/8}=2{15/4} |90° | rowspan="4" |<math>r_{8}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |- style="background: seashell;" | |84° |96° |} The list of 600-cell chords <math>r_{i}</math> can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. The short chord and long chord each have their characteristic {30/n}-gon. Each row identifies a discrete isoclinic rotation of the 600-cell in invariant central planes containing the edges of the short chord {30}-gon, over the isocline chords of the long chord {30}-gon, the rotation's Clifford polygon. Each distinct pair of complementary chord lengths is identified with a distinct [[w:600-cell#Polyhedral sections|polyhedral section of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance <math>\phi^{-1}</math> is a icosahedron vertex figure, and the largest section at radial distance <math>\sqrt{2}</math> is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. Because [[w:3-sphere|<math>\mathbb{S}^3</math>]] is spherical, at radial distances greater than <math>\sqrt{2}</math> the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance <math>\sqrt{2+\phi}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>\sqrt{2}</math></small>]] We can rotate the 600-cell isoclinically in the great square rotation characteristic of the 16-cell, with the same effect on 15 disjoint 16-cells. Each 90° displacement takes 15 pairs of completely orthogonal invariant great square planes to each other. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The rotational curve over each 90° chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> form a circular fibration of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once. The 600-cell has another distinct isoclinic rotation in invariant great square planes, which has period 30 and visits every vertex of a 600-cell Petrie polygon. This [''great square left rotation characteristic of the 600-cell]'' takes place over <math>r_7</math> edge chords and <math>r_8</math> isocline chords. The {30/7} edge polygon is a skew helix of circumference <math>14\pi</math> with each <math>r_7</math> edge belonging to a distinct great square. The four {30/7} polygrams contribute one edge each to 30 great squares. Each 90° displacement takes every 16-cell to another 16-cell. The vertices of the invariant great squares each make seven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 90° {30/7} edge makes seven 12° turns. Four Clifford parallel {30/7} geodesics of circumference <math>16\pi</math> form a circular quadruple helix that intersects each 600-cell vertex once. The {30/8}=2{15/4} Clifford polygon is a compound of two skew {15/4} pentadecagrams of circumference <math>16\pi</math> with each <math>r_8</math> isocline chord belonging to a distinct 16-cell. The four {30/8} polygrams contribute one edge each to 30 great squares. The rotational curve over each 90° {30/8} isocline chord makes eight 12° turns. Four Clifford parallel {30/8} geodesics of circumference <math>16\pi</math> form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>\sqrt{3}</math></small> ]] We can rotate the 600-cell isoclinically in the great hexagon rotation characteristic of the 24-cell, over <math>\sqrt{1}</math> edge chords and <math>\sqrt{3}</math> isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° revolution each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> form a circular fibration of ten twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. The 600-cell has another distinct isoclinic rotation in invariant great hexagon planes, over <math>r_{4}=\sqrt{1}</math> edge chords and <math>r_{11}=\sqrt{3}</math> isocline chords This [''invariant great hexagon left rotation characteristic of the 600-cell]'' has period 30 and visits every vertex of a 600-cell Petrie polygon. Its {30/11} Clifford polygon is a skew helix where each <math>r_{11}</math> isocline chord is the <math>\sqrt{3}</math> diagonal of a great hexagon of a distinct 24-cell. The vertices of the invariant great hexagons of this rotation each make eleven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 120° <math>r_{11}</math> isocline chord makes eleven 12° turns. Four Clifford parallel {30/11} geodesic isoclines of circumference <math>22\pi</math> over <math>r_{11}</math> chords form a circular quadruple helix that intersects each 600-cell vertex once. We can rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its 36° <math>r_{3}</math> edges, over 144° <math>r_{12}</math> isocline chords. This ''invariant great decagon rotation characteristic of the 600-cell'' has period 5 and takes disjoint 24-cells to each other. The rotational curve over each <math>r_{12}</math> chord of its {5/2} Clifford polygon makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference <math>4\pi</math> over five <math>r_{12}</math> chords form a circular fibration of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. The rotation of the 600-cell 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. The 24-cells revolve within the 600-cell, as the 16-cells revolve within the 24-cells. All 120 vertices move at once on four Clifford parallel geodesic isoclines, displaced 144° in different directions. The 600-cell has another distinct isoclinic rotation in invariant great decagon planes containing its 36° <math>r_{2}</math> edges, over 144° <math>r_{13}</math> isocline chords. This [''great decagon left rotation characteristic of the 600-cell]'' has period 30 and visits every vertex of a 600-cell Petrie polygon. The rotational curve over each 144° <math>r_{13}</math> isocline chord makes thirteen 12° turns. Four Clifford parallel {30/13} geodesic isoclines of circumference <math>26\pi</math> form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} == The 5-cell 4-simplex == {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="9" |30 chords (15 180° pairs) make 15 distinct section polyhedra |- ! colspan="3" |Short chord ! Section ! colspan="3" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px]]<br>{30/15}=15{2} |180° | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} |{{radic|4}} |- style="background: palegreen;" | |0 |2 |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° | rowspan="3" |[[File:Regular_polygon_30.svg|100px]]<br>{30/1} | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px]]<br>{30/14} |164.5~° | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} |{{radic|3.927~}} |- style="background: palegreen;" | |0.270~ |1.982~ |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px]]<br>{30/2}=2{15} | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px]]<br>{30/13} |154.8~° | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} |{{radic|3.809~}} |- style="background: gainsboro;" | |0.437~ |1.952~ |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px]]<br>{30/3}=3{10} | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px]]<br>{30/12}=6{5/2} |144° | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: yellow;" | |0.618~ |1.902~ |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |138.6~° | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} |{{radic|3.5}} |- style="background: gainsboro;" | |0.707~ |1.871~ |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px]]<br>{30/4}=2{15/2} | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px]]<br>{30/11} |135.5~° | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} |{{radic|3.427~}} |- style="background: palegreen;" | |0.757~ |1.851~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |130.9~° | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} |{{radic|3.309~}} |- style="background: gainsboro;" | |0.831~ |1.819~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° | rowspan="3" | | rowspan="3" | | rowspan="3" | |124° | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} |{{radic|3.118~}} |- style="background: gainsboro;" | |0.939~ |1.766~ |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px]]<br>{30/5}=5{6} | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px]]<br>{30/10}=10{3} |120° | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |113.9~° | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} |{{radic|2.809~}} |- style="background: gainsboro;" | |1.091~ |1.676~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |110.2~° | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} |{{radic|2.691~}} |- style="background: gainsboro;" | |1.144~ |1.640~ |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px]]<br>{30/6}=6{5} | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px]]<br>{30/9}=3{10/3} |108° | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} |{{radic|2.618~}} |- style="background: yellow;" | |1.176~ |1.618~ |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px]]<br>{30/8}=2{15/4} |104.5~° | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} |{{radic|2.5}} |- style="background: palegreen;" | |1.224~ |1.581~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |98.9~° | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} |{{radic|2.309~}} |- style="background: gainsboro;" | |1.300~ |1.520~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |95.5~° | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} |{{radic|2.191~}} |- style="background: gainsboro;" | |1.345~ |1.480~ |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} |90° | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |} The [[User:Dc.samizdat/Golden chords of the 120-cell#Thirty distinguished distances|list of thirty 120-cell chords]] <math>c_{t}</math> can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct [[w:120-cell#Concentric_hulls|polyhedral section of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance <math>c_1</math> is a tetrahedron vertex figure, and the largest section at radial distance <math>c_{15}</math> is a central section bisecting the 120-cell. Because [[w:3-sphere|<math>\mathbb{S}^3</math>]] is spherical, at radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance <math>c_{29}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. Only 8 of the 30 chords in the table occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. Since all those polytopes except the 5-cell occur in the 600-cell, and the 600-cell and the 120-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. ... {{Clear}} == Finally the 120-cell == The [[120-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{5,3,3\}</math></small>. It has 600 vertices, 1200 edges, 720 pentagon faces, and 120 dodecahedron cells. It is the four-dimensional analogue of the dodecahedron. The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. ... {{Clear}} == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.] The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} jh9zod2ly1yn2ibyarnmiqbbmmoxmu4 2818502 2818501 2026-07-19T00:34:10Z Dc.samizdat 2856930 /* The 600-cell */ 2818502 wikitext text/x-wiki = Golden chords of the 120-cell = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - June 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by 5² non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). Consequently 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which contains 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). Consequently the 8-point (16-cell) compounds by 3 times 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} == The 16-cell 4-orthoplex == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=\sqrt{2}+1</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. Their 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}+1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-1</math> . Over the third <math>r_3</math> chord it also moves a distance of <math>-1</math>. Fontaine and Hurley also demonstrated the significance of <math>1/r_i</math> in Steinbach's Diagonal Product Formula, which says that every chord length is the sum of certain smaller chord lengths. The smaller chords are certain diagonals of the same regular polygon of a smaller edge length, specifically edge length <math>1/r_i</math> rather than <math>1</math>. If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{3,3,4\}</math></small>. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <math>\sqrt{2}</math> edges except opposite pairs. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Each chord is a distinct 4-vector with a length and a direction. 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, where vertices circle over the chords of an <math>r_i</math> polygon. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. The angle between two <math>r_i</math> chords is <math>180^\circ - i \times 45^\circ</math>. [[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 {8/1} 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 the 16-cell form an ''edge polygon'' {8/2}=2{4}. The 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 <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> (it is <math>6\pi</math> in this case), and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> chords, each vertex makes twelve 45° 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 revolution 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. We shall refer to this isoclinic rotation as the ''great square rotation characteristic of the 16-cell'', and note once again that it is Fontaine and Hurley's counterclockwise rotation over the <math>r_3</math> {8/3} star polygon, which constructs <math>1/r_3</math>. == The 8-cell tesseract == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. [[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract <small><math>\{4,3,3\}</math></small> performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]] The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{4,3,3\}</math></small>. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedrons]]. 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 the great square rotation characteristic of the 16-cell, with the same effect on both alternate-position 16-cells. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords. They form a circular double helix which intersects each vertex of the tesseract once. The double helix is an 8-rung ladder twisted around 3 times, and bent into a circle in the fourth dimension with its ends joined. Each rung is a <math>\sqrt{3}</math> chord. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron. The 24-cell has the same chord set as the 4-hypercube tesseract: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]] The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters. The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords: :<math>r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The <math>r_1</math> and <math>r_5</math> chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. The <math>r_1</math> <small><math>\sqrt{1}</math></small> chords form a 8 Petrie dodecagons which zig-zag back and forth, in the left and right rotational directions, between two Clifford parallel great hexagons formed by <math>r_2</math> chords. The 8 Petrie dodecagons can be divided four ways into 2 disjoint dodecagons {24/2}=2{12}. The <math>r_2</math> <small><math>\sqrt{1}</math></small> chords form 16 great hexagons, which can be divided four different ways into 4 Clifford parallel great hexagons {24/4}=4{6}. The <math>r_3</math> <small><math>\sqrt{2}</math></small> chords form 18 great squares, which can be divided three different ways into 6 Clifford parallel great squares {24/6}=6{4}, including one pair of completely orthogonal great squares from each of the three 16-cells. The <math>r_4</math> <small><math>\sqrt{3}</math></small> chords form 32 great triangles, which can be divided four different ways into 8 disjoint great triangles {24/8}=8{3} inscribed in 4 Clifford parallel great hexagons. The <math>r_5</math> <small><math>\sqrt{3}</math></small> chords form 8 circular helix Clifford polygons, visible as a green {12/5} dodecagram in the orthogonal projection. An isoclinic rotation of the 24-cell in 4 invariant <math>r_2</math> hexagon planes moves the vertices along 2 Clifford parallel circular isoclines {24/2}=2{12/5} over <math>r_5</math> chords. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges of Clifford parallel great squares, also isocline chords in great square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in great hexagon rotations. The green {12/5} dodecagram is a Clifford polygon.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} shows three octagram isoclines of <small><math>\sqrt{2}</math> </small>chords in the 24-cell]] We can rotate the 24-cell isoclinically in 6 Clifford parallel invariant great square planes containing 16-cell edges, in the great square rotation characteristic of the 16-cell, with the same effect on all three 16-cells. In 720° each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The rotational curve over each 90° <small><math>\sqrt{2}</math></small> chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <small><math>\sqrt{2}</math></small> chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. The triple helix is an 8-step circular staircase that twists around 3 times, and is bent into a torus in the fourth dimension. Each staircase step is a great triangle of <small><math>\sqrt{3}</math></small> chords. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} shows 2 dodecagram isoclines of <small><math>\sqrt{3}</math></small> chords in the 24-cell]]We can rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing 24-cell edges, over <math>r_{5}</math> isocline chords. This is the ''great hexagon rotation characteristic of the 24-cell'', also Fontaine and Hurley's counterclockwise rotation over the <math>r_5</math> {12/5} star polygon which constructs <math>1/r_5</math>. A complete 24-cell great circle edge plane revolution requires 720° like a complete 16-cell great circle edge plane revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. An isoclinic rotation by 60° in an invariant central plane containing a 24-cell edge takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as 4 invariant great hexagon planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square. All 24 vertices move at once on Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 12 <math>r_5</math> <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane 5 times in a moving invariant rotation plane. The rotational curve over each <math>r_5</math> 120° chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the course of a 720° revolution each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="6" |6 distinct 180° chord pairs make 6 distinct isoclinic rotations |- ! colspan="3" |Edge chord ! colspan="3" |Isocline chord |- style="background: gainsboro;" | | rowspan="4" |<math>t_1</math> |60° | rowspan="4" |[[File:Regular_polygon_24.svg|100px]]<br>{24/1}={24} | rowspan="4" |[[File:Regular_star_polygon_24-11.svg|100px]]<br>{24/11} |120° | rowspan="4" |<math>t_{11}</math> |- style="background: gainsboro;" | |{{radic|1}} |{{radic|3}} |- style="background: gainsboro;" | |1 |1.732~ |- style="background: gainsboro;" | |15° |165° |- style="background: palegreen;" | | rowspan="4" |<math>t_2</math> |60° | rowspan="4" |[[File:Regular_star_figure_2(12,1).svg|100px]]<br>{24/2}=2{12} | rowspan="4" |[[File:Regular_star_figure_2(12,5).svg|100px]]<br>{24/10}=2{12/5} |120° | rowspan="4" |<math>t_{10}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | |30° |150° |- style="background: seashell;" | | rowspan="4" |<math>t_3</math> |90° | rowspan="4" |[[File:Regular_star_figure_3(8,1).svg|100px]]<br>{24/3}=3{8} | rowspan="4" |[[File:Regular_star_figure_3(8,3).svg|100px]]<br>{24/9}=3{8/3} |90° | rowspan="4" |<math>t_{9}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |- style="background: seashell;" | |45° |135° |- style="background: palegreen;" | | rowspan="4" |<math>t_4</math> |60° | rowspan="4" |[[File:Regular_star_figure_4(6,1).svg|100px]]<br>{24/4}=4{6} | rowspan="4" |[[File:Regular_star_figure_8(3,1).svg|100px]]<br>{24/8}=8{3} |120° | rowspan="4" |<math>t_{8}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | |60° |120° |- style="background: gainsboro;" | | rowspan="4" |<math>t_5</math> |60° | rowspan="4" |[[File:Regular_star_polygon_24-5.svg|100px]]<br>{24/5} | rowspan="4" |[[File:Regular_star_polygon_24-7.svg|100px]]<br>{24/7} |120° | rowspan="4" |<math>t_{7}</math> |- style="background: gainsboro;" | |{{radic|1}} |{{radic|3}} |- style="background: gainsboro;" | |1 |1.732~ |- style="background: gainsboro;" | |75° |105° |- style="background: seashell;" | | rowspan="4" |<math>t_6</math> |90° | rowspan="4" |[[File:Regular_star_figure_6(4,1).svg|100px]]<br>{24/6}=6{4} | rowspan="4" |[[File:Regular_star_figure_6(4,1).svg|100px]]<br>{24/6}=6{4} |90° | rowspan="4" |<math>t_{6}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |- style="background: seashell;" | |90° |90° |} By examining the chords <math>r_i</math> of the 24-cell's Petrie {12}-gon we have found two distinct isoclinic rotations, the great square rotation characteristic of the 16-cell and the great hexagon rotation characteristic of the 24-cell. If we examine the chords <math>t_i</math> of the 24-cell's {24}-gon we find these, and also four other distinct isoclinic rotations. Each row of the table is a distinct isoclinic rotation of the 24-cell characterized by a pair of chords that sum to 180°. The 24 edge chords lie in invariant planes of the rotation. The 24 isocline chords form the rotation's Clifford {24}-gon and lie in invariant planes completely orthogonal to the corresponding edge chord's invariant plane. The rotational angle between successive edge chords and the rotational angle between successive isocline chords also sum to 180°. ... {{Clear}} == The 600-cell == [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are: [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</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. {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="7" |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra |- ! colspan="3" |Short edge chord ! Section ! colspan="3" |Long isocline chord |- style="background: palegreen;" | | rowspan="4" |<math>r_0</math> |0° | rowspan="4" | | rowspan="4" | | rowspan="4" |[[File:Regular_star_figure_15(2,1).svg|100px]]<br>{30/15}=15{2} |180° | rowspan="4" |<math>r_{15}</math> |- style="background: palegreen;" | |{{radic|0}} |{{radic|4}} |- style="background: palegreen;" | |0 |2 |- style="background: palegreen;" | |0° |180° |- style="background: palegreen;" | | rowspan="4" |<math>r_1</math> |36° | rowspan="4" |[[File:Regular_polygon_30.svg|100px]]<br>{30/1} | rowspan="4" | | rowspan="4" |[[File:Regular_star_figure_2(15,7).svg|100px]]<br>{30/14}=2{15/7} |144° | rowspan="4" |<math>r_{14}</math> |- style="background: palegreen;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: palegreen;" | |0.618~ |1.902~ |- style="background: palegreen;" | |12° |168° |- style="background: gainsboro;" | | rowspan="4" |<math>r_2</math> |36° | rowspan="4" |[[File:Regular_star_figure_2(15,1).svg|100px]]<br>{30/2}=2{15} | rowspan="4" | | rowspan="4" |[[File:Regular_star_polygon_30-13.svg|100px]]<br>{30/13} |144° | rowspan="4" |<math>r_{13}</math> |- style="background: gainsboro;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: gainsboro;" | |0.618~ |1.902~ |- style="background: gainsboro;" | |24° |156° |- style="background: yellow;" | | rowspan="4" |<math>r_3</math> |36° | rowspan="4" |[[File:Regular_star_figure_3(10,1).svg|100px]]<br>{30/3}=3{10} | rowspan="4" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="4" |[[File:Regular_star_figure_6(5,2).svg|100px]]<br>{30/12}=6{5/2} |144° | rowspan="4" |<math>r_{12}</math> |- style="background: yellow;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: yellow;" | |0.618~ |1.902~ |- style="background: yellow;" | |36° |144° |- style="background: palegreen;" | | rowspan="4" |<math>r_4</math> |60° | rowspan="4" |[[File:Regular_star_figure_2(15,2).svg|100px]]<br>{30/4}=2{15/2} | rowspan="4" | | rowspan="4" |[[File:Regular_star_polygon_30-11.svg|100px]]<br>{30/11} |120° | rowspan="4" |<math>r_{11}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | |48° |132° |- style="background: palegreen;" | | rowspan="4" |<math>r_5</math> |60° | rowspan="4" |[[File:Regular_star_figure_5(6,1).svg|100px]]<br>{30/5}=5{6} | rowspan="4" |[[File:V2 dodecahedron.png|100px]]<br>Dodecahedron | rowspan="4" |[[File:Regular_star_figure_10(3,1).svg|100px]]<br>{30/10}=10{3} |120° | rowspan="4" |<math>r_{10}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | |60° |120° |- style="background: yellow;" | | rowspan="4" |<math>r_{6}</math> |72° | rowspan="4" |[[File:Regular_star_figure_6(5,1).svg|100px]]<br>{30/6}=6{5} | rowspan="4" |[[File:V3 icosahedron.png|100px]]<br>Icosahedron | rowspan="4" |[[File:Regular_star_figure_3(10,3).svg|100px]]<br>{30/9}=3{10/3} |108° | rowspan="4" |<math>r_{9}</math> |- style="background: yellow;" | |{{radic|1.382~}} |{{radic|2.618~}} |- style="background: yellow;" | |1.176~ |1.618~ |- style="background: yellow;" | |72° |108° |- style="background: seashell;" | | rowspan="4" |<math>r_{7}</math> |90° | rowspan="4" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} | rowspan="4" |[[File:V4 icosidodecahedron.png|100px]]<br>Icosidodecahedron | rowspan="4" |[[File:Regular_star_figure_2(15,4).svg|100px]]<br>{30/8}=2{15/4} |90° | rowspan="4" |<math>r_{8}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |- style="background: seashell;" | |84° |96° |} The list of 600-cell chords <math>r_{i}</math> can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. The short chord and long chord each have their characteristic {30/n}-gon. Each row identifies a discrete isoclinic rotation of the 600-cell in invariant central planes containing the edges of the short chord {30}-gon, over the isocline chords of the long chord {30}-gon, the rotation's Clifford polygon. Each distinct pair of complementary chord lengths is identified with a distinct [[w:600-cell#Polyhedral sections|polyhedral section of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance <math>\phi^{-1}</math> is a icosahedron vertex figure, and the largest section at radial distance <math>\sqrt{2}</math> is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. Because [[w:3-sphere|<math>\mathbb{S}^3</math>]] is spherical, at radial distances greater than <math>\sqrt{2}</math> the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance <math>\sqrt{2+\phi}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>\sqrt{2}</math></small>]] We can rotate the 600-cell isoclinically in the great square rotation characteristic of the 16-cell, with the same effect on 15 disjoint 16-cells. Each 90° displacement takes 15 pairs of completely orthogonal invariant great square planes to each other. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The rotational curve over each 90° chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> form a circular fibration of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once. The 600-cell has another distinct isoclinic rotation in invariant great square planes, which has period 30 and visits every vertex of a 600-cell Petrie polygon. This [''great square left rotation characteristic of the 600-cell]'' takes place over <math>r_7</math> edge chords and <math>r_8</math> isocline chords. The {30/7} edge polygon is a skew helix of circumference <math>14\pi</math> with each <math>r_7</math> edge belonging to a distinct great square. The four {30/7} polygrams contribute one edge each to 30 great squares. Each 90° displacement takes every 16-cell to another 16-cell. The vertices of the invariant great squares each make seven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 90° {30/7} edge makes seven 12° turns. Four Clifford parallel {30/7} geodesics of circumference <math>16\pi</math> form a circular quadruple helix that intersects each 600-cell vertex once. The {30/8}=2{15/4} Clifford polygon is a compound of two skew {15/4} pentadecagrams of circumference <math>16\pi</math> with each <math>r_8</math> isocline chord belonging to a distinct 16-cell. The four {30/8} polygrams contribute one edge each to 30 great squares. The rotational curve over each 90° {30/8} isocline chord makes eight 12° turns. Four Clifford parallel {30/8} geodesics of circumference <math>16\pi</math> form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>\sqrt{3}</math></small> ]] We can rotate the 600-cell isoclinically in the great hexagon rotation characteristic of the 24-cell, over <math>\sqrt{1}</math> edge chords and <math>\sqrt{3}</math> isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° revolution each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> form a circular fibration of ten twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. The 600-cell has another distinct isoclinic rotation in invariant great hexagon planes, over <math>r_{4}=\sqrt{1}</math> edge chords and <math>r_{11}=\sqrt{3}</math> isocline chords This [''invariant great hexagon left rotation characteristic of the 600-cell]'' has period 30 and visits every vertex of a 600-cell Petrie polygon. Its {30/11} Clifford polygon is a skew helix where each <math>r_{11}</math> isocline chord is the <math>\sqrt{3}</math> diagonal of a great hexagon of a distinct 24-cell. The vertices of the invariant great hexagons of this rotation each make eleven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 120° <math>r_{11}</math> isocline chord makes eleven 12° turns. Four Clifford parallel {30/11} geodesic isoclines of circumference <math>22\pi</math> over <math>r_{11}</math> chords form a circular quadruple helix that intersects each 600-cell vertex once. We can rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its 36° <math>r_{3}</math> edges, over 144° <math>r_{12}</math> isocline chords. This ''invariant great decagon rotation characteristic of the 600-cell'' has period 5 and takes disjoint 24-cells to each other. The rotational curve over each <math>r_{12}</math> chord of its {5/2} Clifford polygon makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference <math>4\pi</math> over five <math>r_{12}</math> chords form a circular fibration of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. The rotation of the 600-cell 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. The 24-cells revolve within the 600-cell, as the 16-cells revolve within the 24-cells. All 120 vertices move at once on four Clifford parallel geodesic isoclines, displaced 144° in different directions. The 600-cell has another distinct isoclinic rotation in invariant great decagon planes containing its 36° <math>r_{2}</math> edges, over 144° <math>r_{13}</math> isocline chords. This [''great decagon left rotation characteristic of the 600-cell]'' has period 30 and visits every vertex of a 600-cell Petrie polygon. The rotational curve over each 144° <math>r_{13}</math> isocline chord makes thirteen 12° turns. Four Clifford parallel {30/13} geodesic isoclines of circumference <math>26\pi</math> form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} == The 5-cell 4-simplex == {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="9" |30 chords (15 180° pairs) make 15 distinct section polyhedra |- ! colspan="3" |Short chord ! Section ! colspan="3" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px]]<br>{30/15}=15{2} |180° | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} |{{radic|4}} |- style="background: palegreen;" | |0 |2 |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° | rowspan="3" |[[File:Regular_polygon_30.svg|100px]]<br>{30/1} | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px]]<br>{30/14} |164.5~° | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} |{{radic|3.927~}} |- style="background: palegreen;" | |0.270~ |1.982~ |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px]]<br>{30/2}=2{15} | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px]]<br>{30/13} |154.8~° | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} |{{radic|3.809~}} |- style="background: gainsboro;" | |0.437~ |1.952~ |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px]]<br>{30/3}=3{10} | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px]]<br>{30/12}=6{5/2} |144° | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: yellow;" | |0.618~ |1.902~ |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |138.6~° | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} |{{radic|3.5}} |- style="background: gainsboro;" | |0.707~ |1.871~ |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px]]<br>{30/4}=2{15/2} | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px]]<br>{30/11} |135.5~° | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} |{{radic|3.427~}} |- style="background: palegreen;" | |0.757~ |1.851~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |130.9~° | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} |{{radic|3.309~}} |- style="background: gainsboro;" | |0.831~ |1.819~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° | rowspan="3" | | rowspan="3" | | rowspan="3" | |124° | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} |{{radic|3.118~}} |- style="background: gainsboro;" | |0.939~ |1.766~ |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px]]<br>{30/5}=5{6} | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px]]<br>{30/10}=10{3} |120° | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |113.9~° | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} |{{radic|2.809~}} |- style="background: gainsboro;" | |1.091~ |1.676~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |110.2~° | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} |{{radic|2.691~}} |- style="background: gainsboro;" | |1.144~ |1.640~ |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px]]<br>{30/6}=6{5} | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px]]<br>{30/9}=3{10/3} |108° | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} |{{radic|2.618~}} |- style="background: yellow;" | |1.176~ |1.618~ |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px]]<br>{30/8}=2{15/4} |104.5~° | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} |{{radic|2.5}} |- style="background: palegreen;" | |1.224~ |1.581~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |98.9~° | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} |{{radic|2.309~}} |- style="background: gainsboro;" | |1.300~ |1.520~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |95.5~° | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} |{{radic|2.191~}} |- style="background: gainsboro;" | |1.345~ |1.480~ |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} |90° | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |} The [[User:Dc.samizdat/Golden chords of the 120-cell#Thirty distinguished distances|list of thirty 120-cell chords]] <math>c_{t}</math> can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct [[w:120-cell#Concentric_hulls|polyhedral section of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance <math>c_1</math> is a tetrahedron vertex figure, and the largest section at radial distance <math>c_{15}</math> is a central section bisecting the 120-cell. Because [[w:3-sphere|<math>\mathbb{S}^3</math>]] is spherical, at radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance <math>c_{29}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. Only 8 of the 30 chords in the table occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. Since all those polytopes except the 5-cell occur in the 600-cell, and the 600-cell and the 120-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. ... {{Clear}} == Finally the 120-cell == The [[120-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{5,3,3\}</math></small>. It has 600 vertices, 1200 edges, 720 pentagon faces, and 120 dodecahedron cells. It is the four-dimensional analogue of the dodecahedron. The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. ... {{Clear}} == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.] The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} 4ft09dtrrg3qv5hhk6cahv9kxxw4zmy 2818503 2818502 2026-07-19T00:37:59Z Dc.samizdat 2856930 /* The 600-cell */ 2818503 wikitext text/x-wiki = Golden chords of the 120-cell = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - June 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by 5² non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). Consequently 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which contains 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). Consequently the 8-point (16-cell) compounds by 3 times 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} == The 16-cell 4-orthoplex == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=\sqrt{2}+1</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. Their 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}+1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-1</math> . Over the third <math>r_3</math> chord it also moves a distance of <math>-1</math>. Fontaine and Hurley also demonstrated the significance of <math>1/r_i</math> in Steinbach's Diagonal Product Formula, which says that every chord length is the sum of certain smaller chord lengths. The smaller chords are certain diagonals of the same regular polygon of a smaller edge length, specifically edge length <math>1/r_i</math> rather than <math>1</math>. If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{3,3,4\}</math></small>. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <math>\sqrt{2}</math> edges except opposite pairs. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Each chord is a distinct 4-vector with a length and a direction. 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, where vertices circle over the chords of an <math>r_i</math> polygon. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. The angle between two <math>r_i</math> chords is <math>180^\circ - i \times 45^\circ</math>. [[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 {8/1} 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 the 16-cell form an ''edge polygon'' {8/2}=2{4}. The 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 <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> (it is <math>6\pi</math> in this case), and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> chords, each vertex makes twelve 45° 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 revolution 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. We shall refer to this isoclinic rotation as the ''great square rotation characteristic of the 16-cell'', and note once again that it is Fontaine and Hurley's counterclockwise rotation over the <math>r_3</math> {8/3} star polygon, which constructs <math>1/r_3</math>. == The 8-cell tesseract == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. [[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract <small><math>\{4,3,3\}</math></small> performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]] The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{4,3,3\}</math></small>. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedrons]]. 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 the great square rotation characteristic of the 16-cell, with the same effect on both alternate-position 16-cells. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords. They form a circular double helix which intersects each vertex of the tesseract once. The double helix is an 8-rung ladder twisted around 3 times, and bent into a circle in the fourth dimension with its ends joined. Each rung is a <math>\sqrt{3}</math> chord. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron. The 24-cell has the same chord set as the 4-hypercube tesseract: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]] The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters. The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords: :<math>r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The <math>r_1</math> and <math>r_5</math> chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. The <math>r_1</math> <small><math>\sqrt{1}</math></small> chords form a 8 Petrie dodecagons which zig-zag back and forth, in the left and right rotational directions, between two Clifford parallel great hexagons formed by <math>r_2</math> chords. The 8 Petrie dodecagons can be divided four ways into 2 disjoint dodecagons {24/2}=2{12}. The <math>r_2</math> <small><math>\sqrt{1}</math></small> chords form 16 great hexagons, which can be divided four different ways into 4 Clifford parallel great hexagons {24/4}=4{6}. The <math>r_3</math> <small><math>\sqrt{2}</math></small> chords form 18 great squares, which can be divided three different ways into 6 Clifford parallel great squares {24/6}=6{4}, including one pair of completely orthogonal great squares from each of the three 16-cells. The <math>r_4</math> <small><math>\sqrt{3}</math></small> chords form 32 great triangles, which can be divided four different ways into 8 disjoint great triangles {24/8}=8{3} inscribed in 4 Clifford parallel great hexagons. The <math>r_5</math> <small><math>\sqrt{3}</math></small> chords form 8 circular helix Clifford polygons, visible as a green {12/5} dodecagram in the orthogonal projection. An isoclinic rotation of the 24-cell in 4 invariant <math>r_2</math> hexagon planes moves the vertices along 2 Clifford parallel circular isoclines {24/2}=2{12/5} over <math>r_5</math> chords. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges of Clifford parallel great squares, also isocline chords in great square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in great hexagon rotations. The green {12/5} dodecagram is a Clifford polygon.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} shows three octagram isoclines of <small><math>\sqrt{2}</math> </small>chords in the 24-cell]] We can rotate the 24-cell isoclinically in 6 Clifford parallel invariant great square planes containing 16-cell edges, in the great square rotation characteristic of the 16-cell, with the same effect on all three 16-cells. In 720° each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The rotational curve over each 90° <small><math>\sqrt{2}</math></small> chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <small><math>\sqrt{2}</math></small> chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. The triple helix is an 8-step circular staircase that twists around 3 times, and is bent into a torus in the fourth dimension. Each staircase step is a great triangle of <small><math>\sqrt{3}</math></small> chords. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} shows 2 dodecagram isoclines of <small><math>\sqrt{3}</math></small> chords in the 24-cell]]We can rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing 24-cell edges, over <math>r_{5}</math> isocline chords. This is the ''great hexagon rotation characteristic of the 24-cell'', also Fontaine and Hurley's counterclockwise rotation over the <math>r_5</math> {12/5} star polygon which constructs <math>1/r_5</math>. A complete 24-cell great circle edge plane revolution requires 720° like a complete 16-cell great circle edge plane revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. An isoclinic rotation by 60° in an invariant central plane containing a 24-cell edge takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as 4 invariant great hexagon planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square. All 24 vertices move at once on Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 12 <math>r_5</math> <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane 5 times in a moving invariant rotation plane. The rotational curve over each <math>r_5</math> 120° chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the course of a 720° revolution each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="6" |6 distinct 180° chord pairs make 6 distinct isoclinic rotations |- ! colspan="3" |Edge chord ! colspan="3" |Isocline chord |- style="background: gainsboro;" | | rowspan="4" |<math>t_1</math> |60° | rowspan="4" |[[File:Regular_polygon_24.svg|100px]]<br>{24/1}={24} | rowspan="4" |[[File:Regular_star_polygon_24-11.svg|100px]]<br>{24/11} |120° | rowspan="4" |<math>t_{11}</math> |- style="background: gainsboro;" | |{{radic|1}} |{{radic|3}} |- style="background: gainsboro;" | |1 |1.732~ |- style="background: gainsboro;" | |15° |165° |- style="background: palegreen;" | | rowspan="4" |<math>t_2</math> |60° | rowspan="4" |[[File:Regular_star_figure_2(12,1).svg|100px]]<br>{24/2}=2{12} | rowspan="4" |[[File:Regular_star_figure_2(12,5).svg|100px]]<br>{24/10}=2{12/5} |120° | rowspan="4" |<math>t_{10}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | |30° |150° |- style="background: seashell;" | | rowspan="4" |<math>t_3</math> |90° | rowspan="4" |[[File:Regular_star_figure_3(8,1).svg|100px]]<br>{24/3}=3{8} | rowspan="4" |[[File:Regular_star_figure_3(8,3).svg|100px]]<br>{24/9}=3{8/3} |90° | rowspan="4" |<math>t_{9}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |- style="background: seashell;" | |45° |135° |- style="background: palegreen;" | | rowspan="4" |<math>t_4</math> |60° | rowspan="4" |[[File:Regular_star_figure_4(6,1).svg|100px]]<br>{24/4}=4{6} | rowspan="4" |[[File:Regular_star_figure_8(3,1).svg|100px]]<br>{24/8}=8{3} |120° | rowspan="4" |<math>t_{8}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | |60° |120° |- style="background: gainsboro;" | | rowspan="4" |<math>t_5</math> |60° | rowspan="4" |[[File:Regular_star_polygon_24-5.svg|100px]]<br>{24/5} | rowspan="4" |[[File:Regular_star_polygon_24-7.svg|100px]]<br>{24/7} |120° | rowspan="4" |<math>t_{7}</math> |- style="background: gainsboro;" | |{{radic|1}} |{{radic|3}} |- style="background: gainsboro;" | |1 |1.732~ |- style="background: gainsboro;" | |75° |105° |- style="background: seashell;" | | rowspan="4" |<math>t_6</math> |90° | rowspan="4" |[[File:Regular_star_figure_6(4,1).svg|100px]]<br>{24/6}=6{4} | rowspan="4" |[[File:Regular_star_figure_6(4,1).svg|100px]]<br>{24/6}=6{4} |90° | rowspan="4" |<math>t_{6}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |- style="background: seashell;" | |90° |90° |} By examining the chords <math>r_i</math> of the 24-cell's Petrie {12}-gon we have found two distinct isoclinic rotations, the great square rotation characteristic of the 16-cell and the great hexagon rotation characteristic of the 24-cell. If we examine the chords <math>t_i</math> of the 24-cell's {24}-gon we find these, and also four other distinct isoclinic rotations. Each row of the table is a distinct isoclinic rotation of the 24-cell characterized by a pair of chords that sum to 180°. The 24 edge chords lie in invariant planes of the rotation. The 24 isocline chords form the rotation's Clifford {24}-gon and lie in invariant planes completely orthogonal to a corresponding edge chord's invariant plane. The rotational angle between successive edge chords and the rotational angle between successive isocline chords also sum to 180°. ... {{Clear}} == The 600-cell == [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are: [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</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. {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="7" |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra |- ! colspan="3" |Short edge chord ! Section ! colspan="3" |Long isocline chord |- style="background: palegreen;" | | rowspan="4" |<math>r_0</math> |0° | rowspan="4" | | rowspan="4" | | rowspan="4" |[[File:Regular_star_figure_15(2,1).svg|100px]]<br>{30/15}=15{2} |180° | rowspan="4" |<math>r_{15}</math> |- style="background: palegreen;" | |{{radic|0}} |{{radic|4}} |- style="background: palegreen;" | |0 |2 |- style="background: palegreen;" | |0° |180° |- style="background: palegreen;" | | rowspan="4" |<math>r_1</math> |36° | rowspan="4" |[[File:Regular_polygon_30.svg|100px]]<br>{30/1} | rowspan="4" | | rowspan="4" |[[File:Regular_star_figure_2(15,7).svg|100px]]<br>{30/14}=2{15/7} |144° | rowspan="4" |<math>r_{14}</math> |- style="background: palegreen;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: palegreen;" | |0.618~ |1.902~ |- style="background: palegreen;" | |12° |168° |- style="background: gainsboro;" | | rowspan="4" |<math>r_2</math> |36° | rowspan="4" |[[File:Regular_star_figure_2(15,1).svg|100px]]<br>{30/2}=2{15} | rowspan="4" | | rowspan="4" |[[File:Regular_star_polygon_30-13.svg|100px]]<br>{30/13} |144° | rowspan="4" |<math>r_{13}</math> |- style="background: gainsboro;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: gainsboro;" | |0.618~ |1.902~ |- style="background: gainsboro;" | |24° |156° |- style="background: yellow;" | | rowspan="4" |<math>r_3</math> |36° | rowspan="4" |[[File:Regular_star_figure_3(10,1).svg|100px]]<br>{30/3}=3{10} | rowspan="4" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="4" |[[File:Regular_star_figure_6(5,2).svg|100px]]<br>{30/12}=6{5/2} |144° | rowspan="4" |<math>r_{12}</math> |- style="background: yellow;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: yellow;" | |0.618~ |1.902~ |- style="background: yellow;" | |36° |144° |- style="background: palegreen;" | | rowspan="4" |<math>r_4</math> |60° | rowspan="4" |[[File:Regular_star_figure_2(15,2).svg|100px]]<br>{30/4}=2{15/2} | rowspan="4" | | rowspan="4" |[[File:Regular_star_polygon_30-11.svg|100px]]<br>{30/11} |120° | rowspan="4" |<math>r_{11}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | |48° |132° |- style="background: palegreen;" | | rowspan="4" |<math>r_5</math> |60° | rowspan="4" |[[File:Regular_star_figure_5(6,1).svg|100px]]<br>{30/5}=5{6} | rowspan="4" |[[File:V2 dodecahedron.png|100px]]<br>Dodecahedron | rowspan="4" |[[File:Regular_star_figure_10(3,1).svg|100px]]<br>{30/10}=10{3} |120° | rowspan="4" |<math>r_{10}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | |60° |120° |- style="background: yellow;" | | rowspan="4" |<math>r_{6}</math> |72° | rowspan="4" |[[File:Regular_star_figure_6(5,1).svg|100px]]<br>{30/6}=6{5} | rowspan="4" |[[File:V3 icosahedron.png|100px]]<br>Icosahedron | rowspan="4" |[[File:Regular_star_figure_3(10,3).svg|100px]]<br>{30/9}=3{10/3} |108° | rowspan="4" |<math>r_{9}</math> |- style="background: yellow;" | |{{radic|1.382~}} |{{radic|2.618~}} |- style="background: yellow;" | |1.176~ |1.618~ |- style="background: yellow;" | |72° |108° |- style="background: seashell;" | | rowspan="4" |<math>r_{7}</math> |90° | rowspan="4" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} | rowspan="4" |[[File:V4 icosidodecahedron.png|100px]]<br>Icosidodecahedron | rowspan="4" |[[File:Regular_star_figure_2(15,4).svg|100px]]<br>{30/8}=2{15/4} |90° | rowspan="4" |<math>r_{8}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |- style="background: seashell;" | |84° |96° |} The list of 600-cell chords <math>r_{i}</math> can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. The short chord and long chord each have their characteristic {30/n}-gon. Each row identifies a discrete isoclinic rotation of the 600-cell in invariant central planes containing the edges of the short chord {30}-gon, over the isocline chords of the long chord {30}-gon, the rotation's Clifford polygon. Each distinct pair of complementary chord lengths is identified with a distinct [[w:600-cell#Polyhedral sections|polyhedral section of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance <math>\phi^{-1}</math> is a icosahedron vertex figure, and the largest section at radial distance <math>\sqrt{2}</math> is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. Because [[w:3-sphere|<math>\mathbb{S}^3</math>]] is spherical, at radial distances greater than <math>\sqrt{2}</math> the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance <math>\sqrt{2+\phi}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>\sqrt{2}</math></small>]] We can rotate the 600-cell isoclinically in the great square rotation characteristic of the 16-cell, with the same effect on 15 disjoint 16-cells. Each 90° displacement takes 15 pairs of completely orthogonal invariant great square planes to each other. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The rotational curve over each 90° chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> form a circular fibration of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once. The 600-cell has another distinct isoclinic rotation in invariant great square planes, which has period 30 and visits every vertex of a 600-cell Petrie polygon. This [''great square left rotation characteristic of the 600-cell]'' takes place over <math>r_7</math> edge chords and <math>r_8</math> isocline chords. The {30/7} edge polygon is a skew helix of circumference <math>14\pi</math> with each <math>r_7</math> edge belonging to a distinct great square. The four {30/7} polygrams contribute one edge each to 30 great squares. Each 90° displacement takes every 16-cell to another 16-cell. The vertices of the invariant great squares each make seven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 90° {30/7} edge makes seven 12° turns. Four Clifford parallel {30/7} geodesics of circumference <math>16\pi</math> form a circular quadruple helix that intersects each 600-cell vertex once. The {30/8}=2{15/4} Clifford polygon is a compound of two skew {15/4} pentadecagrams of circumference <math>16\pi</math> with each <math>r_8</math> isocline chord belonging to a distinct 16-cell. The four {30/8} polygrams contribute one edge each to 30 great squares. The rotational curve over each 90° {30/8} isocline chord makes eight 12° turns. Four Clifford parallel {30/8} geodesics of circumference <math>16\pi</math> form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>\sqrt{3}</math></small> ]] We can rotate the 600-cell isoclinically in the great hexagon rotation characteristic of the 24-cell, over <math>\sqrt{1}</math> edge chords and <math>\sqrt{3}</math> isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° revolution each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> form a circular fibration of ten twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. The 600-cell has another distinct isoclinic rotation in invariant great hexagon planes, over <math>r_{4}=\sqrt{1}</math> edge chords and <math>r_{11}=\sqrt{3}</math> isocline chords This [''invariant great hexagon left rotation characteristic of the 600-cell]'' has period 30 and visits every vertex of a 600-cell Petrie polygon. Its {30/11} Clifford polygon is a skew helix where each <math>r_{11}</math> isocline chord is the <math>\sqrt{3}</math> diagonal of a great hexagon of a distinct 24-cell. The vertices of the invariant great hexagons of this rotation each make eleven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 120° <math>r_{11}</math> isocline chord makes eleven 12° turns. Four Clifford parallel {30/11} geodesic isoclines of circumference <math>22\pi</math> over <math>r_{11}</math> chords form a circular quadruple helix that intersects each 600-cell vertex once. We can rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its 36° <math>r_{3}</math> edges, over 144° <math>r_{12}</math> isocline chords. This ''invariant great decagon rotation characteristic of the 600-cell'' has period 5 and takes disjoint 24-cells to each other. The rotational curve over each <math>r_{12}</math> chord of its {5/2} Clifford polygon makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference <math>4\pi</math> over five <math>r_{12}</math> chords form a circular fibration of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. The rotation of the 600-cell 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. The 24-cells revolve within the 600-cell, as the 16-cells revolve within the 24-cells. All 120 vertices move at once on four Clifford parallel geodesic isoclines, displaced 144° in different directions. The 600-cell has another distinct isoclinic rotation in invariant great decagon planes containing its 36° <math>r_{2}</math> edges, over 144° <math>r_{13}</math> isocline chords. This [''great decagon left rotation characteristic of the 600-cell]'' has period 30 and visits every vertex of a 600-cell Petrie polygon. The rotational curve over each 144° <math>r_{13}</math> isocline chord makes thirteen 12° turns. Four Clifford parallel {30/13} geodesic isoclines of circumference <math>26\pi</math> form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} == The 5-cell 4-simplex == {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="9" |30 chords (15 180° pairs) make 15 distinct section polyhedra |- ! colspan="3" |Short chord ! Section ! colspan="3" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px]]<br>{30/15}=15{2} |180° | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} |{{radic|4}} |- style="background: palegreen;" | |0 |2 |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° | rowspan="3" |[[File:Regular_polygon_30.svg|100px]]<br>{30/1} | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px]]<br>{30/14} |164.5~° | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} |{{radic|3.927~}} |- style="background: palegreen;" | |0.270~ |1.982~ |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px]]<br>{30/2}=2{15} | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px]]<br>{30/13} |154.8~° | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} |{{radic|3.809~}} |- style="background: gainsboro;" | |0.437~ |1.952~ |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px]]<br>{30/3}=3{10} | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px]]<br>{30/12}=6{5/2} |144° | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: yellow;" | |0.618~ |1.902~ |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |138.6~° | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} |{{radic|3.5}} |- style="background: gainsboro;" | |0.707~ |1.871~ |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px]]<br>{30/4}=2{15/2} | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px]]<br>{30/11} |135.5~° | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} |{{radic|3.427~}} |- style="background: palegreen;" | |0.757~ |1.851~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |130.9~° | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} |{{radic|3.309~}} |- style="background: gainsboro;" | |0.831~ |1.819~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° | rowspan="3" | | rowspan="3" | | rowspan="3" | |124° | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} |{{radic|3.118~}} |- style="background: gainsboro;" | |0.939~ |1.766~ |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px]]<br>{30/5}=5{6} | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px]]<br>{30/10}=10{3} |120° | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |113.9~° | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} |{{radic|2.809~}} |- style="background: gainsboro;" | |1.091~ |1.676~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |110.2~° | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} |{{radic|2.691~}} |- style="background: gainsboro;" | |1.144~ |1.640~ |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px]]<br>{30/6}=6{5} | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px]]<br>{30/9}=3{10/3} |108° | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} |{{radic|2.618~}} |- style="background: yellow;" | |1.176~ |1.618~ |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px]]<br>{30/8}=2{15/4} |104.5~° | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} |{{radic|2.5}} |- style="background: palegreen;" | |1.224~ |1.581~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |98.9~° | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} |{{radic|2.309~}} |- style="background: gainsboro;" | |1.300~ |1.520~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |95.5~° | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} |{{radic|2.191~}} |- style="background: gainsboro;" | |1.345~ |1.480~ |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} |90° | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |} The [[User:Dc.samizdat/Golden chords of the 120-cell#Thirty distinguished distances|list of thirty 120-cell chords]] <math>c_{t}</math> can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct [[w:120-cell#Concentric_hulls|polyhedral section of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance <math>c_1</math> is a tetrahedron vertex figure, and the largest section at radial distance <math>c_{15}</math> is a central section bisecting the 120-cell. Because [[w:3-sphere|<math>\mathbb{S}^3</math>]] is spherical, at radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance <math>c_{29}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. Only 8 of the 30 chords in the table occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. Since all those polytopes except the 5-cell occur in the 600-cell, and the 600-cell and the 120-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. ... {{Clear}} == Finally the 120-cell == The [[120-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{5,3,3\}</math></small>. It has 600 vertices, 1200 edges, 720 pentagon faces, and 120 dodecahedron cells. It is the four-dimensional analogue of the dodecahedron. The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. ... {{Clear}} == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.] The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} 7241a0bxzmhsa3aoae9z6vtyuyhx696 2 329439 2818521 2815488 2026-07-19T11:52:25Z Fortuna imperatrix mundi 1058581 + 2818521 wikitext text/x-wiki ==Background== ===Richard II=== ===Elizabeth I=== ====Essex's rebellion==== ==IaR2== Lambare's accoiunt of his intervieew with the queen is well known. well-known ===Political anxiety and tension=== ===Censorship=== ==Consequences and aftermath== == Notes == {{reflist|group=note}} ==References== {{Reflist|20em}} ==Bibliography== {{refbegin|30em|indent=yes}} * [https://link.springer.com/chapter/10.1057/9780230307261_2 Orgel, S. (2011). Prologue: I am Richard II. In: Petrina, A., Tosi, L. (eds) Representations of Elizabeth I in Early Modern Culture. Palgrave Macmillan, London. https://doi.org/10.1057/9780230307261_2] * [https://broadlytextual.com/2017/12/15/i-am-richard-ii-know-ye-not-that-drama-and-political-anxiety-in-shakespeares-london/ Hixon, E., Syracuse Univ] * Bate, Jonathan (2008). Soul of the Age. London: Penguin. pp. 256–286. ISBN 978-0-670-91482-1. * [https://theconversation.com/richard-ii-by-william-shakespeare-why-the-divine-right-of-kings-still-matters-186648 McFarlane, K., Univ South Australia] * [https://muse.jhu.edu/article/31090/summary Lemon, Rebecca. "The Faulty Verdict in "The Crown v. John Hayward"." SEL Studies in English Literature 1500-1900, vol. 41 no. 1, 2001, p. 109-132. Project MUSE, https://dx.doi.org/10.1353/sel.2001.0009] * [https://brill.com/display/book/9789401211666/B9789401211666-s009.xml?language=en&srsltid=AfmBOoqTUaTQr0msNWRdtEZZ9qQ3rNcVkSoBvDPXuGv7nRvHrC8t5OzH Kizelbach, U. (2014), In The Pragmatics of Early Modern Politics: Power and Kingship in Shakespeare’s History Plays. Leiden, The Netherlands: Brill. https://doi.org/10.1163/9789401211666_009] * [https://www.google.co.uk/books/edition/Richard_II/z-kOAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Lopez, R2] * [https://www.manchesterhive.com/display/9781526130532/9781526130532.00008.xml?print rgel, S. (2017). "I am Richard II". In Spectacular Performances. Manchester, England: Manchester University Press. Retrieved Apr 22, 2026, from https://doi.org/10.7765/9781526130532.00008] * [https://www.researchgate.net/publication/271041996_Was_Elizabeth_I_Richard_II_The_Authenticity_of_Lambarde%27s_%27Conversation%27 Scott-Warren, Jason. (2012). Was Elizabeth I Richard II?: The Authenticity of Lambarde's 'Conversation'. The Review of English Studies. 64. 208-230. 10.1093/res/hgs062] * Stanley Wells, “Introduction” in Richard II, The New Penguin Shakespeare, ed. Stanley Wells (London: Penguin Books, 1969), 13. * [https://www.cambridge.org/core/books/abs/shakespeare-survey/shakespeare-and-history-divergencies-and-agreements/B065A23215FD86BEE8E1CFD51DC7C1FB Ives EW. Shakespeare and History: Divergencies and Agreements. In: Wells S, ed. Shakespeare Survey. Shakespeare Survey. Cambridge University Press; 1986:19-36] * [https://www.jstor.org/stable/457855 Albright, Evelyn May. “Shakespeare’s Richard II, Hayward’s History of Henry IV, and the Essex Conspiracy.” PMLA, vol. 46, no. 3, 1931, pp. 694–719. JSTOR, https://doi.org/10.2307/457855. Accessed 22 Apr. 2026] * [https://dsc.duq.edu/cgi/viewcontent.cgi?article=2800&context=etd Morris, A. (2019). "within the hollow crown": Performing Kingship in Richard II and Henry IV Part One (Master's thesis, Duquesne University). Retrieved from https://dsc.duq.edu/etd/1771] * [https://digitalcommons.lib.uconn.edu/cgi/viewcontent.cgi?article=1122&context=srhonors_theses Scannell, Sarah J., "Shakespeare's Richard II and Henry V and Political Rebellions in the Reign of Queen Elizabeth I" (2010). Honors Scholar Theses, 138] * [https://compass.onlinelibrary.wiley.com/doi/abs/10.1111/j.1741-4113.2011.00873.x Luecking Frost, L. (2012), “A Kyng That Ruled All By Lust”: Richard II in Elizabethan Literature. Literature Compass, 9: 183-198. https://doi.org/10.1111/j.1741-4113.2011.00873.x] * [https://www.academia.edu/8347491/A_Law_Case_in_Verse_Venus_and_Adonis_and_the_Authorship_Question Stritmatter, R., A Law Case in Verse: Venus and Adonis and the Authorship Question, 2004 University of Tennessee Law Review] * [https://www.researchgate.net/publication/304647742_Prologue_I_am_Richard_II Orgel, Stephen. (2011). Prologue: I am Richard II. 10.1057/9780230307261_2] * [https://go.gale.com/ps/i.do?id=GALE%7CA314252957&sid=googleScholar&v=2.1&it=r&linkaccess=abs&issn=15256863&sw=w&p=AONE&userGroupName=anon%7Ebf79dda0&aty=open-web-entry Egan, Michael. "The Essex Rebellion and Richard II: why wasn't Shakespeare arrested?" Shakespeare Oxford Newsletter, vol. 48, no. 3, summer-fall 2012, p. 20] * [https://www.cambridge.org/core/books/abs/richard-ii/stagehistory/D62D9CA061F03B2AE8D56E856288EEDA 1. Child H, Shakespeare W. THE STAGE-HISTORY. In: Dover Wilson J, ed. Richard II: The Cambridge Dover Wilson Shakespeare. Cambridge Library Collection - Literary Studies. Cambridge University Press; 2009:lxxvii-xcii] * [https://www.openstarts.units.it/entities/publication/16d8c447-62f4-48e8-8347-5d56af674aa3/details Snelling, D., 'Prospero on the Coast of Bohemia. "Nihil sed rara et miranda amabat". Diary of Melchior Goldast upon the late Emperor Rudolph II' 1994] * [https://www.jstor.org/stable/2902265 Gajowski, Evelyn. Shakespeare in Performance: King Richard IIby Margaret Shewring. Shakespeare Quarterly, vol. 49, no. 3, 1998, pp. 328–30. JSTOR, https://doi.org/10.2307/2902265. Accessed 22 Apr. 2026] * [https://muse.jhu.edu/article/217842/summary Mosse, Ramona Franziska. Review of Richard II. Shakespeare Bulletin, vol. 25 no. 2, 2007, p. 73-77. Project MUSE, https://dx.doi.org/10.1353/shb.2007.0036] * [https://www.journals.uchicago.edu/doi/abs/10.1086/665894?journalCode=rq Petrina and Tosi. Representations of Elizabeth I in Early Modern Culture Elizabeth Pentland Renaissance Quarterly 2012 65:1, 274-276] * [https://journals.co.za/doi/10.4314/sisa.v26i1.4 The construction of a king : waste, effeminacy and queerness in Shakespeare's Richard II Higginbotham, D.] * [https://academic.oup.com/book/2363/chapter-abstract/142533948?redirectedFrom=fulltext Erskine-Hill, Howard, 'The Second Tetralogy and After', Poetry and the Realm of Politics: Shakespeare to Dryden (Oxford, 1996; online edn, Oxford Academic, 3 Oct. 2011), https://doi.org/10.1093/acprof:oso/9780198117315.003.0004] * [https://www.academia.edu/123917862/Shakespeare_s_Richard_II_in_the_minds_of_the_Elizabethan_audiences Shakespeare’s Richard II in the minds of the Elizabethan audiences] * [https://journals.sagepub.com/doi/10.1177/03064220221084555 Plays, protests and the censor’s pencil] * [https://www.indexoncensorship.org/2022/03/plays-protests-and-the-censors-pencil/ Plays, protests and the censor’s pencil A brittle Gloriana: staging the deposition of Queen Elizabeth I]. * Saul, N., Richard II * [https://www.academia.edu/3236651/_A_god_on_earth_thou_art_Richard_II_and_the_nature_of_Kingship “A god on earth thou art”: Richard II and the nature of Kingship] * [https://www.jstor.org/stable/457398 Albright, Evelyn May. “Shakespeare’s Richard II and the Essex Conspiracy.” PMLA, vol. 42, no. 3, 1927, pp. 686–720. JSTOR, https://doi.org/10.2307/457398. Accessed 22 Apr. 2026] * [https://www.berghahnjournals.com/view/journals/critical-survey/37/3/cs370301.xml Holderness, G. (2025). Introduction: Shakespeare's Wars of the Roses. Critical Survey, 37(3), 1-4. Retrieved Apr 22, 2026, from https://doi.org/10.3167/cs.2025.370301] * [https://www.folger.edu/podcasts/shakespeare-unlimited/tragedy-richard-ii-henry-iv-helen-castor/ Folger: Shakespeare Unlimited: Episode 249] * [https://www.rachaeldickzen.com/blog/2021/1/10/charlesiii The Historical and Shakespearean References in King Charles III] * Kermode, The Age of Shakespeare, pg.53. * [https://heinonline.org/HOL/login-hol?redirect_url=https%3A%2F%2Fheinonline.org%2FHOL%2Fmojo%3F%252Fmojo%3D%26auth_token%3DyhcMOu8W7PCc5OrmTqVnr2qFPpoiOdhM5uQmkQyWufU%253D%26casa_token%3D%26div%3D5%26g_sent%3D1%26paywall_url%3D%26referrer%3Dhttps%253A%252F%252Fheinonline.org%252FHOL%252FPage%253Fhandle%253Dhein.journals%252Fneastlr1%26timestamp%3D1776855600%26url%3D%252FHOL%252FPage%253Fhandle%253Dhein.journals%252Fneastlr1%2526div%253D5%2526id%253D%2526page%253D Jones, H., ''I AM RICHARD II, KNOW YE NOT THAT?' - WORKS. OF WILLIAM SHAKESPEARE AND THE ART OF. GOVERNANCE'''] * Hattaway, M., William Shakespeare: Richard II (Literature Insights, 9781847600349) * [https://books.openedition.org/psn/8305 Findlay, A., “Good sometimes queen” (V.1.37). Richard II, Mary Stuart and the Politics of Queenship, 205-219] * [https://pure.manchester.ac.uk/ws/portalfiles/portal/54508829/FULL_TEXT.PDF Pritchard, K., 'Legitimacy, Illegitimacy and Sovereignty in Shakespeare’s British Plays' (2011 University of Manchester PhD] * CHATELANAT, Marine. Histrionic Future Kings: The Politics of Metadrama in Shakespeare’s Richard III, Richard II, and Henry IV Part 1. Master, 2023. * [https://www.persee.fr/doc/xvii_0291-3798_2004_num_59_1_1993 Mayer, J-C., 'The "Parliament Sceane" in Shakespeare's King Richard II', XVII-XVIII. Revue de la Société d'études anglo-américaines des XVIIe et XVIIIe siècles Année 2004 59 pp. 27-42] * [https://www.emerald.com/md/article-abstract/38/5/315/285132/Managers-as-kings-Shakespeare-on-modern-leadership?redirectedFrom=fulltext Egan M (2000), "Managers as kings: Shakespeare on modern leadership". Management Decision, Vol. 38 No. 5 pp. 315–327, doi: https://doi.org/10.1108/00251740010340490] * [https://d-nb.info/1279081511/34 Quabeck (2022): William Shakespeare, in: Thomas Gutmann, Eberhard Ortland, Klaus Stierstorfer, eds., Encyclopedia of Law and Literature (last edited 26 October 2022), doi: 10.17879/12009577118] * [https://lawliberty.org/book-review/was-the-bard-writing-about-the-donald/ Sullivan, V., 'Was the Bard Writing about The Donald?'] * [https://deveresociety.co.uk/wp-content/uploads/2022/11/FINAL_NL_Apr2022-Haste-July-Letterresponse.pdf Haste, I., 'THE PLAY ON THE EVE OF THE ESSEX REBELLION'] * [https://www.cherylmtaylor.com/2017/11/08/a-companion-to-shakespeares-richard-ii/ A Companion to Shakespeare’s Richard II] * [https://www.city-journal.org/article/shakespeares-richards Dalrymple, T., Shakespeare’s Richards: The playwright’s two historical dramas offer contrasts between political pathologies] * [https://hfriedberg.web.wesleyan.edu/engl205/wshakespeare/plotsandrebelions.htm ELIZABETHAN PLOTS AND REBELLIONS] * [https://www.jstor.org/stable/2869708 Rackin, Phyllis. “The Role of the Audience In Shakespeare’s Richard II.” Shakespeare Quarterly, vol. 36, no. 3, 1985, pp. 262–81. JSTOR, https://doi.org/10.2307/2869708. Accessed 22 Apr. 2026.] * [https://www.psychologytoday.com/ie/blog/whats-in-name/201603/group-think-and-academia-shocking-shakespeare-shenanigans Group Think and Academia: Shocking Shakespeare Shenanigans] * [https://journals.sagepub.com/doi/10.1177/01847678221099975 Hiscopck, A., '"Why I should welcome such a guest as grief [?": Lodging and dislodging in Shakespeare's Richard II'] * [https://academic.oup.com/book/51779/chapter-abstract/420487796?redirectedFrom=fulltext 'Epigraph', Imagining Time in the English Chronicle Play: Historical Futures, 1590-1660 (Oxford, 2023; online edn, Oxford Academic, 19 Oct. 2023), https://doi.org/10.1093/oso/9780198872658.002.0009] * [https://journals.openedition.org/interfaces/319?lang=en Blake & Shakespeare] * [https://www.cambridge.org/core/books/abs/shakespeare-and-the-visual-imagination/richard-iiand-the-politics-of-perspective/23A5D4615E2A8BF9380BAF1ECE99B46D Sillars S. Richard IIand the Politics of Perspective. In: Shakespeare and the Visual Imagination. Cambridge University Press; 2015:133-162.] * [https://www.persee.fr/doc/xvii_0291-3798_2004_num_59_1_1992 Le miroir des rois: de Richard II à Henry VIII] * [https://www.cambridge.org/core/books/abs/king-richard-ll/introduction/25419329A06BC1C2E4D3DDBF331B7F91 Shakespeare W, McEachern C. Introduction. In: Gurr A, ed. King Richard Ll. The New Cambridge Shakespeare. Cambridge University Press; 2018:1-66.] * [https://www.researchgate.net/publication/372862689_A_Critical_Discourse_Study_of_Shakespeare%27s_Theological_Conceptions_in_Acts_IV_and_V_of_Richard_II_The_%27Divine_Mandate%27_of_Richard_Kingship_Falls_Apart A Critical Discourse Study of Shakespeare’s Theological Conceptions in Acts IV and V of Richard II: The ‘Divine Mandate’ of Richard Kingship Falls Apart] * [https://www.jstor.org/stable/10.1525/lal.2004.16.3.313 Richard II, Bracton, and the End of Political Theology] * [https://dokumen.pub/the-art-of-law-in-shakespeare-9781509905478-9781509905508-9781509905492.html The Art of Law in Shakespeare] * [https://www.cambridge.org/core/books/abs/cambridge-companion-to-shakespeare-and-religion/providence-and-divine-right-in-the-english-histories/DBE9BFEBF5C199F9DD72D31ED9C3B041 Mayer J-C. Providence and Divine Right in the English Histories. In: Hamlin H, ed. The Cambridge Companion to Shakespeare and Religion. Cambridge Companions to Literature. Cambridge University Press; 2019:151-167] * [https://www.researchgate.net/publication/315698000_THE_CHOMSKY-FOUCAULT_DEBATE_WHERE_DOES_THE_BARD_STAND_IN_RICHARD_II Salami, Ali & Riahi Nouri, Amir. (2016). THE CHOMSKY-FOUCAULT DEBATE: WHERE DOES THE BARD STAND IN RICHARD II?] * [https://dokumen.pub/shakespeare-richard-ii-9781350363168-9780333416709.html Richard II] * [https://www.jstor.org/stable/457907?seq=1 Heffner, Ray. “Shakespeare, Hayward, and Essex.” PMLA, vol. 45, no. 3, 1930, pp. 754–80. JSTOR, https://doi.org/10.2307/457907. Accessed 22 Apr. 2026] * [https://www.cambridge.org/core/books/abs/cambridge-companion-to-shakespeare-and-war/shakespeare-on-civil-and-dynastic-wars/00D0269CFF998876DB961FB01C22954D Shakespeare on Civil and Dynastic Wars] * [https://brill.com/display/book/edcoll/9789004356900/B9789004356900_012.xml?srsltid=AfmBOopJMjwJAQnUGME9FmT6Gwg0ih9pArbXBdTJbN6YAqbdXqbkGyuo Is a Dramatic Theory of History Possible? Shakespeare’s Richard ii and ‘Historical Truth’] * [https://www.jstor.org/stable/24475607 THE CHILD'S TWO BODIES: SHAKESPEARE, SOVEREIGNTY, AND THE END OF SUCCESSION] * [https://dokumen.pub/english-renaissance-drama-an-introduction-to-theatre-and-theatres-in-shakespeares-time-9781847600202-9781847600714.html English Renaissance Drama : An Introduction to Theatre and Theatres in Shakespeare’s Time] * [https://www.cambridge.org/core/books/abs/believing-in-shakespeare/person-plot-place/BD4EF4CBA6E7D8E2ECE5519FA1D08C16 1. McEachern C. Person, Plot, Place. In: Believing in Shakespeare: Studies in Longing. Cambridge University Press; 2018:181-296] * [https://www.enotes.com/topics/richard-2/criticism/richard-ii-vol-91/criticism-production-reviews-1000228/nigel-saul-review-date-12-may-2000 With an Eye to the Present] * [https://www.academia.edu/128303633/Shakespeare_s_Other_Eden_Richard_II_by_Ouroboros_Theatre_Company_2013_ Shakespeare’s Other Eden Richard II by Ouroboros Theatre Company (2013)] * [https://dokumen.pub/shakespeare-upstart-crow-to-sweet-swan-1592-1623-9781472555496-9781408130148-9781408139189.html Shakespeare: Upstart Crow to Sweet Swan 1592-1623] * [https://www.jstor.org/stable/41917215 RACKIN, PHYLLIS. “Temporality, Anachronism, and Presence in Shakespeare’s English Histories.” Renaissance Drama, vol. 17, 1986, pp. 101–23. JSTOR, http://www.jstor.org/stable/41917215. Accessed 22 Apr. 2026.] * [https://www.cambridge.org/core/journals/greece-and-rome/article/abs/alexander-philotas-and-the-origins-of-modern-historiography/D90568C0A1131952625E0A3B44F43EC8 Stoneman R. Alexander, Philotas, and the origins of modern historiography. Greece and Rome. 2013;60(2):296-312. doi:10.1017/S0017383513000119] * [https://www.academia.edu/35800720/Shakespeares_Richard_II_and_Elizabethan_politics Shakespeare's Richard II and Elizabethan politics] * [https://www.cambridge.org/core/books/abs/rethinking-historicism-from-shakespeare-to-milton/old-new-now/2E8CBEFFEFB11D8B843BAF2C108E9768 Introduction - Old, New, Now] * [https://www.academia.edu/24392476/Richard_II_hamartia_and_climax Richard II hamartia and climax] * [https://www.cambridge.org/core/books/abs/babylonian-jews-and-sasanian-imperialism-in-late-antiquity/kings-and-religion-in-the-talmud-and-in-the-imagination-of-sasanian-communities/E2DCC2F3E0FBA74ABE543DDD2DD7F1A7 Kings and Religion in the Talmud and in the Imagination of Sasanian Communities] * Stephen Greenblatt, “Introduction,” in The Power of Forms in the English Renaissance, ed. Stephen Greenblatt (Norman, OK: 1982), 3-6] * [https://www.academia.edu/30472558/Americas_Donald_Trump_Shakespeares_Richard_II_15_Dec_2016 America's Donald Trump, Shakespeare's Richard II] * [https://www.academia.edu/5321346/Looking_Back_Shakespeare_s_Indebtedness_to_Chaucer_and_the_Representations_of_Chivalry_in_King_Richard_II_The_Two_Noble_Kinsmen_and_the_Knight_s_Tale Looking Back: Shakespeare’s Indebtedness to Chaucer and the Representations of Chivalry in King Richard II, The Two Noble Kinsmen and the Knight’s Tale] * [https://www.academia.edu/92423912/The_Good_Usurper_in_the_eyes_of_God_and_the_people_An_analysis_of_the_role_of_the_usurper_in_Shakespeares_Richard_II_and_Henry_IV_Part_1 The Good Usurper in the eyes of God and the people: An analysis of the role of the usurper in Shakespeare's Richard II and Henry IV Part 1] * [https://www.academia.edu/91789871/The_Bacon_Shakespeare_Manuscript_Hitherto_known_as_the_Northumberland_Manuscript_which_originally_Contained_Copies_of_his_Shakespeare_Plays_Richard_II_and_Richard_III THE BACON-SHAKESPEARE MANUSCRIPT (HITHERTO KNOWN AS THE NORTHUMBERLAND MANUSCRIPT) WHICH ORIGINALLY CONTAINED COPIES OF HIS SHAKESPEARE PLAYS RICHARD II AND RICHARD III] * [https://www.google.co.uk/books/edition/Richard_II/f4gGCAAAQBAJ?hl=en&gbpv=1&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&pg=PA88&printsec=frontcover Richard II: Critical Essays] * [https://www.google.co.uk/books/edition/A_Companion_to_Shakespeare_s_Works_Volum/JlDNEAAAQBAJ?hl=en&gbpv=1&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&pg=PA400&printsec=frontcover A Companion to Shakespeare's Works, Volume II] * [https://www.google.co.uk/books/edition/Shakespeare_Reread/Z6JhDwAAQBAJ?hl=en&gbpv=1&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&pg=PA239&printsec=frontcover Shakespeare Reread] * [https://www.google.co.uk/books/edition/Shakespeare_in_the_Present/yTKWEAAAQBAJ?hl=en&gbpv=1&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&pg=PT8&printsec=frontcover Shakespeare in the Present] * [https://www.google.co.uk/books/edition/Studying_Shakespeare/1N4FBAAAQBAJ?hl=en&gbpv=1&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&pg=PA129&printsec=frontcover Studying Shakespeare A Practical Introduction] * [https://www.google.co.uk/books/edition/Critical_Essays_on_William_Faulkner/kAGBEAAAQBAJ?hl=en&gbpv=1&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&pg=PT178&printsec=frontcover Critical Essays on William Faulkner] * [https://www.google.co.uk/books/edition/Shakespeare/QY6aEQAAQBAJ?hl=en&gbpv=1&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&pg=PA216&printsec=frontcover Shakespeare: An Anthology of Criticism and Theory 1945-2000] * [https://www.google.co.uk/books/edition/The_Reign_of_Richard_II/cv0WAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Dodd, G., R2] * [https://www.google.co.uk/books/edition/Shakespeare_and_the_Institution_of_Theat/Nkt9DAAAQBAJ?hl=en&gbpv=1&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&pg=PA108&printsec=frontcover Shakespeare and the Institution of Theatre] * [https://www.google.co.uk/books/edition/The_Theory_of_the_King_s_Two_Bodies_in_t/8SRXAAAAYAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Theory of the King's Two Bodies in the Age of Shakespeare] * [https://www.google.co.uk/books/edition/Stealing_the_Story/RaJlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Stealing the Story: Shakespeare's Self-Conscious Use of the Mimetic Tradition in the Tragedies] * [https://www.google.co.uk/books/edition/The_Greenwood_Companion_to_Shakespeare_O/JkcgAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Greenwood Companion to Shakespeare: Overviews and the history plays] * [https://www.google.co.uk/books/edition/British_and_Irish_Literature_and_Its_Tim/xH4jAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover British and Irish Literature and Its Times] * [https://www.google.co.uk/books/edition/Shakespeare_s_Philosophy_of_History_Reve/1jwgAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare's Philosophy of History Revealed in a Detailed Analysis of Henry V and Examined in Other History Plays] * [https://www.google.co.uk/books/edition/Shakespeare_and_His_Contemporaries/7em7coOthxQC?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare and His Contemporaries] * [https://www.google.co.uk/books/edition/The_Politics_of_the_Public_Sphere_in_Ear/0sGHAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Politics of the Public Sphere in Early Modern England] * [https://www.google.co.uk/books/edition/Shakespeare_and_the_Catholic_Religion/Fl4gAQAAIAAJ?hl=en&gbpv=0&bsq=%22I%20am%20Richard%20II,%20know%20ye%20not%20that?%22 Shakespeare and the Catholic religion] * [https://www.google.co.uk/books/edition/Proceedings_of_the_British_Academy_Volum/8CsoAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Lectures] * [https://www.google.co.uk/books/edition/Slander_and_Censorship_in_Late_Sixteenth/wb_oAvuJgbAC?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Slander and Censorship in Late Sixteenth Century Literature] * [https://www.google.co.uk/books/edition/Elizabethan_Literature_and_the_Law_of_Fr/vBFdAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Elizabethan Literature and the Law of Fraudulent Conveyance] * [https://www.google.co.uk/books/edition/Hamlet_History_and_commentary/yxgrAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Hamlet History etc] * [https://www.google.co.uk/books/edition/William_Shakespeare_A_Popular_Life/1BuaAAAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover William Shakespeare: a popular life] * [https://www.google.co.uk/books/edition/A_State_of_Mind/pCiIAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover A State of Mind?] * [https://www.google.co.uk/books/edition/Shakespeare_s_Dramatic_Genres/J5FlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare's Dramatic Genres] * [https://www.google.co.uk/books/edition/Literature_Criticism_from_1400_to_1800/O6VkAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Literature Criticism from 1400 to 1800 Volume 89] * [https://www.google.co.uk/books/edition/Shakespeare_the_Papist/LPwNAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare the Papist] * [https://www.google.co.uk/books/edition/Goslicius_Ideal_Senator_and_His_Cultural/1jc7AQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Goslicius' Ideal Senator and His Cultural Impact Over the Centuries] * [https://www.google.co.uk/books/edition/Shakespeare_by_Another_Name/FqllAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover "Shakespeare" by Another Name] * [https://www.google.co.uk/books/edition/Shakespeare_s_Friends/AlZlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare's Friends] * [https://www.google.co.uk/books/edition/Dr_Simon_Forman/qHceAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Dr Simon Forman] * [https://www.google.co.uk/books/edition/William_Shakespeare_the_Wars_of_the_Rose/dZFlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover William Shakespeare, the Wars of the Roses and the historians] * [https://www.google.co.uk/books/edition/Shakespearean_Criticism/2TdlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespearean Criticism] * [https://www.google.co.uk/books/edition/Poetry_History_and_Antiquarianism_in_Ref/jwIhAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Poetry, History, and Antiquarianism in Reformation England] * [https://www.google.co.uk/books/edition/Shakespeare_and_the_Prince_of_Love/D0QgAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare and the Prince of Love] * [https://www.google.co.uk/books/edition/Paper_Bullets_of_the_Brain/o0YgAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Paper Bullets of the Brain] * [https://www.google.co.uk/books/edition/As_You_Like_It/GiBaAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover As You Like It: Third Series] * [https://www.google.co.uk/books/edition/William_Shakespeare/WCqaAAAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover William Shakespeare, Dominic Shellard] * [https://www.google.co.uk/books/edition/Poets_and_God/0XZlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Poets and God] * [https://www.google.co.uk/books/edition/Law_and_Literature/Ax5MAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Law and Literature Volume 16] * [https://www.google.co.uk/books/edition/The_Embodied_Word/FV8sAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Embodied Word] * [https://www.google.co.uk/books/edition/The_Case_for_Shakespeare/WaRlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Case for Shakespeare: The End of the Authorship Question] * [https://www.google.co.uk/books/edition/Explorations_in_Renaissance_Culture/_SYrAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Explorations in Renaissance Culture Volumes 33-34] * [https://www.google.co.uk/books/edition/The_Touch_of_the_Real/ewdaAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Touch of the Real: Essays in Early Modern Culture in Honour of Stephen Greenblatt] * [https://www.google.co.uk/books/edition/Wotton_and_His_Worlds/ZfgNAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Wotton and his Worlds] * [https://www.google.co.uk/books/edition/Theatre_and_Religion/wo1lAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Theatre and Religion Lancastrian Shakespeare] * [https://www.google.co.uk/books/edition/Trying_Treason/TOKxAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Trying Treason] * [https://www.google.co.uk/books/edition/Willing_Subjects/IEX0sGwT1QQC?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Willing Subjects] * [https://www.google.co.uk/books/edition/Symbolism/Bt0ZAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Symbolism] * [https://www.google.co.uk/books/edition/Performing_Shakespeare/35pQAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Performing Shakespeare] * [https://www.google.co.uk/books/edition/Soul_of_the_Age/e0UgAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Soul of the Age] * [https://www.google.co.uk/books/edition/England/aD9nAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover England] * [https://www.google.co.uk/books/edition/Elizabeth_I/-GtnAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Elizabeth I] * [https://www.google.co.uk/books/edition/King_Richard_II/oGUMX4RntjgC?hl=en&gbpv=1&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&pg=PA25&printsec=frontcover King Richard II] * [https://www.google.co.uk/books/edition/Shakespeare_and_the_Legal_Imagination/OXPvBqQLw-4C?hl=en&gbpv=1&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&pg=PA38&printsec=frontcover Shakespeare and the legal imagination] * [https://www.google.co.uk/books/edition/Shakespeare_s_Theatre/GxN3ue9_r3oC?hl=en&gbpv=1&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&pg=PA69&printsec=frontcover Shakespeare's Theatre] * [https://www.google.co.uk/books/edition/Critical_Essays_on_Shakespeare_s_Richard/AaYoAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Critical Essays on Shakespeare's Richard II] * [https://www.google.co.uk/books/edition/The_Reign_of_Richard_II_Essays_in_Honour/y3xnAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Reign of Richard II: Essays in Honour of May McKisack] * [https://www.google.co.uk/books/edition/Poetry_and_the_Realm_of_Politics/oQFaAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Poetry and the Realm of Politics] * [https://www.google.co.uk/books/edition/Shakespeare/BM0mAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare] * [https://www.google.co.uk/books/edition/Shakespearean_Politics/oTdlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespearean Politics] * [https://www.google.co.uk/books/edition/Shakespeare_the_Theatrical_Dimension/wl4gAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare, the Theatrical Dimension] * [https://www.google.co.uk/books/edition/Who_was_Kit_Marlowe/zQ1aAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Kit Marlowe etc] * [https://www.google.co.uk/books/edition/From_Page_to_Performance/beQKAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover From Page to Performance] * [https://www.google.co.uk/books/edition/Exploring_Tudor_England/ax56AAAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Exploring Tudor England] * [https://www.google.co.uk/books/edition/The_Movement_Towards_Subversion/vyJaAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Movement Towards Subversion] * [https://www.google.co.uk/books/edition/Shakespeare_s_Typological_Satire/G5BlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare's Typological Satire] * [https://www.google.co.uk/books/edition/Shakespeare_Recycled/zzNlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare Recycled] * [https://www.google.co.uk/books/edition/Reinventing_the_Middle_Ages_the_Renaissa/fXFnAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Reinventing the Middle Ages & the Renaissance] * [https://www.google.co.uk/books/edition/The_Mysterious_William_Shakespeare/WnllAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Mysterious William Shakespeare] * [https://www.google.co.uk/books/edition/Richard_II/GHhlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Richard II Critical Essays] * [https://www.google.co.uk/books/edition/William_Shakespeare/WJvC6gu_I0gC?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover William Shakespeare: Records and Images] * [https://www.google.co.uk/books/edition/Shakespeare_and_the_Actors/HYtlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare and the Actors] * [https://www.google.co.uk/books/edition/Shakespeare_the_Man/BVdlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare the man] * [https://www.google.co.uk/books/edition/Henry_V/zXllAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Henry V: A Guide to the Play] * [https://www.google.co.uk/books/edition/Shakespearean_Contingencies/Cw1NAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespearean Contingencies] * [https://www.google.co.uk/books/edition/Renaissance_Drama/E60kAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Renaissance Drama 1990] * [https://www.google.co.uk/books/edition/Language_Discourse_Sign/uH4oAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Language, Discourse, Sign] * [https://www.google.co.uk/books/edition/Mock_Kings_in_Medieval_Society_and_Renai/T98KAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Mock kings in medieval society and Renaissance drama] * [https://www.google.co.uk/books/edition/Shakespeare_Invention_of_the_Human/ojHirImrtYoC?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare: Invention of the Human] * [https://www.google.co.uk/books/edition/Shakespeare/wn5lAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare] * [https://www.google.co.uk/books/edition/Persons_in_Groups/rQ24AAAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Persons in Groups] * [https://www.google.co.uk/books/edition/All_Semblative_a_Woman_s_Part/0DlaAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover All Semblative a Woman's Part?] * [https://www.google.co.uk/books/edition/Crossing_the_Mirror/qRZNAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Crossing the Mirror] * [https://www.google.co.uk/books/edition/De_Vere_is_Shakespeare/dKJlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover De Vere is Shakespeare] * [https://www.google.co.uk/books/edition/William_Lambarde_Elizabethan_Antiquary_1/x1RnAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover William Lambarde, Elizabethan Antiquary, 1536-1601] * [https://www.google.co.uk/books/edition/The_Power_of_Forms_in_the_English_Renais/cPtZAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Power of Forms in the English Renaissance] * [https://www.google.co.uk/books/edition/Ravishment_and_Rememberance/G31LAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Ravishment and Rememberance] * [https://www.google.co.uk/books/edition/Shakespeare_and_His_Theatre/8A5ZQq3uOVQC?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare and His Theatre] * [https://www.google.co.uk/books/edition/Critical_Hermeneutics_and_Shakespeare_s/O10gAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Critical Hermeneutics and Shakespeare's History Plays] * [https://www.google.co.uk/books/edition/Christian_England/K-WfAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Christian England] * [https://www.google.co.uk/books/edition/Shakespeare_s_Religious_Background/xDSaAAAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare's Religious Background] * [https://www.google.co.uk/books/edition/Shylock/N4RlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shylock] * [https://www.google.co.uk/books/edition/The_Shakespeare_Legacy/MM5XAAAAYAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Shakespeare Legacy] * [https://www.google.co.uk/books/edition/Renaissance_Genres/0uFZAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Renaissance Genres] * [https://www.google.co.uk/books/edition/Cannibals_Witches_and_Divorce/qZRpAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Cannibals, Witches, and Divorce] * [https://www.google.co.uk/books/edition/The_Problem_of_Religious_Knowledge/C29LAAAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Problem of Religious Knowledge] * [https://www.google.co.uk/books/edition/Essex_and_the_Great_Revolt_of_1381/J8RzAAAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Essex and the Great Revolt of 1381] * [https://www.google.co.uk/books/edition/Transactions_of_the_London_and_Middlesex/4dtJAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover TLMAS] * [https://www.google.co.uk/books/edition/Shakespeare_Politics_and_the_State/Mn9lAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare, Politics and the State] * [https://www.google.co.uk/books/edition/Allegories_of_Power_in_the_England_of_El/LIYgAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Allegories of Power in the England of Elizabeth] * [https://www.google.co.uk/books/edition/William_Shakespeare/rIVlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover William Shakespeare] * [https://www.google.co.uk/books/edition/Women_s_Matters/PDRlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Women's Matters] * [https://www.google.co.uk/books/edition/The_Weak_King_Dilemma_in_the_Shakespeare/0bJlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Weak King Dilemma in the Shakespearean History Play] * [https://www.google.co.uk/books/edition/The_Book_Known_as_Q/S2tlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Book Known as Q] * [https://www.google.co.uk/books/edition/Fields_of_Vision/OD0eAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Fields of Vision] * [https://www.google.co.uk/books/edition/Ungodly_Delights/RKgcAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Ungodly Delights] * [https://www.google.co.uk/books/edition/The_Shakespeare_Handbook/rLRlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Shakespeare HandboOK] * [https://www.google.co.uk/books/edition/Humanities/y5FZAAAAYAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Humanities] * [https://www.google.co.uk/books/edition/Richard_II_by_William_Shakespeare/Bb3yAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Richard II by William Shakespeare] * [https://www.google.co.uk/books/edition/King_Richard_II/50NnAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover King Richard II] * [https://www.google.co.uk/books/edition/Murder_Under_Trust_Or_The_Topical_Macbet/0oNlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Murder under trust] * [https://www.google.co.uk/books/edition/The_Shakespearean_Kings/tHBlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Shakespearean Kings] * [https://www.google.co.uk/books/edition/America_the_Mabr_e_y_Experience/mRQ3AAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover America, the Mabr(e)y Experience: Resistance, Revolution & Civil War] * [https://www.google.co.uk/books/edition/Richard_II/ZDEkAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Richard II: An Annotated Bibliography, Volume 2] * [https://www.google.co.uk/books/edition/The_Batsford_Companion_to_Medieval_Engla/ev78b9EJQy0C?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Batsford Companion to Medieval England] * [https://www.google.co.uk/books/edition/Shakespeare_s_Unruly_Women/FKFlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare's Unruly Women] * [https://www.google.co.uk/books/edition/Shakespeare_and_Others/iFEgAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare and Others] * [https://www.google.co.uk/books/edition/Kings_and_Chroniclers/L1wpAAAAYAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Kings and Chroniclers] * [https://www.google.co.uk/books/edition/A_Kingdom_for_a_Stage/UzxlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover A Kingdom for a Stage] * [https://www.google.co.uk/books/edition/The_House_of_Commons/Ezz4OZuYVFYC?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The History of Parliament: The House of Commons 1558-1603 (3 v.)] * [https://www.google.co.uk/books/edition/Shakespeare_Soul_of_the_Age/nMYCAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover shakespeare, Soul of the Age] * [https://www.google.co.uk/books/edition/After_Poststructuralism/TOaEAAAAIAAJ?hl=en&gbpv=0&bsq=%22I%20am%20Richard%20II,%20know%20ye%20not%20that?%22 After Poststructuralism: Interdisciplinarity and Literary Theory] * [https://www.google.co.uk/books/edition/The_Unschooled_Mind/C7WnYtt219IC?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Unschooled Mind] * [https://www.google.co.uk/books/edition/Elizabeth_I/hHZnAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Eliz I] * [https://www.google.co.uk/books/edition/Dramas_of_Christian_Time/mnIqAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Dramas of Christian Time] * [https://www.google.co.uk/books/edition/Elizabeth_I/XjQmAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Elizabeth I: The Shrewdness of Virtue] * [https://www.google.co.uk/books/edition/John_Dryden/9Q1aAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover John Dryden] * [https://www.google.co.uk/books/edition/Shakespeare_and_Early_Modern_Political_T/DUwhAwAAQBAJ?hl=en&gbpv=1&dq=%22shakespeare%22+%2B+%22political+propaganda%22&pg=PA259&printsec=frontcover Shakespeare and Early Modern Political Thought] * [https://www.google.co.uk/books/edition/The_English_History_Play_in_the_age_of_S/5TT-AQAAQBAJ?hl=en&gbpv=1&dq=%22shakespeare%22+%2B+%22political+propaganda%22&pg=PA158&printsec=frontcover The English History Play in the Age of Shakespeare] * [https://www.google.co.uk/books/edition/Shakespeare_and_the_Political/rEcREQAAQBAJ?hl=en&gbpv=1&dq=%22shakespeare%22+%2B+%22political+propaganda%22&pg=PA215&printsec=frontcover Shakespeare and the Political] * [https://www.google.co.uk/books/edition/William_Shakespeare_Subject_of_the_Crown/a7G6DAAAQBAJ?hl=en&gbpv=1&dq=%22shakespeare%22+%2B+%22political+propaganda%22&pg=PT18&printsec=frontcover William Shakespeare - Subject of the Crown?] * [https://www.google.com/search?q=%22shakespeare%22+%2B+%22political+propaganda%22&client=firefox-b-d&hs=4AQ&sca_esv=6d4ade7bd26771c9&udm=36&biw=2510&bih=1307&tbs=cdr%3A1%2Ccd_min%3A2000%2Ccd_max%3A2099&sxsrf=ANbL-n6I6Pkwl7mmdHK6N1xPQXLbGBIOSg%3A1776853062010&ei=RqDoaZUvztiFsg_I5bToDw&ved=0ahUKEwiV6tC8nYGUAxVObEEAHcgyDf0Q4dUDCBM&uact=5&oq=%22shakespeare%22+%2B+%22political+propaganda%22&gs_lp=EhBnd3Mtd2l6LW1vZGVsZXNzIiYic2hha2VzcGVhcmUiICsgInBvbGl0aWNhbCBwcm9wYWdhbmRhIjIIECEYoAEYwwRInQlQxgZYuwdwAXgAkAEAmAF_oAHPAaoBAzEuMbgBA8gBAPgBAZgCAqACVsICCxAAGIAEGKIEGLADmAMAiAYBkAYCkgcBMqAHowOyBwExuAdTwgcDMC4yyAcEgAgB&sclient=gws-wiz-modeless The Nazi Appropriation of Shakespeare: Cultural Politics in] {{refend}} p8zr7dg1qoeiwjik3l8614xudxn9znp 2818522 2818521 2026-07-19T11:52:41Z Fortuna imperatrix mundi 1058581 dup 2818522 wikitext text/x-wiki ==Background== ===Richard II=== ===Elizabeth I=== ====Essex's rebellion==== ==IaR2== Lambare's accoiunt of his intervieew with the queen is well known. ===Political anxiety and tension=== ===Censorship=== ==Consequences and aftermath== == Notes == {{reflist|group=note}} ==References== {{Reflist|20em}} ==Bibliography== {{refbegin|30em|indent=yes}} * [https://link.springer.com/chapter/10.1057/9780230307261_2 Orgel, S. (2011). Prologue: I am Richard II. In: Petrina, A., Tosi, L. (eds) Representations of Elizabeth I in Early Modern Culture. Palgrave Macmillan, London. https://doi.org/10.1057/9780230307261_2] * [https://broadlytextual.com/2017/12/15/i-am-richard-ii-know-ye-not-that-drama-and-political-anxiety-in-shakespeares-london/ Hixon, E., Syracuse Univ] * Bate, Jonathan (2008). Soul of the Age. London: Penguin. pp. 256–286. ISBN 978-0-670-91482-1. * [https://theconversation.com/richard-ii-by-william-shakespeare-why-the-divine-right-of-kings-still-matters-186648 McFarlane, K., Univ South Australia] * [https://muse.jhu.edu/article/31090/summary Lemon, Rebecca. "The Faulty Verdict in "The Crown v. John Hayward"." SEL Studies in English Literature 1500-1900, vol. 41 no. 1, 2001, p. 109-132. Project MUSE, https://dx.doi.org/10.1353/sel.2001.0009] * [https://brill.com/display/book/9789401211666/B9789401211666-s009.xml?language=en&srsltid=AfmBOoqTUaTQr0msNWRdtEZZ9qQ3rNcVkSoBvDPXuGv7nRvHrC8t5OzH Kizelbach, U. (2014), In The Pragmatics of Early Modern Politics: Power and Kingship in Shakespeare’s History Plays. Leiden, The Netherlands: Brill. https://doi.org/10.1163/9789401211666_009] * [https://www.google.co.uk/books/edition/Richard_II/z-kOAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Lopez, R2] * [https://www.manchesterhive.com/display/9781526130532/9781526130532.00008.xml?print rgel, S. (2017). "I am Richard II". In Spectacular Performances. Manchester, England: Manchester University Press. Retrieved Apr 22, 2026, from https://doi.org/10.7765/9781526130532.00008] * [https://www.researchgate.net/publication/271041996_Was_Elizabeth_I_Richard_II_The_Authenticity_of_Lambarde%27s_%27Conversation%27 Scott-Warren, Jason. (2012). Was Elizabeth I Richard II?: The Authenticity of Lambarde's 'Conversation'. The Review of English Studies. 64. 208-230. 10.1093/res/hgs062] * Stanley Wells, “Introduction” in Richard II, The New Penguin Shakespeare, ed. Stanley Wells (London: Penguin Books, 1969), 13. * [https://www.cambridge.org/core/books/abs/shakespeare-survey/shakespeare-and-history-divergencies-and-agreements/B065A23215FD86BEE8E1CFD51DC7C1FB Ives EW. Shakespeare and History: Divergencies and Agreements. In: Wells S, ed. Shakespeare Survey. Shakespeare Survey. Cambridge University Press; 1986:19-36] * [https://www.jstor.org/stable/457855 Albright, Evelyn May. “Shakespeare’s Richard II, Hayward’s History of Henry IV, and the Essex Conspiracy.” PMLA, vol. 46, no. 3, 1931, pp. 694–719. JSTOR, https://doi.org/10.2307/457855. Accessed 22 Apr. 2026] * [https://dsc.duq.edu/cgi/viewcontent.cgi?article=2800&context=etd Morris, A. (2019). "within the hollow crown": Performing Kingship in Richard II and Henry IV Part One (Master's thesis, Duquesne University). Retrieved from https://dsc.duq.edu/etd/1771] * [https://digitalcommons.lib.uconn.edu/cgi/viewcontent.cgi?article=1122&context=srhonors_theses Scannell, Sarah J., "Shakespeare's Richard II and Henry V and Political Rebellions in the Reign of Queen Elizabeth I" (2010). Honors Scholar Theses, 138] * [https://compass.onlinelibrary.wiley.com/doi/abs/10.1111/j.1741-4113.2011.00873.x Luecking Frost, L. (2012), “A Kyng That Ruled All By Lust”: Richard II in Elizabethan Literature. Literature Compass, 9: 183-198. https://doi.org/10.1111/j.1741-4113.2011.00873.x] * [https://www.academia.edu/8347491/A_Law_Case_in_Verse_Venus_and_Adonis_and_the_Authorship_Question Stritmatter, R., A Law Case in Verse: Venus and Adonis and the Authorship Question, 2004 University of Tennessee Law Review] * [https://www.researchgate.net/publication/304647742_Prologue_I_am_Richard_II Orgel, Stephen. (2011). Prologue: I am Richard II. 10.1057/9780230307261_2] * [https://go.gale.com/ps/i.do?id=GALE%7CA314252957&sid=googleScholar&v=2.1&it=r&linkaccess=abs&issn=15256863&sw=w&p=AONE&userGroupName=anon%7Ebf79dda0&aty=open-web-entry Egan, Michael. "The Essex Rebellion and Richard II: why wasn't Shakespeare arrested?" Shakespeare Oxford Newsletter, vol. 48, no. 3, summer-fall 2012, p. 20] * [https://www.cambridge.org/core/books/abs/richard-ii/stagehistory/D62D9CA061F03B2AE8D56E856288EEDA 1. Child H, Shakespeare W. THE STAGE-HISTORY. In: Dover Wilson J, ed. Richard II: The Cambridge Dover Wilson Shakespeare. Cambridge Library Collection - Literary Studies. Cambridge University Press; 2009:lxxvii-xcii] * [https://www.openstarts.units.it/entities/publication/16d8c447-62f4-48e8-8347-5d56af674aa3/details Snelling, D., 'Prospero on the Coast of Bohemia. "Nihil sed rara et miranda amabat". Diary of Melchior Goldast upon the late Emperor Rudolph II' 1994] * [https://www.jstor.org/stable/2902265 Gajowski, Evelyn. Shakespeare in Performance: King Richard IIby Margaret Shewring. Shakespeare Quarterly, vol. 49, no. 3, 1998, pp. 328–30. JSTOR, https://doi.org/10.2307/2902265. Accessed 22 Apr. 2026] * [https://muse.jhu.edu/article/217842/summary Mosse, Ramona Franziska. Review of Richard II. Shakespeare Bulletin, vol. 25 no. 2, 2007, p. 73-77. Project MUSE, https://dx.doi.org/10.1353/shb.2007.0036] * [https://www.journals.uchicago.edu/doi/abs/10.1086/665894?journalCode=rq Petrina and Tosi. Representations of Elizabeth I in Early Modern Culture Elizabeth Pentland Renaissance Quarterly 2012 65:1, 274-276] * [https://journals.co.za/doi/10.4314/sisa.v26i1.4 The construction of a king : waste, effeminacy and queerness in Shakespeare's Richard II Higginbotham, D.] * [https://academic.oup.com/book/2363/chapter-abstract/142533948?redirectedFrom=fulltext Erskine-Hill, Howard, 'The Second Tetralogy and After', Poetry and the Realm of Politics: Shakespeare to Dryden (Oxford, 1996; online edn, Oxford Academic, 3 Oct. 2011), https://doi.org/10.1093/acprof:oso/9780198117315.003.0004] * [https://www.academia.edu/123917862/Shakespeare_s_Richard_II_in_the_minds_of_the_Elizabethan_audiences Shakespeare’s Richard II in the minds of the Elizabethan audiences] * [https://journals.sagepub.com/doi/10.1177/03064220221084555 Plays, protests and the censor’s pencil] * [https://www.indexoncensorship.org/2022/03/plays-protests-and-the-censors-pencil/ Plays, protests and the censor’s pencil A brittle Gloriana: staging the deposition of Queen Elizabeth I]. * Saul, N., Richard II * [https://www.academia.edu/3236651/_A_god_on_earth_thou_art_Richard_II_and_the_nature_of_Kingship “A god on earth thou art”: Richard II and the nature of Kingship] * [https://www.jstor.org/stable/457398 Albright, Evelyn May. “Shakespeare’s Richard II and the Essex Conspiracy.” PMLA, vol. 42, no. 3, 1927, pp. 686–720. JSTOR, https://doi.org/10.2307/457398. Accessed 22 Apr. 2026] * [https://www.berghahnjournals.com/view/journals/critical-survey/37/3/cs370301.xml Holderness, G. (2025). Introduction: Shakespeare's Wars of the Roses. Critical Survey, 37(3), 1-4. Retrieved Apr 22, 2026, from https://doi.org/10.3167/cs.2025.370301] * [https://www.folger.edu/podcasts/shakespeare-unlimited/tragedy-richard-ii-henry-iv-helen-castor/ Folger: Shakespeare Unlimited: Episode 249] * [https://www.rachaeldickzen.com/blog/2021/1/10/charlesiii The Historical and Shakespearean References in King Charles III] * Kermode, The Age of Shakespeare, pg.53. * [https://heinonline.org/HOL/login-hol?redirect_url=https%3A%2F%2Fheinonline.org%2FHOL%2Fmojo%3F%252Fmojo%3D%26auth_token%3DyhcMOu8W7PCc5OrmTqVnr2qFPpoiOdhM5uQmkQyWufU%253D%26casa_token%3D%26div%3D5%26g_sent%3D1%26paywall_url%3D%26referrer%3Dhttps%253A%252F%252Fheinonline.org%252FHOL%252FPage%253Fhandle%253Dhein.journals%252Fneastlr1%26timestamp%3D1776855600%26url%3D%252FHOL%252FPage%253Fhandle%253Dhein.journals%252Fneastlr1%2526div%253D5%2526id%253D%2526page%253D Jones, H., ''I AM RICHARD II, KNOW YE NOT THAT?' - WORKS. OF WILLIAM SHAKESPEARE AND THE ART OF. GOVERNANCE'''] * Hattaway, M., William Shakespeare: Richard II (Literature Insights, 9781847600349) * [https://books.openedition.org/psn/8305 Findlay, A., “Good sometimes queen” (V.1.37). Richard II, Mary Stuart and the Politics of Queenship, 205-219] * [https://pure.manchester.ac.uk/ws/portalfiles/portal/54508829/FULL_TEXT.PDF Pritchard, K., 'Legitimacy, Illegitimacy and Sovereignty in Shakespeare’s British Plays' (2011 University of Manchester PhD] * CHATELANAT, Marine. Histrionic Future Kings: The Politics of Metadrama in Shakespeare’s Richard III, Richard II, and Henry IV Part 1. Master, 2023. * [https://www.persee.fr/doc/xvii_0291-3798_2004_num_59_1_1993 Mayer, J-C., 'The "Parliament Sceane" in Shakespeare's King Richard II', XVII-XVIII. Revue de la Société d'études anglo-américaines des XVIIe et XVIIIe siècles Année 2004 59 pp. 27-42] * [https://www.emerald.com/md/article-abstract/38/5/315/285132/Managers-as-kings-Shakespeare-on-modern-leadership?redirectedFrom=fulltext Egan M (2000), "Managers as kings: Shakespeare on modern leadership". Management Decision, Vol. 38 No. 5 pp. 315–327, doi: https://doi.org/10.1108/00251740010340490] * [https://d-nb.info/1279081511/34 Quabeck (2022): William Shakespeare, in: Thomas Gutmann, Eberhard Ortland, Klaus Stierstorfer, eds., Encyclopedia of Law and Literature (last edited 26 October 2022), doi: 10.17879/12009577118] * [https://lawliberty.org/book-review/was-the-bard-writing-about-the-donald/ Sullivan, V., 'Was the Bard Writing about The Donald?'] * [https://deveresociety.co.uk/wp-content/uploads/2022/11/FINAL_NL_Apr2022-Haste-July-Letterresponse.pdf Haste, I., 'THE PLAY ON THE EVE OF THE ESSEX REBELLION'] * [https://www.cherylmtaylor.com/2017/11/08/a-companion-to-shakespeares-richard-ii/ A Companion to Shakespeare’s Richard II] * [https://www.city-journal.org/article/shakespeares-richards Dalrymple, T., Shakespeare’s Richards: The playwright’s two historical dramas offer contrasts between political pathologies] * [https://hfriedberg.web.wesleyan.edu/engl205/wshakespeare/plotsandrebelions.htm ELIZABETHAN PLOTS AND REBELLIONS] * [https://www.jstor.org/stable/2869708 Rackin, Phyllis. “The Role of the Audience In Shakespeare’s Richard II.” Shakespeare Quarterly, vol. 36, no. 3, 1985, pp. 262–81. JSTOR, https://doi.org/10.2307/2869708. Accessed 22 Apr. 2026.] * [https://www.psychologytoday.com/ie/blog/whats-in-name/201603/group-think-and-academia-shocking-shakespeare-shenanigans Group Think and Academia: Shocking Shakespeare Shenanigans] * [https://journals.sagepub.com/doi/10.1177/01847678221099975 Hiscopck, A., '"Why I should welcome such a guest as grief [?": Lodging and dislodging in Shakespeare's Richard II'] * [https://academic.oup.com/book/51779/chapter-abstract/420487796?redirectedFrom=fulltext 'Epigraph', Imagining Time in the English Chronicle Play: Historical Futures, 1590-1660 (Oxford, 2023; online edn, Oxford Academic, 19 Oct. 2023), https://doi.org/10.1093/oso/9780198872658.002.0009] * [https://journals.openedition.org/interfaces/319?lang=en Blake & Shakespeare] * [https://www.cambridge.org/core/books/abs/shakespeare-and-the-visual-imagination/richard-iiand-the-politics-of-perspective/23A5D4615E2A8BF9380BAF1ECE99B46D Sillars S. Richard IIand the Politics of Perspective. In: Shakespeare and the Visual Imagination. Cambridge University Press; 2015:133-162.] * [https://www.persee.fr/doc/xvii_0291-3798_2004_num_59_1_1992 Le miroir des rois: de Richard II à Henry VIII] * [https://www.cambridge.org/core/books/abs/king-richard-ll/introduction/25419329A06BC1C2E4D3DDBF331B7F91 Shakespeare W, McEachern C. Introduction. In: Gurr A, ed. King Richard Ll. The New Cambridge Shakespeare. Cambridge University Press; 2018:1-66.] * [https://www.researchgate.net/publication/372862689_A_Critical_Discourse_Study_of_Shakespeare%27s_Theological_Conceptions_in_Acts_IV_and_V_of_Richard_II_The_%27Divine_Mandate%27_of_Richard_Kingship_Falls_Apart A Critical Discourse Study of Shakespeare’s Theological Conceptions in Acts IV and V of Richard II: The ‘Divine Mandate’ of Richard Kingship Falls Apart] * [https://www.jstor.org/stable/10.1525/lal.2004.16.3.313 Richard II, Bracton, and the End of Political Theology] * [https://dokumen.pub/the-art-of-law-in-shakespeare-9781509905478-9781509905508-9781509905492.html The Art of Law in Shakespeare] * [https://www.cambridge.org/core/books/abs/cambridge-companion-to-shakespeare-and-religion/providence-and-divine-right-in-the-english-histories/DBE9BFEBF5C199F9DD72D31ED9C3B041 Mayer J-C. Providence and Divine Right in the English Histories. In: Hamlin H, ed. The Cambridge Companion to Shakespeare and Religion. Cambridge Companions to Literature. Cambridge University Press; 2019:151-167] * [https://www.researchgate.net/publication/315698000_THE_CHOMSKY-FOUCAULT_DEBATE_WHERE_DOES_THE_BARD_STAND_IN_RICHARD_II Salami, Ali & Riahi Nouri, Amir. (2016). THE CHOMSKY-FOUCAULT DEBATE: WHERE DOES THE BARD STAND IN RICHARD II?] * [https://dokumen.pub/shakespeare-richard-ii-9781350363168-9780333416709.html Richard II] * [https://www.jstor.org/stable/457907?seq=1 Heffner, Ray. “Shakespeare, Hayward, and Essex.” PMLA, vol. 45, no. 3, 1930, pp. 754–80. JSTOR, https://doi.org/10.2307/457907. Accessed 22 Apr. 2026] * [https://www.cambridge.org/core/books/abs/cambridge-companion-to-shakespeare-and-war/shakespeare-on-civil-and-dynastic-wars/00D0269CFF998876DB961FB01C22954D Shakespeare on Civil and Dynastic Wars] * [https://brill.com/display/book/edcoll/9789004356900/B9789004356900_012.xml?srsltid=AfmBOopJMjwJAQnUGME9FmT6Gwg0ih9pArbXBdTJbN6YAqbdXqbkGyuo Is a Dramatic Theory of History Possible? Shakespeare’s Richard ii and ‘Historical Truth’] * [https://www.jstor.org/stable/24475607 THE CHILD'S TWO BODIES: SHAKESPEARE, SOVEREIGNTY, AND THE END OF SUCCESSION] * [https://dokumen.pub/english-renaissance-drama-an-introduction-to-theatre-and-theatres-in-shakespeares-time-9781847600202-9781847600714.html English Renaissance Drama : An Introduction to Theatre and Theatres in Shakespeare’s Time] * [https://www.cambridge.org/core/books/abs/believing-in-shakespeare/person-plot-place/BD4EF4CBA6E7D8E2ECE5519FA1D08C16 1. McEachern C. Person, Plot, Place. In: Believing in Shakespeare: Studies in Longing. Cambridge University Press; 2018:181-296] * [https://www.enotes.com/topics/richard-2/criticism/richard-ii-vol-91/criticism-production-reviews-1000228/nigel-saul-review-date-12-may-2000 With an Eye to the Present] * [https://www.academia.edu/128303633/Shakespeare_s_Other_Eden_Richard_II_by_Ouroboros_Theatre_Company_2013_ Shakespeare’s Other Eden Richard II by Ouroboros Theatre Company (2013)] * [https://dokumen.pub/shakespeare-upstart-crow-to-sweet-swan-1592-1623-9781472555496-9781408130148-9781408139189.html Shakespeare: Upstart Crow to Sweet Swan 1592-1623] * [https://www.jstor.org/stable/41917215 RACKIN, PHYLLIS. “Temporality, Anachronism, and Presence in Shakespeare’s English Histories.” Renaissance Drama, vol. 17, 1986, pp. 101–23. JSTOR, http://www.jstor.org/stable/41917215. Accessed 22 Apr. 2026.] * [https://www.cambridge.org/core/journals/greece-and-rome/article/abs/alexander-philotas-and-the-origins-of-modern-historiography/D90568C0A1131952625E0A3B44F43EC8 Stoneman R. Alexander, Philotas, and the origins of modern historiography. Greece and Rome. 2013;60(2):296-312. doi:10.1017/S0017383513000119] * [https://www.academia.edu/35800720/Shakespeares_Richard_II_and_Elizabethan_politics Shakespeare's Richard II and Elizabethan politics] * [https://www.cambridge.org/core/books/abs/rethinking-historicism-from-shakespeare-to-milton/old-new-now/2E8CBEFFEFB11D8B843BAF2C108E9768 Introduction - Old, New, Now] * [https://www.academia.edu/24392476/Richard_II_hamartia_and_climax Richard II hamartia and climax] * [https://www.cambridge.org/core/books/abs/babylonian-jews-and-sasanian-imperialism-in-late-antiquity/kings-and-religion-in-the-talmud-and-in-the-imagination-of-sasanian-communities/E2DCC2F3E0FBA74ABE543DDD2DD7F1A7 Kings and Religion in the Talmud and in the Imagination of Sasanian Communities] * Stephen Greenblatt, “Introduction,” in The Power of Forms in the English Renaissance, ed. Stephen Greenblatt (Norman, OK: 1982), 3-6] * [https://www.academia.edu/30472558/Americas_Donald_Trump_Shakespeares_Richard_II_15_Dec_2016 America's Donald Trump, Shakespeare's Richard II] * [https://www.academia.edu/5321346/Looking_Back_Shakespeare_s_Indebtedness_to_Chaucer_and_the_Representations_of_Chivalry_in_King_Richard_II_The_Two_Noble_Kinsmen_and_the_Knight_s_Tale Looking Back: Shakespeare’s Indebtedness to Chaucer and the Representations of Chivalry in King Richard II, The Two Noble Kinsmen and the Knight’s Tale] * [https://www.academia.edu/92423912/The_Good_Usurper_in_the_eyes_of_God_and_the_people_An_analysis_of_the_role_of_the_usurper_in_Shakespeares_Richard_II_and_Henry_IV_Part_1 The Good Usurper in the eyes of God and the people: An analysis of the role of the usurper in Shakespeare's Richard II and Henry IV Part 1] * [https://www.academia.edu/91789871/The_Bacon_Shakespeare_Manuscript_Hitherto_known_as_the_Northumberland_Manuscript_which_originally_Contained_Copies_of_his_Shakespeare_Plays_Richard_II_and_Richard_III THE BACON-SHAKESPEARE MANUSCRIPT (HITHERTO KNOWN AS THE NORTHUMBERLAND MANUSCRIPT) WHICH ORIGINALLY CONTAINED COPIES OF HIS SHAKESPEARE PLAYS RICHARD II AND RICHARD III] * [https://www.google.co.uk/books/edition/Richard_II/f4gGCAAAQBAJ?hl=en&gbpv=1&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&pg=PA88&printsec=frontcover Richard II: Critical Essays] * [https://www.google.co.uk/books/edition/A_Companion_to_Shakespeare_s_Works_Volum/JlDNEAAAQBAJ?hl=en&gbpv=1&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&pg=PA400&printsec=frontcover A Companion to Shakespeare's Works, Volume II] * [https://www.google.co.uk/books/edition/Shakespeare_Reread/Z6JhDwAAQBAJ?hl=en&gbpv=1&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&pg=PA239&printsec=frontcover Shakespeare Reread] * [https://www.google.co.uk/books/edition/Shakespeare_in_the_Present/yTKWEAAAQBAJ?hl=en&gbpv=1&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&pg=PT8&printsec=frontcover Shakespeare in the Present] * [https://www.google.co.uk/books/edition/Studying_Shakespeare/1N4FBAAAQBAJ?hl=en&gbpv=1&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&pg=PA129&printsec=frontcover Studying Shakespeare A Practical Introduction] * [https://www.google.co.uk/books/edition/Critical_Essays_on_William_Faulkner/kAGBEAAAQBAJ?hl=en&gbpv=1&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&pg=PT178&printsec=frontcover Critical Essays on William Faulkner] * [https://www.google.co.uk/books/edition/Shakespeare/QY6aEQAAQBAJ?hl=en&gbpv=1&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&pg=PA216&printsec=frontcover Shakespeare: An Anthology of Criticism and Theory 1945-2000] * [https://www.google.co.uk/books/edition/The_Reign_of_Richard_II/cv0WAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Dodd, G., R2] * [https://www.google.co.uk/books/edition/Shakespeare_and_the_Institution_of_Theat/Nkt9DAAAQBAJ?hl=en&gbpv=1&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&pg=PA108&printsec=frontcover Shakespeare and the Institution of Theatre] * [https://www.google.co.uk/books/edition/The_Theory_of_the_King_s_Two_Bodies_in_t/8SRXAAAAYAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Theory of the King's Two Bodies in the Age of Shakespeare] * [https://www.google.co.uk/books/edition/Stealing_the_Story/RaJlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Stealing the Story: Shakespeare's Self-Conscious Use of the Mimetic Tradition in the Tragedies] * [https://www.google.co.uk/books/edition/The_Greenwood_Companion_to_Shakespeare_O/JkcgAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Greenwood Companion to Shakespeare: Overviews and the history plays] * [https://www.google.co.uk/books/edition/British_and_Irish_Literature_and_Its_Tim/xH4jAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover British and Irish Literature and Its Times] * [https://www.google.co.uk/books/edition/Shakespeare_s_Philosophy_of_History_Reve/1jwgAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare's Philosophy of History Revealed in a Detailed Analysis of Henry V and Examined in Other History Plays] * [https://www.google.co.uk/books/edition/Shakespeare_and_His_Contemporaries/7em7coOthxQC?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare and His Contemporaries] * [https://www.google.co.uk/books/edition/The_Politics_of_the_Public_Sphere_in_Ear/0sGHAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Politics of the Public Sphere in Early Modern England] * [https://www.google.co.uk/books/edition/Shakespeare_and_the_Catholic_Religion/Fl4gAQAAIAAJ?hl=en&gbpv=0&bsq=%22I%20am%20Richard%20II,%20know%20ye%20not%20that?%22 Shakespeare and the Catholic religion] * [https://www.google.co.uk/books/edition/Proceedings_of_the_British_Academy_Volum/8CsoAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Lectures] * [https://www.google.co.uk/books/edition/Slander_and_Censorship_in_Late_Sixteenth/wb_oAvuJgbAC?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Slander and Censorship in Late Sixteenth Century Literature] * [https://www.google.co.uk/books/edition/Elizabethan_Literature_and_the_Law_of_Fr/vBFdAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Elizabethan Literature and the Law of Fraudulent Conveyance] * [https://www.google.co.uk/books/edition/Hamlet_History_and_commentary/yxgrAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Hamlet History etc] * [https://www.google.co.uk/books/edition/William_Shakespeare_A_Popular_Life/1BuaAAAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover William Shakespeare: a popular life] * [https://www.google.co.uk/books/edition/A_State_of_Mind/pCiIAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover A State of Mind?] * [https://www.google.co.uk/books/edition/Shakespeare_s_Dramatic_Genres/J5FlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare's Dramatic Genres] * [https://www.google.co.uk/books/edition/Literature_Criticism_from_1400_to_1800/O6VkAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Literature Criticism from 1400 to 1800 Volume 89] * [https://www.google.co.uk/books/edition/Shakespeare_the_Papist/LPwNAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare the Papist] * [https://www.google.co.uk/books/edition/Goslicius_Ideal_Senator_and_His_Cultural/1jc7AQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Goslicius' Ideal Senator and His Cultural Impact Over the Centuries] * [https://www.google.co.uk/books/edition/Shakespeare_by_Another_Name/FqllAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover "Shakespeare" by Another Name] * [https://www.google.co.uk/books/edition/Shakespeare_s_Friends/AlZlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare's Friends] * [https://www.google.co.uk/books/edition/Dr_Simon_Forman/qHceAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Dr Simon Forman] * [https://www.google.co.uk/books/edition/William_Shakespeare_the_Wars_of_the_Rose/dZFlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover William Shakespeare, the Wars of the Roses and the historians] * [https://www.google.co.uk/books/edition/Shakespearean_Criticism/2TdlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespearean Criticism] * [https://www.google.co.uk/books/edition/Poetry_History_and_Antiquarianism_in_Ref/jwIhAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Poetry, History, and Antiquarianism in Reformation England] * [https://www.google.co.uk/books/edition/Shakespeare_and_the_Prince_of_Love/D0QgAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare and the Prince of Love] * [https://www.google.co.uk/books/edition/Paper_Bullets_of_the_Brain/o0YgAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Paper Bullets of the Brain] * [https://www.google.co.uk/books/edition/As_You_Like_It/GiBaAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover As You Like It: Third Series] * [https://www.google.co.uk/books/edition/William_Shakespeare/WCqaAAAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover William Shakespeare, Dominic Shellard] * [https://www.google.co.uk/books/edition/Poets_and_God/0XZlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Poets and God] * [https://www.google.co.uk/books/edition/Law_and_Literature/Ax5MAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Law and Literature Volume 16] * [https://www.google.co.uk/books/edition/The_Embodied_Word/FV8sAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Embodied Word] * [https://www.google.co.uk/books/edition/The_Case_for_Shakespeare/WaRlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Case for Shakespeare: The End of the Authorship Question] * [https://www.google.co.uk/books/edition/Explorations_in_Renaissance_Culture/_SYrAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Explorations in Renaissance Culture Volumes 33-34] * [https://www.google.co.uk/books/edition/The_Touch_of_the_Real/ewdaAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Touch of the Real: Essays in Early Modern Culture in Honour of Stephen Greenblatt] * [https://www.google.co.uk/books/edition/Wotton_and_His_Worlds/ZfgNAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Wotton and his Worlds] * [https://www.google.co.uk/books/edition/Theatre_and_Religion/wo1lAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Theatre and Religion Lancastrian Shakespeare] * [https://www.google.co.uk/books/edition/Trying_Treason/TOKxAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Trying Treason] * [https://www.google.co.uk/books/edition/Willing_Subjects/IEX0sGwT1QQC?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Willing Subjects] * [https://www.google.co.uk/books/edition/Symbolism/Bt0ZAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Symbolism] * [https://www.google.co.uk/books/edition/Performing_Shakespeare/35pQAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Performing Shakespeare] * [https://www.google.co.uk/books/edition/Soul_of_the_Age/e0UgAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Soul of the Age] * [https://www.google.co.uk/books/edition/England/aD9nAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover England] * [https://www.google.co.uk/books/edition/Elizabeth_I/-GtnAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Elizabeth I] * [https://www.google.co.uk/books/edition/King_Richard_II/oGUMX4RntjgC?hl=en&gbpv=1&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&pg=PA25&printsec=frontcover King Richard II] * [https://www.google.co.uk/books/edition/Shakespeare_and_the_Legal_Imagination/OXPvBqQLw-4C?hl=en&gbpv=1&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&pg=PA38&printsec=frontcover Shakespeare and the legal imagination] * [https://www.google.co.uk/books/edition/Shakespeare_s_Theatre/GxN3ue9_r3oC?hl=en&gbpv=1&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&pg=PA69&printsec=frontcover Shakespeare's Theatre] * [https://www.google.co.uk/books/edition/Critical_Essays_on_Shakespeare_s_Richard/AaYoAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Critical Essays on Shakespeare's Richard II] * [https://www.google.co.uk/books/edition/The_Reign_of_Richard_II_Essays_in_Honour/y3xnAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Reign of Richard II: Essays in Honour of May McKisack] * [https://www.google.co.uk/books/edition/Poetry_and_the_Realm_of_Politics/oQFaAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Poetry and the Realm of Politics] * [https://www.google.co.uk/books/edition/Shakespeare/BM0mAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare] * [https://www.google.co.uk/books/edition/Shakespearean_Politics/oTdlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespearean Politics] * [https://www.google.co.uk/books/edition/Shakespeare_the_Theatrical_Dimension/wl4gAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare, the Theatrical Dimension] * [https://www.google.co.uk/books/edition/Who_was_Kit_Marlowe/zQ1aAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Kit Marlowe etc] * [https://www.google.co.uk/books/edition/From_Page_to_Performance/beQKAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover From Page to Performance] * [https://www.google.co.uk/books/edition/Exploring_Tudor_England/ax56AAAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Exploring Tudor England] * [https://www.google.co.uk/books/edition/The_Movement_Towards_Subversion/vyJaAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Movement Towards Subversion] * [https://www.google.co.uk/books/edition/Shakespeare_s_Typological_Satire/G5BlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare's Typological Satire] * [https://www.google.co.uk/books/edition/Shakespeare_Recycled/zzNlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare Recycled] * [https://www.google.co.uk/books/edition/Reinventing_the_Middle_Ages_the_Renaissa/fXFnAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Reinventing the Middle Ages & the Renaissance] * [https://www.google.co.uk/books/edition/The_Mysterious_William_Shakespeare/WnllAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Mysterious William Shakespeare] * [https://www.google.co.uk/books/edition/Richard_II/GHhlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Richard II Critical Essays] * [https://www.google.co.uk/books/edition/William_Shakespeare/WJvC6gu_I0gC?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover William Shakespeare: Records and Images] * [https://www.google.co.uk/books/edition/Shakespeare_and_the_Actors/HYtlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare and the Actors] * [https://www.google.co.uk/books/edition/Shakespeare_the_Man/BVdlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare the man] * [https://www.google.co.uk/books/edition/Henry_V/zXllAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Henry V: A Guide to the Play] * [https://www.google.co.uk/books/edition/Shakespearean_Contingencies/Cw1NAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespearean Contingencies] * [https://www.google.co.uk/books/edition/Renaissance_Drama/E60kAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Renaissance Drama 1990] * [https://www.google.co.uk/books/edition/Language_Discourse_Sign/uH4oAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Language, Discourse, Sign] * [https://www.google.co.uk/books/edition/Mock_Kings_in_Medieval_Society_and_Renai/T98KAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Mock kings in medieval society and Renaissance drama] * [https://www.google.co.uk/books/edition/Shakespeare_Invention_of_the_Human/ojHirImrtYoC?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare: Invention of the Human] * [https://www.google.co.uk/books/edition/Shakespeare/wn5lAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare] * [https://www.google.co.uk/books/edition/Persons_in_Groups/rQ24AAAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Persons in Groups] * [https://www.google.co.uk/books/edition/All_Semblative_a_Woman_s_Part/0DlaAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover All Semblative a Woman's Part?] * [https://www.google.co.uk/books/edition/Crossing_the_Mirror/qRZNAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Crossing the Mirror] * [https://www.google.co.uk/books/edition/De_Vere_is_Shakespeare/dKJlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover De Vere is Shakespeare] * [https://www.google.co.uk/books/edition/William_Lambarde_Elizabethan_Antiquary_1/x1RnAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover William Lambarde, Elizabethan Antiquary, 1536-1601] * [https://www.google.co.uk/books/edition/The_Power_of_Forms_in_the_English_Renais/cPtZAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Power of Forms in the English Renaissance] * [https://www.google.co.uk/books/edition/Ravishment_and_Rememberance/G31LAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Ravishment and Rememberance] * [https://www.google.co.uk/books/edition/Shakespeare_and_His_Theatre/8A5ZQq3uOVQC?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare and His Theatre] * [https://www.google.co.uk/books/edition/Critical_Hermeneutics_and_Shakespeare_s/O10gAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Critical Hermeneutics and Shakespeare's History Plays] * [https://www.google.co.uk/books/edition/Christian_England/K-WfAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Christian England] * [https://www.google.co.uk/books/edition/Shakespeare_s_Religious_Background/xDSaAAAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare's Religious Background] * [https://www.google.co.uk/books/edition/Shylock/N4RlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shylock] * [https://www.google.co.uk/books/edition/The_Shakespeare_Legacy/MM5XAAAAYAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Shakespeare Legacy] * [https://www.google.co.uk/books/edition/Renaissance_Genres/0uFZAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Renaissance Genres] * [https://www.google.co.uk/books/edition/Cannibals_Witches_and_Divorce/qZRpAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Cannibals, Witches, and Divorce] * [https://www.google.co.uk/books/edition/The_Problem_of_Religious_Knowledge/C29LAAAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Problem of Religious Knowledge] * [https://www.google.co.uk/books/edition/Essex_and_the_Great_Revolt_of_1381/J8RzAAAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Essex and the Great Revolt of 1381] * [https://www.google.co.uk/books/edition/Transactions_of_the_London_and_Middlesex/4dtJAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover TLMAS] * [https://www.google.co.uk/books/edition/Shakespeare_Politics_and_the_State/Mn9lAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare, Politics and the State] * [https://www.google.co.uk/books/edition/Allegories_of_Power_in_the_England_of_El/LIYgAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Allegories of Power in the England of Elizabeth] * [https://www.google.co.uk/books/edition/William_Shakespeare/rIVlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover William Shakespeare] * [https://www.google.co.uk/books/edition/Women_s_Matters/PDRlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Women's Matters] * [https://www.google.co.uk/books/edition/The_Weak_King_Dilemma_in_the_Shakespeare/0bJlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Weak King Dilemma in the Shakespearean History Play] * [https://www.google.co.uk/books/edition/The_Book_Known_as_Q/S2tlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Book Known as Q] * [https://www.google.co.uk/books/edition/Fields_of_Vision/OD0eAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Fields of Vision] * [https://www.google.co.uk/books/edition/Ungodly_Delights/RKgcAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Ungodly Delights] * [https://www.google.co.uk/books/edition/The_Shakespeare_Handbook/rLRlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Shakespeare HandboOK] * [https://www.google.co.uk/books/edition/Humanities/y5FZAAAAYAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Humanities] * [https://www.google.co.uk/books/edition/Richard_II_by_William_Shakespeare/Bb3yAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Richard II by William Shakespeare] * [https://www.google.co.uk/books/edition/King_Richard_II/50NnAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover King Richard II] * [https://www.google.co.uk/books/edition/Murder_Under_Trust_Or_The_Topical_Macbet/0oNlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Murder under trust] * [https://www.google.co.uk/books/edition/The_Shakespearean_Kings/tHBlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Shakespearean Kings] * [https://www.google.co.uk/books/edition/America_the_Mabr_e_y_Experience/mRQ3AAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover America, the Mabr(e)y Experience: Resistance, Revolution & Civil War] * [https://www.google.co.uk/books/edition/Richard_II/ZDEkAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Richard II: An Annotated Bibliography, Volume 2] * [https://www.google.co.uk/books/edition/The_Batsford_Companion_to_Medieval_Engla/ev78b9EJQy0C?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Batsford Companion to Medieval England] * [https://www.google.co.uk/books/edition/Shakespeare_s_Unruly_Women/FKFlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare's Unruly Women] * [https://www.google.co.uk/books/edition/Shakespeare_and_Others/iFEgAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare and Others] * [https://www.google.co.uk/books/edition/Kings_and_Chroniclers/L1wpAAAAYAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Kings and Chroniclers] * [https://www.google.co.uk/books/edition/A_Kingdom_for_a_Stage/UzxlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover A Kingdom for a Stage] * [https://www.google.co.uk/books/edition/The_House_of_Commons/Ezz4OZuYVFYC?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The History of Parliament: The House of Commons 1558-1603 (3 v.)] * [https://www.google.co.uk/books/edition/Shakespeare_Soul_of_the_Age/nMYCAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover shakespeare, Soul of the Age] * [https://www.google.co.uk/books/edition/After_Poststructuralism/TOaEAAAAIAAJ?hl=en&gbpv=0&bsq=%22I%20am%20Richard%20II,%20know%20ye%20not%20that?%22 After Poststructuralism: Interdisciplinarity and Literary Theory] * [https://www.google.co.uk/books/edition/The_Unschooled_Mind/C7WnYtt219IC?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Unschooled Mind] * [https://www.google.co.uk/books/edition/Elizabeth_I/hHZnAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Eliz I] * [https://www.google.co.uk/books/edition/Dramas_of_Christian_Time/mnIqAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Dramas of Christian Time] * [https://www.google.co.uk/books/edition/Elizabeth_I/XjQmAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Elizabeth I: The Shrewdness of Virtue] * [https://www.google.co.uk/books/edition/John_Dryden/9Q1aAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover John Dryden] * [https://www.google.co.uk/books/edition/Shakespeare_and_Early_Modern_Political_T/DUwhAwAAQBAJ?hl=en&gbpv=1&dq=%22shakespeare%22+%2B+%22political+propaganda%22&pg=PA259&printsec=frontcover Shakespeare and Early Modern Political Thought] * [https://www.google.co.uk/books/edition/The_English_History_Play_in_the_age_of_S/5TT-AQAAQBAJ?hl=en&gbpv=1&dq=%22shakespeare%22+%2B+%22political+propaganda%22&pg=PA158&printsec=frontcover The English History Play in the Age of Shakespeare] * [https://www.google.co.uk/books/edition/Shakespeare_and_the_Political/rEcREQAAQBAJ?hl=en&gbpv=1&dq=%22shakespeare%22+%2B+%22political+propaganda%22&pg=PA215&printsec=frontcover Shakespeare and the Political] * [https://www.google.co.uk/books/edition/William_Shakespeare_Subject_of_the_Crown/a7G6DAAAQBAJ?hl=en&gbpv=1&dq=%22shakespeare%22+%2B+%22political+propaganda%22&pg=PT18&printsec=frontcover William Shakespeare - Subject of the Crown?] * [https://www.google.com/search?q=%22shakespeare%22+%2B+%22political+propaganda%22&client=firefox-b-d&hs=4AQ&sca_esv=6d4ade7bd26771c9&udm=36&biw=2510&bih=1307&tbs=cdr%3A1%2Ccd_min%3A2000%2Ccd_max%3A2099&sxsrf=ANbL-n6I6Pkwl7mmdHK6N1xPQXLbGBIOSg%3A1776853062010&ei=RqDoaZUvztiFsg_I5bToDw&ved=0ahUKEwiV6tC8nYGUAxVObEEAHcgyDf0Q4dUDCBM&uact=5&oq=%22shakespeare%22+%2B+%22political+propaganda%22&gs_lp=EhBnd3Mtd2l6LW1vZGVsZXNzIiYic2hha2VzcGVhcmUiICsgInBvbGl0aWNhbCBwcm9wYWdhbmRhIjIIECEYoAEYwwRInQlQxgZYuwdwAXgAkAEAmAF_oAHPAaoBAzEuMbgBA8gBAPgBAZgCAqACVsICCxAAGIAEGKIEGLADmAMAiAYBkAYCkgcBMqAHowOyBwExuAdTwgcDMC4yyAcEgAgB&sclient=gws-wiz-modeless The Nazi Appropriation of Shakespeare: Cultural Politics in] {{refend}} ja9y68rmkeqmku8osy7aon9g9kquxaw 2818523 2818522 2026-07-19T11:53:50Z Fortuna imperatrix mundi 1058581 sfn 2818523 wikitext text/x-wiki ==Background== ===Richard II=== ===Elizabeth I=== ====Essex's rebellion==== ==IaR2== Lambare's accoiunt of his intervieew with the queen is well known.{{sfn|Orgel|2011|pp=11–43}} ===Political anxiety and tension=== ===Censorship=== ==Consequences and aftermath== == Notes == {{reflist|group=note}} ==References== {{Reflist|20em}} ==Bibliography== {{refbegin|30em|indent=yes}} * [https://link.springer.com/chapter/10.1057/9780230307261_2 Orgel, S. (2011). Prologue: I am Richard II. In: Petrina, A., Tosi, L. (eds) Representations of Elizabeth I in Early Modern Culture. Palgrave Macmillan, London. https://doi.org/10.1057/9780230307261_2] * [https://broadlytextual.com/2017/12/15/i-am-richard-ii-know-ye-not-that-drama-and-political-anxiety-in-shakespeares-london/ Hixon, E., Syracuse Univ] * Bate, Jonathan (2008). Soul of the Age. London: Penguin. pp. 256–286. ISBN 978-0-670-91482-1. * [https://theconversation.com/richard-ii-by-william-shakespeare-why-the-divine-right-of-kings-still-matters-186648 McFarlane, K., Univ South Australia] * [https://muse.jhu.edu/article/31090/summary Lemon, Rebecca. "The Faulty Verdict in "The Crown v. John Hayward"." SEL Studies in English Literature 1500-1900, vol. 41 no. 1, 2001, p. 109-132. Project MUSE, https://dx.doi.org/10.1353/sel.2001.0009] * [https://brill.com/display/book/9789401211666/B9789401211666-s009.xml?language=en&srsltid=AfmBOoqTUaTQr0msNWRdtEZZ9qQ3rNcVkSoBvDPXuGv7nRvHrC8t5OzH Kizelbach, U. (2014), In The Pragmatics of Early Modern Politics: Power and Kingship in Shakespeare’s History Plays. Leiden, The Netherlands: Brill. https://doi.org/10.1163/9789401211666_009] * [https://www.google.co.uk/books/edition/Richard_II/z-kOAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Lopez, R2] * [https://www.manchesterhive.com/display/9781526130532/9781526130532.00008.xml?print rgel, S. (2017). "I am Richard II". In Spectacular Performances. Manchester, England: Manchester University Press. Retrieved Apr 22, 2026, from https://doi.org/10.7765/9781526130532.00008] * [https://www.researchgate.net/publication/271041996_Was_Elizabeth_I_Richard_II_The_Authenticity_of_Lambarde%27s_%27Conversation%27 Scott-Warren, Jason. (2012). Was Elizabeth I Richard II?: The Authenticity of Lambarde's 'Conversation'. The Review of English Studies. 64. 208-230. 10.1093/res/hgs062] * Stanley Wells, “Introduction” in Richard II, The New Penguin Shakespeare, ed. Stanley Wells (London: Penguin Books, 1969), 13. * [https://www.cambridge.org/core/books/abs/shakespeare-survey/shakespeare-and-history-divergencies-and-agreements/B065A23215FD86BEE8E1CFD51DC7C1FB Ives EW. Shakespeare and History: Divergencies and Agreements. In: Wells S, ed. Shakespeare Survey. Shakespeare Survey. Cambridge University Press; 1986:19-36] * [https://www.jstor.org/stable/457855 Albright, Evelyn May. “Shakespeare’s Richard II, Hayward’s History of Henry IV, and the Essex Conspiracy.” PMLA, vol. 46, no. 3, 1931, pp. 694–719. JSTOR, https://doi.org/10.2307/457855. Accessed 22 Apr. 2026] * [https://dsc.duq.edu/cgi/viewcontent.cgi?article=2800&context=etd Morris, A. (2019). "within the hollow crown": Performing Kingship in Richard II and Henry IV Part One (Master's thesis, Duquesne University). Retrieved from https://dsc.duq.edu/etd/1771] * [https://digitalcommons.lib.uconn.edu/cgi/viewcontent.cgi?article=1122&context=srhonors_theses Scannell, Sarah J., "Shakespeare's Richard II and Henry V and Political Rebellions in the Reign of Queen Elizabeth I" (2010). Honors Scholar Theses, 138] * [https://compass.onlinelibrary.wiley.com/doi/abs/10.1111/j.1741-4113.2011.00873.x Luecking Frost, L. (2012), “A Kyng That Ruled All By Lust”: Richard II in Elizabethan Literature. Literature Compass, 9: 183-198. https://doi.org/10.1111/j.1741-4113.2011.00873.x] * [https://www.academia.edu/8347491/A_Law_Case_in_Verse_Venus_and_Adonis_and_the_Authorship_Question Stritmatter, R., A Law Case in Verse: Venus and Adonis and the Authorship Question, 2004 University of Tennessee Law Review] * [https://www.researchgate.net/publication/304647742_Prologue_I_am_Richard_II Orgel, Stephen. (2011). Prologue: I am Richard II. 10.1057/9780230307261_2] * [https://go.gale.com/ps/i.do?id=GALE%7CA314252957&sid=googleScholar&v=2.1&it=r&linkaccess=abs&issn=15256863&sw=w&p=AONE&userGroupName=anon%7Ebf79dda0&aty=open-web-entry Egan, Michael. "The Essex Rebellion and Richard II: why wasn't Shakespeare arrested?" Shakespeare Oxford Newsletter, vol. 48, no. 3, summer-fall 2012, p. 20] * [https://www.cambridge.org/core/books/abs/richard-ii/stagehistory/D62D9CA061F03B2AE8D56E856288EEDA 1. Child H, Shakespeare W. THE STAGE-HISTORY. In: Dover Wilson J, ed. Richard II: The Cambridge Dover Wilson Shakespeare. Cambridge Library Collection - Literary Studies. Cambridge University Press; 2009:lxxvii-xcii] * [https://www.openstarts.units.it/entities/publication/16d8c447-62f4-48e8-8347-5d56af674aa3/details Snelling, D., 'Prospero on the Coast of Bohemia. "Nihil sed rara et miranda amabat". Diary of Melchior Goldast upon the late Emperor Rudolph II' 1994] * [https://www.jstor.org/stable/2902265 Gajowski, Evelyn. Shakespeare in Performance: King Richard IIby Margaret Shewring. Shakespeare Quarterly, vol. 49, no. 3, 1998, pp. 328–30. JSTOR, https://doi.org/10.2307/2902265. Accessed 22 Apr. 2026] * [https://muse.jhu.edu/article/217842/summary Mosse, Ramona Franziska. Review of Richard II. Shakespeare Bulletin, vol. 25 no. 2, 2007, p. 73-77. Project MUSE, https://dx.doi.org/10.1353/shb.2007.0036] * [https://www.journals.uchicago.edu/doi/abs/10.1086/665894?journalCode=rq Petrina and Tosi. Representations of Elizabeth I in Early Modern Culture Elizabeth Pentland Renaissance Quarterly 2012 65:1, 274-276] * [https://journals.co.za/doi/10.4314/sisa.v26i1.4 The construction of a king : waste, effeminacy and queerness in Shakespeare's Richard II Higginbotham, D.] * [https://academic.oup.com/book/2363/chapter-abstract/142533948?redirectedFrom=fulltext Erskine-Hill, Howard, 'The Second Tetralogy and After', Poetry and the Realm of Politics: Shakespeare to Dryden (Oxford, 1996; online edn, Oxford Academic, 3 Oct. 2011), https://doi.org/10.1093/acprof:oso/9780198117315.003.0004] * [https://www.academia.edu/123917862/Shakespeare_s_Richard_II_in_the_minds_of_the_Elizabethan_audiences Shakespeare’s Richard II in the minds of the Elizabethan audiences] * [https://journals.sagepub.com/doi/10.1177/03064220221084555 Plays, protests and the censor’s pencil] * [https://www.indexoncensorship.org/2022/03/plays-protests-and-the-censors-pencil/ Plays, protests and the censor’s pencil A brittle Gloriana: staging the deposition of Queen Elizabeth I]. * Saul, N., Richard II * [https://www.academia.edu/3236651/_A_god_on_earth_thou_art_Richard_II_and_the_nature_of_Kingship “A god on earth thou art”: Richard II and the nature of Kingship] * [https://www.jstor.org/stable/457398 Albright, Evelyn May. “Shakespeare’s Richard II and the Essex Conspiracy.” PMLA, vol. 42, no. 3, 1927, pp. 686–720. JSTOR, https://doi.org/10.2307/457398. Accessed 22 Apr. 2026] * [https://www.berghahnjournals.com/view/journals/critical-survey/37/3/cs370301.xml Holderness, G. (2025). Introduction: Shakespeare's Wars of the Roses. Critical Survey, 37(3), 1-4. Retrieved Apr 22, 2026, from https://doi.org/10.3167/cs.2025.370301] * [https://www.folger.edu/podcasts/shakespeare-unlimited/tragedy-richard-ii-henry-iv-helen-castor/ Folger: Shakespeare Unlimited: Episode 249] * [https://www.rachaeldickzen.com/blog/2021/1/10/charlesiii The Historical and Shakespearean References in King Charles III] * Kermode, The Age of Shakespeare, pg.53. * [https://heinonline.org/HOL/login-hol?redirect_url=https%3A%2F%2Fheinonline.org%2FHOL%2Fmojo%3F%252Fmojo%3D%26auth_token%3DyhcMOu8W7PCc5OrmTqVnr2qFPpoiOdhM5uQmkQyWufU%253D%26casa_token%3D%26div%3D5%26g_sent%3D1%26paywall_url%3D%26referrer%3Dhttps%253A%252F%252Fheinonline.org%252FHOL%252FPage%253Fhandle%253Dhein.journals%252Fneastlr1%26timestamp%3D1776855600%26url%3D%252FHOL%252FPage%253Fhandle%253Dhein.journals%252Fneastlr1%2526div%253D5%2526id%253D%2526page%253D Jones, H., ''I AM RICHARD II, KNOW YE NOT THAT?' - WORKS. OF WILLIAM SHAKESPEARE AND THE ART OF. GOVERNANCE'''] * Hattaway, M., William Shakespeare: Richard II (Literature Insights, 9781847600349) * [https://books.openedition.org/psn/8305 Findlay, A., “Good sometimes queen” (V.1.37). Richard II, Mary Stuart and the Politics of Queenship, 205-219] * [https://pure.manchester.ac.uk/ws/portalfiles/portal/54508829/FULL_TEXT.PDF Pritchard, K., 'Legitimacy, Illegitimacy and Sovereignty in Shakespeare’s British Plays' (2011 University of Manchester PhD] * CHATELANAT, Marine. Histrionic Future Kings: The Politics of Metadrama in Shakespeare’s Richard III, Richard II, and Henry IV Part 1. Master, 2023. * [https://www.persee.fr/doc/xvii_0291-3798_2004_num_59_1_1993 Mayer, J-C., 'The "Parliament Sceane" in Shakespeare's King Richard II', XVII-XVIII. Revue de la Société d'études anglo-américaines des XVIIe et XVIIIe siècles Année 2004 59 pp. 27-42] * [https://www.emerald.com/md/article-abstract/38/5/315/285132/Managers-as-kings-Shakespeare-on-modern-leadership?redirectedFrom=fulltext Egan M (2000), "Managers as kings: Shakespeare on modern leadership". Management Decision, Vol. 38 No. 5 pp. 315–327, doi: https://doi.org/10.1108/00251740010340490] * [https://d-nb.info/1279081511/34 Quabeck (2022): William Shakespeare, in: Thomas Gutmann, Eberhard Ortland, Klaus Stierstorfer, eds., Encyclopedia of Law and Literature (last edited 26 October 2022), doi: 10.17879/12009577118] * [https://lawliberty.org/book-review/was-the-bard-writing-about-the-donald/ Sullivan, V., 'Was the Bard Writing about The Donald?'] * [https://deveresociety.co.uk/wp-content/uploads/2022/11/FINAL_NL_Apr2022-Haste-July-Letterresponse.pdf Haste, I., 'THE PLAY ON THE EVE OF THE ESSEX REBELLION'] * [https://www.cherylmtaylor.com/2017/11/08/a-companion-to-shakespeares-richard-ii/ A Companion to Shakespeare’s Richard II] * [https://www.city-journal.org/article/shakespeares-richards Dalrymple, T., Shakespeare’s Richards: The playwright’s two historical dramas offer contrasts between political pathologies] * [https://hfriedberg.web.wesleyan.edu/engl205/wshakespeare/plotsandrebelions.htm ELIZABETHAN PLOTS AND REBELLIONS] * [https://www.jstor.org/stable/2869708 Rackin, Phyllis. “The Role of the Audience In Shakespeare’s Richard II.” Shakespeare Quarterly, vol. 36, no. 3, 1985, pp. 262–81. JSTOR, https://doi.org/10.2307/2869708. Accessed 22 Apr. 2026.] * [https://www.psychologytoday.com/ie/blog/whats-in-name/201603/group-think-and-academia-shocking-shakespeare-shenanigans Group Think and Academia: Shocking Shakespeare Shenanigans] * [https://journals.sagepub.com/doi/10.1177/01847678221099975 Hiscopck, A., '"Why I should welcome such a guest as grief [?": Lodging and dislodging in Shakespeare's Richard II'] * [https://academic.oup.com/book/51779/chapter-abstract/420487796?redirectedFrom=fulltext 'Epigraph', Imagining Time in the English Chronicle Play: Historical Futures, 1590-1660 (Oxford, 2023; online edn, Oxford Academic, 19 Oct. 2023), https://doi.org/10.1093/oso/9780198872658.002.0009] * [https://journals.openedition.org/interfaces/319?lang=en Blake & Shakespeare] * [https://www.cambridge.org/core/books/abs/shakespeare-and-the-visual-imagination/richard-iiand-the-politics-of-perspective/23A5D4615E2A8BF9380BAF1ECE99B46D Sillars S. Richard IIand the Politics of Perspective. In: Shakespeare and the Visual Imagination. Cambridge University Press; 2015:133-162.] * [https://www.persee.fr/doc/xvii_0291-3798_2004_num_59_1_1992 Le miroir des rois: de Richard II à Henry VIII] * [https://www.cambridge.org/core/books/abs/king-richard-ll/introduction/25419329A06BC1C2E4D3DDBF331B7F91 Shakespeare W, McEachern C. Introduction. In: Gurr A, ed. King Richard Ll. The New Cambridge Shakespeare. Cambridge University Press; 2018:1-66.] * [https://www.researchgate.net/publication/372862689_A_Critical_Discourse_Study_of_Shakespeare%27s_Theological_Conceptions_in_Acts_IV_and_V_of_Richard_II_The_%27Divine_Mandate%27_of_Richard_Kingship_Falls_Apart A Critical Discourse Study of Shakespeare’s Theological Conceptions in Acts IV and V of Richard II: The ‘Divine Mandate’ of Richard Kingship Falls Apart] * [https://www.jstor.org/stable/10.1525/lal.2004.16.3.313 Richard II, Bracton, and the End of Political Theology] * [https://dokumen.pub/the-art-of-law-in-shakespeare-9781509905478-9781509905508-9781509905492.html The Art of Law in Shakespeare] * [https://www.cambridge.org/core/books/abs/cambridge-companion-to-shakespeare-and-religion/providence-and-divine-right-in-the-english-histories/DBE9BFEBF5C199F9DD72D31ED9C3B041 Mayer J-C. Providence and Divine Right in the English Histories. In: Hamlin H, ed. The Cambridge Companion to Shakespeare and Religion. Cambridge Companions to Literature. Cambridge University Press; 2019:151-167] * [https://www.researchgate.net/publication/315698000_THE_CHOMSKY-FOUCAULT_DEBATE_WHERE_DOES_THE_BARD_STAND_IN_RICHARD_II Salami, Ali & Riahi Nouri, Amir. (2016). THE CHOMSKY-FOUCAULT DEBATE: WHERE DOES THE BARD STAND IN RICHARD II?] * [https://dokumen.pub/shakespeare-richard-ii-9781350363168-9780333416709.html Richard II] * [https://www.jstor.org/stable/457907?seq=1 Heffner, Ray. “Shakespeare, Hayward, and Essex.” PMLA, vol. 45, no. 3, 1930, pp. 754–80. JSTOR, https://doi.org/10.2307/457907. Accessed 22 Apr. 2026] * [https://www.cambridge.org/core/books/abs/cambridge-companion-to-shakespeare-and-war/shakespeare-on-civil-and-dynastic-wars/00D0269CFF998876DB961FB01C22954D Shakespeare on Civil and Dynastic Wars] * [https://brill.com/display/book/edcoll/9789004356900/B9789004356900_012.xml?srsltid=AfmBOopJMjwJAQnUGME9FmT6Gwg0ih9pArbXBdTJbN6YAqbdXqbkGyuo Is a Dramatic Theory of History Possible? Shakespeare’s Richard ii and ‘Historical Truth’] * [https://www.jstor.org/stable/24475607 THE CHILD'S TWO BODIES: SHAKESPEARE, SOVEREIGNTY, AND THE END OF SUCCESSION] * [https://dokumen.pub/english-renaissance-drama-an-introduction-to-theatre-and-theatres-in-shakespeares-time-9781847600202-9781847600714.html English Renaissance Drama : An Introduction to Theatre and Theatres in Shakespeare’s Time] * [https://www.cambridge.org/core/books/abs/believing-in-shakespeare/person-plot-place/BD4EF4CBA6E7D8E2ECE5519FA1D08C16 1. McEachern C. Person, Plot, Place. In: Believing in Shakespeare: Studies in Longing. Cambridge University Press; 2018:181-296] * [https://www.enotes.com/topics/richard-2/criticism/richard-ii-vol-91/criticism-production-reviews-1000228/nigel-saul-review-date-12-may-2000 With an Eye to the Present] * [https://www.academia.edu/128303633/Shakespeare_s_Other_Eden_Richard_II_by_Ouroboros_Theatre_Company_2013_ Shakespeare’s Other Eden Richard II by Ouroboros Theatre Company (2013)] * [https://dokumen.pub/shakespeare-upstart-crow-to-sweet-swan-1592-1623-9781472555496-9781408130148-9781408139189.html Shakespeare: Upstart Crow to Sweet Swan 1592-1623] * [https://www.jstor.org/stable/41917215 RACKIN, PHYLLIS. “Temporality, Anachronism, and Presence in Shakespeare’s English Histories.” Renaissance Drama, vol. 17, 1986, pp. 101–23. JSTOR, http://www.jstor.org/stable/41917215. Accessed 22 Apr. 2026.] * [https://www.cambridge.org/core/journals/greece-and-rome/article/abs/alexander-philotas-and-the-origins-of-modern-historiography/D90568C0A1131952625E0A3B44F43EC8 Stoneman R. Alexander, Philotas, and the origins of modern historiography. Greece and Rome. 2013;60(2):296-312. doi:10.1017/S0017383513000119] * [https://www.academia.edu/35800720/Shakespeares_Richard_II_and_Elizabethan_politics Shakespeare's Richard II and Elizabethan politics] * [https://www.cambridge.org/core/books/abs/rethinking-historicism-from-shakespeare-to-milton/old-new-now/2E8CBEFFEFB11D8B843BAF2C108E9768 Introduction - Old, New, Now] * [https://www.academia.edu/24392476/Richard_II_hamartia_and_climax Richard II hamartia and climax] * [https://www.cambridge.org/core/books/abs/babylonian-jews-and-sasanian-imperialism-in-late-antiquity/kings-and-religion-in-the-talmud-and-in-the-imagination-of-sasanian-communities/E2DCC2F3E0FBA74ABE543DDD2DD7F1A7 Kings and Religion in the Talmud and in the Imagination of Sasanian Communities] * Stephen Greenblatt, “Introduction,” in The Power of Forms in the English Renaissance, ed. Stephen Greenblatt (Norman, OK: 1982), 3-6] * [https://www.academia.edu/30472558/Americas_Donald_Trump_Shakespeares_Richard_II_15_Dec_2016 America's Donald Trump, Shakespeare's Richard II] * [https://www.academia.edu/5321346/Looking_Back_Shakespeare_s_Indebtedness_to_Chaucer_and_the_Representations_of_Chivalry_in_King_Richard_II_The_Two_Noble_Kinsmen_and_the_Knight_s_Tale Looking Back: Shakespeare’s Indebtedness to Chaucer and the Representations of Chivalry in King Richard II, The Two Noble Kinsmen and the Knight’s Tale] * [https://www.academia.edu/92423912/The_Good_Usurper_in_the_eyes_of_God_and_the_people_An_analysis_of_the_role_of_the_usurper_in_Shakespeares_Richard_II_and_Henry_IV_Part_1 The Good Usurper in the eyes of God and the people: An analysis of the role of the usurper in Shakespeare's Richard II and Henry IV Part 1] * [https://www.academia.edu/91789871/The_Bacon_Shakespeare_Manuscript_Hitherto_known_as_the_Northumberland_Manuscript_which_originally_Contained_Copies_of_his_Shakespeare_Plays_Richard_II_and_Richard_III THE BACON-SHAKESPEARE MANUSCRIPT (HITHERTO KNOWN AS THE NORTHUMBERLAND MANUSCRIPT) WHICH ORIGINALLY CONTAINED COPIES OF HIS SHAKESPEARE PLAYS RICHARD II AND RICHARD III] * [https://www.google.co.uk/books/edition/Richard_II/f4gGCAAAQBAJ?hl=en&gbpv=1&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&pg=PA88&printsec=frontcover Richard II: Critical Essays] * [https://www.google.co.uk/books/edition/A_Companion_to_Shakespeare_s_Works_Volum/JlDNEAAAQBAJ?hl=en&gbpv=1&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&pg=PA400&printsec=frontcover A Companion to Shakespeare's Works, Volume II] * [https://www.google.co.uk/books/edition/Shakespeare_Reread/Z6JhDwAAQBAJ?hl=en&gbpv=1&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&pg=PA239&printsec=frontcover Shakespeare Reread] * [https://www.google.co.uk/books/edition/Shakespeare_in_the_Present/yTKWEAAAQBAJ?hl=en&gbpv=1&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&pg=PT8&printsec=frontcover Shakespeare in the Present] * [https://www.google.co.uk/books/edition/Studying_Shakespeare/1N4FBAAAQBAJ?hl=en&gbpv=1&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&pg=PA129&printsec=frontcover Studying Shakespeare A Practical Introduction] * [https://www.google.co.uk/books/edition/Critical_Essays_on_William_Faulkner/kAGBEAAAQBAJ?hl=en&gbpv=1&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&pg=PT178&printsec=frontcover Critical Essays on William Faulkner] * [https://www.google.co.uk/books/edition/Shakespeare/QY6aEQAAQBAJ?hl=en&gbpv=1&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&pg=PA216&printsec=frontcover Shakespeare: An Anthology of Criticism and Theory 1945-2000] * [https://www.google.co.uk/books/edition/The_Reign_of_Richard_II/cv0WAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Dodd, G., R2] * [https://www.google.co.uk/books/edition/Shakespeare_and_the_Institution_of_Theat/Nkt9DAAAQBAJ?hl=en&gbpv=1&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&pg=PA108&printsec=frontcover Shakespeare and the Institution of Theatre] * [https://www.google.co.uk/books/edition/The_Theory_of_the_King_s_Two_Bodies_in_t/8SRXAAAAYAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Theory of the King's Two Bodies in the Age of Shakespeare] * [https://www.google.co.uk/books/edition/Stealing_the_Story/RaJlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Stealing the Story: Shakespeare's Self-Conscious Use of the Mimetic Tradition in the Tragedies] * [https://www.google.co.uk/books/edition/The_Greenwood_Companion_to_Shakespeare_O/JkcgAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Greenwood Companion to Shakespeare: Overviews and the history plays] * [https://www.google.co.uk/books/edition/British_and_Irish_Literature_and_Its_Tim/xH4jAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover British and Irish Literature and Its Times] * [https://www.google.co.uk/books/edition/Shakespeare_s_Philosophy_of_History_Reve/1jwgAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare's Philosophy of History Revealed in a Detailed Analysis of Henry V and Examined in Other History Plays] * [https://www.google.co.uk/books/edition/Shakespeare_and_His_Contemporaries/7em7coOthxQC?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare and His Contemporaries] * [https://www.google.co.uk/books/edition/The_Politics_of_the_Public_Sphere_in_Ear/0sGHAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Politics of the Public Sphere in Early Modern England] * [https://www.google.co.uk/books/edition/Shakespeare_and_the_Catholic_Religion/Fl4gAQAAIAAJ?hl=en&gbpv=0&bsq=%22I%20am%20Richard%20II,%20know%20ye%20not%20that?%22 Shakespeare and the Catholic religion] * [https://www.google.co.uk/books/edition/Proceedings_of_the_British_Academy_Volum/8CsoAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Lectures] * [https://www.google.co.uk/books/edition/Slander_and_Censorship_in_Late_Sixteenth/wb_oAvuJgbAC?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Slander and Censorship in Late Sixteenth Century Literature] * [https://www.google.co.uk/books/edition/Elizabethan_Literature_and_the_Law_of_Fr/vBFdAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Elizabethan Literature and the Law of Fraudulent Conveyance] * [https://www.google.co.uk/books/edition/Hamlet_History_and_commentary/yxgrAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Hamlet History etc] * [https://www.google.co.uk/books/edition/William_Shakespeare_A_Popular_Life/1BuaAAAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover William Shakespeare: a popular life] * [https://www.google.co.uk/books/edition/A_State_of_Mind/pCiIAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover A State of Mind?] * [https://www.google.co.uk/books/edition/Shakespeare_s_Dramatic_Genres/J5FlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare's Dramatic Genres] * [https://www.google.co.uk/books/edition/Literature_Criticism_from_1400_to_1800/O6VkAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Literature Criticism from 1400 to 1800 Volume 89] * [https://www.google.co.uk/books/edition/Shakespeare_the_Papist/LPwNAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare the Papist] * [https://www.google.co.uk/books/edition/Goslicius_Ideal_Senator_and_His_Cultural/1jc7AQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Goslicius' Ideal Senator and His Cultural Impact Over the Centuries] * [https://www.google.co.uk/books/edition/Shakespeare_by_Another_Name/FqllAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover "Shakespeare" by Another Name] * [https://www.google.co.uk/books/edition/Shakespeare_s_Friends/AlZlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare's Friends] * [https://www.google.co.uk/books/edition/Dr_Simon_Forman/qHceAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Dr Simon Forman] * [https://www.google.co.uk/books/edition/William_Shakespeare_the_Wars_of_the_Rose/dZFlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover William Shakespeare, the Wars of the Roses and the historians] * [https://www.google.co.uk/books/edition/Shakespearean_Criticism/2TdlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespearean Criticism] * [https://www.google.co.uk/books/edition/Poetry_History_and_Antiquarianism_in_Ref/jwIhAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Poetry, History, and Antiquarianism in Reformation England] * [https://www.google.co.uk/books/edition/Shakespeare_and_the_Prince_of_Love/D0QgAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare and the Prince of Love] * [https://www.google.co.uk/books/edition/Paper_Bullets_of_the_Brain/o0YgAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Paper Bullets of the Brain] * [https://www.google.co.uk/books/edition/As_You_Like_It/GiBaAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover As You Like It: Third Series] * [https://www.google.co.uk/books/edition/William_Shakespeare/WCqaAAAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover William Shakespeare, Dominic Shellard] * [https://www.google.co.uk/books/edition/Poets_and_God/0XZlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Poets and God] * [https://www.google.co.uk/books/edition/Law_and_Literature/Ax5MAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Law and Literature Volume 16] * [https://www.google.co.uk/books/edition/The_Embodied_Word/FV8sAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Embodied Word] * [https://www.google.co.uk/books/edition/The_Case_for_Shakespeare/WaRlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Case for Shakespeare: The End of the Authorship Question] * [https://www.google.co.uk/books/edition/Explorations_in_Renaissance_Culture/_SYrAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Explorations in Renaissance Culture Volumes 33-34] * [https://www.google.co.uk/books/edition/The_Touch_of_the_Real/ewdaAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Touch of the Real: Essays in Early Modern Culture in Honour of Stephen Greenblatt] * [https://www.google.co.uk/books/edition/Wotton_and_His_Worlds/ZfgNAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Wotton and his Worlds] * [https://www.google.co.uk/books/edition/Theatre_and_Religion/wo1lAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Theatre and Religion Lancastrian Shakespeare] * [https://www.google.co.uk/books/edition/Trying_Treason/TOKxAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Trying Treason] * [https://www.google.co.uk/books/edition/Willing_Subjects/IEX0sGwT1QQC?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Willing Subjects] * [https://www.google.co.uk/books/edition/Symbolism/Bt0ZAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Symbolism] * [https://www.google.co.uk/books/edition/Performing_Shakespeare/35pQAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Performing Shakespeare] * [https://www.google.co.uk/books/edition/Soul_of_the_Age/e0UgAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Soul of the Age] * [https://www.google.co.uk/books/edition/England/aD9nAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover England] * [https://www.google.co.uk/books/edition/Elizabeth_I/-GtnAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Elizabeth I] * [https://www.google.co.uk/books/edition/King_Richard_II/oGUMX4RntjgC?hl=en&gbpv=1&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&pg=PA25&printsec=frontcover King Richard II] * [https://www.google.co.uk/books/edition/Shakespeare_and_the_Legal_Imagination/OXPvBqQLw-4C?hl=en&gbpv=1&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&pg=PA38&printsec=frontcover Shakespeare and the legal imagination] * [https://www.google.co.uk/books/edition/Shakespeare_s_Theatre/GxN3ue9_r3oC?hl=en&gbpv=1&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&pg=PA69&printsec=frontcover Shakespeare's Theatre] * [https://www.google.co.uk/books/edition/Critical_Essays_on_Shakespeare_s_Richard/AaYoAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Critical Essays on Shakespeare's Richard II] * [https://www.google.co.uk/books/edition/The_Reign_of_Richard_II_Essays_in_Honour/y3xnAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Reign of Richard II: Essays in Honour of May McKisack] * [https://www.google.co.uk/books/edition/Poetry_and_the_Realm_of_Politics/oQFaAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Poetry and the Realm of Politics] * [https://www.google.co.uk/books/edition/Shakespeare/BM0mAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare] * [https://www.google.co.uk/books/edition/Shakespearean_Politics/oTdlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespearean Politics] * [https://www.google.co.uk/books/edition/Shakespeare_the_Theatrical_Dimension/wl4gAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare, the Theatrical Dimension] * [https://www.google.co.uk/books/edition/Who_was_Kit_Marlowe/zQ1aAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Kit Marlowe etc] * [https://www.google.co.uk/books/edition/From_Page_to_Performance/beQKAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover From Page to Performance] * [https://www.google.co.uk/books/edition/Exploring_Tudor_England/ax56AAAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Exploring Tudor England] * [https://www.google.co.uk/books/edition/The_Movement_Towards_Subversion/vyJaAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Movement Towards Subversion] * [https://www.google.co.uk/books/edition/Shakespeare_s_Typological_Satire/G5BlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare's Typological Satire] * [https://www.google.co.uk/books/edition/Shakespeare_Recycled/zzNlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare Recycled] * [https://www.google.co.uk/books/edition/Reinventing_the_Middle_Ages_the_Renaissa/fXFnAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Reinventing the Middle Ages & the Renaissance] * [https://www.google.co.uk/books/edition/The_Mysterious_William_Shakespeare/WnllAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Mysterious William Shakespeare] * [https://www.google.co.uk/books/edition/Richard_II/GHhlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Richard II Critical Essays] * [https://www.google.co.uk/books/edition/William_Shakespeare/WJvC6gu_I0gC?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover William Shakespeare: Records and Images] * [https://www.google.co.uk/books/edition/Shakespeare_and_the_Actors/HYtlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare and the Actors] * [https://www.google.co.uk/books/edition/Shakespeare_the_Man/BVdlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare the man] * [https://www.google.co.uk/books/edition/Henry_V/zXllAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Henry V: A Guide to the Play] * [https://www.google.co.uk/books/edition/Shakespearean_Contingencies/Cw1NAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespearean Contingencies] * [https://www.google.co.uk/books/edition/Renaissance_Drama/E60kAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Renaissance Drama 1990] * [https://www.google.co.uk/books/edition/Language_Discourse_Sign/uH4oAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Language, Discourse, Sign] * [https://www.google.co.uk/books/edition/Mock_Kings_in_Medieval_Society_and_Renai/T98KAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Mock kings in medieval society and Renaissance drama] * [https://www.google.co.uk/books/edition/Shakespeare_Invention_of_the_Human/ojHirImrtYoC?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare: Invention of the Human] * [https://www.google.co.uk/books/edition/Shakespeare/wn5lAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare] * [https://www.google.co.uk/books/edition/Persons_in_Groups/rQ24AAAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Persons in Groups] * [https://www.google.co.uk/books/edition/All_Semblative_a_Woman_s_Part/0DlaAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover All Semblative a Woman's Part?] * [https://www.google.co.uk/books/edition/Crossing_the_Mirror/qRZNAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Crossing the Mirror] * [https://www.google.co.uk/books/edition/De_Vere_is_Shakespeare/dKJlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover De Vere is Shakespeare] * [https://www.google.co.uk/books/edition/William_Lambarde_Elizabethan_Antiquary_1/x1RnAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover William Lambarde, Elizabethan Antiquary, 1536-1601] * [https://www.google.co.uk/books/edition/The_Power_of_Forms_in_the_English_Renais/cPtZAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Power of Forms in the English Renaissance] * [https://www.google.co.uk/books/edition/Ravishment_and_Rememberance/G31LAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Ravishment and Rememberance] * [https://www.google.co.uk/books/edition/Shakespeare_and_His_Theatre/8A5ZQq3uOVQC?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare and His Theatre] * [https://www.google.co.uk/books/edition/Critical_Hermeneutics_and_Shakespeare_s/O10gAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Critical Hermeneutics and Shakespeare's History Plays] * [https://www.google.co.uk/books/edition/Christian_England/K-WfAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Christian England] * [https://www.google.co.uk/books/edition/Shakespeare_s_Religious_Background/xDSaAAAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare's Religious Background] * [https://www.google.co.uk/books/edition/Shylock/N4RlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shylock] * [https://www.google.co.uk/books/edition/The_Shakespeare_Legacy/MM5XAAAAYAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Shakespeare Legacy] * [https://www.google.co.uk/books/edition/Renaissance_Genres/0uFZAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Renaissance Genres] * [https://www.google.co.uk/books/edition/Cannibals_Witches_and_Divorce/qZRpAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Cannibals, Witches, and Divorce] * [https://www.google.co.uk/books/edition/The_Problem_of_Religious_Knowledge/C29LAAAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Problem of Religious Knowledge] * [https://www.google.co.uk/books/edition/Essex_and_the_Great_Revolt_of_1381/J8RzAAAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Essex and the Great Revolt of 1381] * [https://www.google.co.uk/books/edition/Transactions_of_the_London_and_Middlesex/4dtJAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover TLMAS] * [https://www.google.co.uk/books/edition/Shakespeare_Politics_and_the_State/Mn9lAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare, Politics and the State] * [https://www.google.co.uk/books/edition/Allegories_of_Power_in_the_England_of_El/LIYgAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Allegories of Power in the England of Elizabeth] * [https://www.google.co.uk/books/edition/William_Shakespeare/rIVlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover William Shakespeare] * [https://www.google.co.uk/books/edition/Women_s_Matters/PDRlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Women's Matters] * [https://www.google.co.uk/books/edition/The_Weak_King_Dilemma_in_the_Shakespeare/0bJlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Weak King Dilemma in the Shakespearean History Play] * [https://www.google.co.uk/books/edition/The_Book_Known_as_Q/S2tlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Book Known as Q] * [https://www.google.co.uk/books/edition/Fields_of_Vision/OD0eAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Fields of Vision] * [https://www.google.co.uk/books/edition/Ungodly_Delights/RKgcAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Ungodly Delights] * [https://www.google.co.uk/books/edition/The_Shakespeare_Handbook/rLRlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Shakespeare HandboOK] * [https://www.google.co.uk/books/edition/Humanities/y5FZAAAAYAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Humanities] * [https://www.google.co.uk/books/edition/Richard_II_by_William_Shakespeare/Bb3yAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Richard II by William Shakespeare] * [https://www.google.co.uk/books/edition/King_Richard_II/50NnAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover King Richard II] * [https://www.google.co.uk/books/edition/Murder_Under_Trust_Or_The_Topical_Macbet/0oNlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Murder under trust] * [https://www.google.co.uk/books/edition/The_Shakespearean_Kings/tHBlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Shakespearean Kings] * [https://www.google.co.uk/books/edition/America_the_Mabr_e_y_Experience/mRQ3AAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover America, the Mabr(e)y Experience: Resistance, Revolution & Civil War] * [https://www.google.co.uk/books/edition/Richard_II/ZDEkAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Richard II: An Annotated Bibliography, Volume 2] * [https://www.google.co.uk/books/edition/The_Batsford_Companion_to_Medieval_Engla/ev78b9EJQy0C?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Batsford Companion to Medieval England] * [https://www.google.co.uk/books/edition/Shakespeare_s_Unruly_Women/FKFlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare's Unruly Women] * [https://www.google.co.uk/books/edition/Shakespeare_and_Others/iFEgAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare and Others] * [https://www.google.co.uk/books/edition/Kings_and_Chroniclers/L1wpAAAAYAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Kings and Chroniclers] * [https://www.google.co.uk/books/edition/A_Kingdom_for_a_Stage/UzxlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover A Kingdom for a Stage] * [https://www.google.co.uk/books/edition/The_House_of_Commons/Ezz4OZuYVFYC?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The History of Parliament: The House of Commons 1558-1603 (3 v.)] * [https://www.google.co.uk/books/edition/Shakespeare_Soul_of_the_Age/nMYCAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover shakespeare, Soul of the Age] * [https://www.google.co.uk/books/edition/After_Poststructuralism/TOaEAAAAIAAJ?hl=en&gbpv=0&bsq=%22I%20am%20Richard%20II,%20know%20ye%20not%20that?%22 After Poststructuralism: Interdisciplinarity and Literary Theory] * [https://www.google.co.uk/books/edition/The_Unschooled_Mind/C7WnYtt219IC?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Unschooled Mind] * [https://www.google.co.uk/books/edition/Elizabeth_I/hHZnAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Eliz I] * [https://www.google.co.uk/books/edition/Dramas_of_Christian_Time/mnIqAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Dramas of Christian Time] * [https://www.google.co.uk/books/edition/Elizabeth_I/XjQmAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Elizabeth I: The Shrewdness of Virtue] * [https://www.google.co.uk/books/edition/John_Dryden/9Q1aAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover John Dryden] * [https://www.google.co.uk/books/edition/Shakespeare_and_Early_Modern_Political_T/DUwhAwAAQBAJ?hl=en&gbpv=1&dq=%22shakespeare%22+%2B+%22political+propaganda%22&pg=PA259&printsec=frontcover Shakespeare and Early Modern Political Thought] * [https://www.google.co.uk/books/edition/The_English_History_Play_in_the_age_of_S/5TT-AQAAQBAJ?hl=en&gbpv=1&dq=%22shakespeare%22+%2B+%22political+propaganda%22&pg=PA158&printsec=frontcover The English History Play in the Age of Shakespeare] * [https://www.google.co.uk/books/edition/Shakespeare_and_the_Political/rEcREQAAQBAJ?hl=en&gbpv=1&dq=%22shakespeare%22+%2B+%22political+propaganda%22&pg=PA215&printsec=frontcover Shakespeare and the Political] * [https://www.google.co.uk/books/edition/William_Shakespeare_Subject_of_the_Crown/a7G6DAAAQBAJ?hl=en&gbpv=1&dq=%22shakespeare%22+%2B+%22political+propaganda%22&pg=PT18&printsec=frontcover William Shakespeare - Subject of the Crown?] * [https://www.google.com/search?q=%22shakespeare%22+%2B+%22political+propaganda%22&client=firefox-b-d&hs=4AQ&sca_esv=6d4ade7bd26771c9&udm=36&biw=2510&bih=1307&tbs=cdr%3A1%2Ccd_min%3A2000%2Ccd_max%3A2099&sxsrf=ANbL-n6I6Pkwl7mmdHK6N1xPQXLbGBIOSg%3A1776853062010&ei=RqDoaZUvztiFsg_I5bToDw&ved=0ahUKEwiV6tC8nYGUAxVObEEAHcgyDf0Q4dUDCBM&uact=5&oq=%22shakespeare%22+%2B+%22political+propaganda%22&gs_lp=EhBnd3Mtd2l6LW1vZGVsZXNzIiYic2hha2VzcGVhcmUiICsgInBvbGl0aWNhbCBwcm9wYWdhbmRhIjIIECEYoAEYwwRInQlQxgZYuwdwAXgAkAEAmAF_oAHPAaoBAzEuMbgBA8gBAPgBAZgCAqACVsICCxAAGIAEGKIEGLADmAMAiAYBkAYCkgcBMqAHowOyBwExuAdTwgcDMC4yyAcEgAgB&sclient=gws-wiz-modeless The Nazi Appropriation of Shakespeare: Cultural Politics in] {{refend}} ie24uq7ns78vvfy8wfh1mgif82e640p 2818524 2818523 2026-07-19T11:55:12Z Fortuna imperatrix mundi 1058581 /* Bibliography */ ref 2818524 wikitext text/x-wiki ==Background== ===Richard II=== ===Elizabeth I=== ====Essex's rebellion==== ==IaR2== Lambare's accoiunt of his intervieew with the queen is well known.{{sfn|Orgel|2011|pp=11–43}} ===Political anxiety and tension=== ===Censorship=== ==Consequences and aftermath== == Notes == {{reflist|group=note}} ==References== {{Reflist|20em}} ==Bibliography== {{refbegin|30em|indent=yes}} * {{Cite book|url=https://doi.org/10.1057/9780230307261_2|title=Prologue: I am Richard II|last=Orgel|first=Stephen|date=2011|publisher=Palgrave Macmillan UK|isbn=978-0-230-30726-1|editor-last=Petrina|editor-first=Alessandra|location=London|pages=11–43|language=en|doi=10.1057/9780230307261_2|editor-last2=Tosi|editor-first2=Laura}} * [https://broadlytextual.com/2017/12/15/i-am-richard-ii-know-ye-not-that-drama-and-political-anxiety-in-shakespeares-london/ Hixon, E., Syracuse Univ] * Bate, Jonathan (2008). Soul of the Age. London: Penguin. pp. 256–286. ISBN 978-0-670-91482-1. * [https://theconversation.com/richard-ii-by-william-shakespeare-why-the-divine-right-of-kings-still-matters-186648 McFarlane, K., Univ South Australia] * [https://muse.jhu.edu/article/31090/summary Lemon, Rebecca. "The Faulty Verdict in "The Crown v. John Hayward"." SEL Studies in English Literature 1500-1900, vol. 41 no. 1, 2001, p. 109-132. Project MUSE, https://dx.doi.org/10.1353/sel.2001.0009] * [https://brill.com/display/book/9789401211666/B9789401211666-s009.xml?language=en&srsltid=AfmBOoqTUaTQr0msNWRdtEZZ9qQ3rNcVkSoBvDPXuGv7nRvHrC8t5OzH Kizelbach, U. (2014), In The Pragmatics of Early Modern Politics: Power and Kingship in Shakespeare’s History Plays. Leiden, The Netherlands: Brill. https://doi.org/10.1163/9789401211666_009] * [https://www.google.co.uk/books/edition/Richard_II/z-kOAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Lopez, R2] * [https://www.manchesterhive.com/display/9781526130532/9781526130532.00008.xml?print rgel, S. (2017). "I am Richard II". In Spectacular Performances. Manchester, England: Manchester University Press. Retrieved Apr 22, 2026, from https://doi.org/10.7765/9781526130532.00008] * [https://www.researchgate.net/publication/271041996_Was_Elizabeth_I_Richard_II_The_Authenticity_of_Lambarde%27s_%27Conversation%27 Scott-Warren, Jason. (2012). Was Elizabeth I Richard II?: The Authenticity of Lambarde's 'Conversation'. The Review of English Studies. 64. 208-230. 10.1093/res/hgs062] * Stanley Wells, “Introduction” in Richard II, The New Penguin Shakespeare, ed. Stanley Wells (London: Penguin Books, 1969), 13. * [https://www.cambridge.org/core/books/abs/shakespeare-survey/shakespeare-and-history-divergencies-and-agreements/B065A23215FD86BEE8E1CFD51DC7C1FB Ives EW. Shakespeare and History: Divergencies and Agreements. In: Wells S, ed. Shakespeare Survey. Shakespeare Survey. Cambridge University Press; 1986:19-36] * [https://www.jstor.org/stable/457855 Albright, Evelyn May. “Shakespeare’s Richard II, Hayward’s History of Henry IV, and the Essex Conspiracy.” PMLA, vol. 46, no. 3, 1931, pp. 694–719. JSTOR, https://doi.org/10.2307/457855. Accessed 22 Apr. 2026] * [https://dsc.duq.edu/cgi/viewcontent.cgi?article=2800&context=etd Morris, A. (2019). "within the hollow crown": Performing Kingship in Richard II and Henry IV Part One (Master's thesis, Duquesne University). Retrieved from https://dsc.duq.edu/etd/1771] * [https://digitalcommons.lib.uconn.edu/cgi/viewcontent.cgi?article=1122&context=srhonors_theses Scannell, Sarah J., "Shakespeare's Richard II and Henry V and Political Rebellions in the Reign of Queen Elizabeth I" (2010). Honors Scholar Theses, 138] * [https://compass.onlinelibrary.wiley.com/doi/abs/10.1111/j.1741-4113.2011.00873.x Luecking Frost, L. (2012), “A Kyng That Ruled All By Lust”: Richard II in Elizabethan Literature. Literature Compass, 9: 183-198. https://doi.org/10.1111/j.1741-4113.2011.00873.x] * [https://www.academia.edu/8347491/A_Law_Case_in_Verse_Venus_and_Adonis_and_the_Authorship_Question Stritmatter, R., A Law Case in Verse: Venus and Adonis and the Authorship Question, 2004 University of Tennessee Law Review] * [https://www.researchgate.net/publication/304647742_Prologue_I_am_Richard_II Orgel, Stephen. (2011). Prologue: I am Richard II. 10.1057/9780230307261_2] * [https://go.gale.com/ps/i.do?id=GALE%7CA314252957&sid=googleScholar&v=2.1&it=r&linkaccess=abs&issn=15256863&sw=w&p=AONE&userGroupName=anon%7Ebf79dda0&aty=open-web-entry Egan, Michael. "The Essex Rebellion and Richard II: why wasn't Shakespeare arrested?" Shakespeare Oxford Newsletter, vol. 48, no. 3, summer-fall 2012, p. 20] * [https://www.cambridge.org/core/books/abs/richard-ii/stagehistory/D62D9CA061F03B2AE8D56E856288EEDA 1. Child H, Shakespeare W. THE STAGE-HISTORY. In: Dover Wilson J, ed. Richard II: The Cambridge Dover Wilson Shakespeare. Cambridge Library Collection - Literary Studies. Cambridge University Press; 2009:lxxvii-xcii] * [https://www.openstarts.units.it/entities/publication/16d8c447-62f4-48e8-8347-5d56af674aa3/details Snelling, D., 'Prospero on the Coast of Bohemia. "Nihil sed rara et miranda amabat". Diary of Melchior Goldast upon the late Emperor Rudolph II' 1994] * [https://www.jstor.org/stable/2902265 Gajowski, Evelyn. Shakespeare in Performance: King Richard IIby Margaret Shewring. Shakespeare Quarterly, vol. 49, no. 3, 1998, pp. 328–30. JSTOR, https://doi.org/10.2307/2902265. Accessed 22 Apr. 2026] * [https://muse.jhu.edu/article/217842/summary Mosse, Ramona Franziska. Review of Richard II. Shakespeare Bulletin, vol. 25 no. 2, 2007, p. 73-77. Project MUSE, https://dx.doi.org/10.1353/shb.2007.0036] * [https://www.journals.uchicago.edu/doi/abs/10.1086/665894?journalCode=rq Petrina and Tosi. Representations of Elizabeth I in Early Modern Culture Elizabeth Pentland Renaissance Quarterly 2012 65:1, 274-276] * [https://journals.co.za/doi/10.4314/sisa.v26i1.4 The construction of a king : waste, effeminacy and queerness in Shakespeare's Richard II Higginbotham, D.] * [https://academic.oup.com/book/2363/chapter-abstract/142533948?redirectedFrom=fulltext Erskine-Hill, Howard, 'The Second Tetralogy and After', Poetry and the Realm of Politics: Shakespeare to Dryden (Oxford, 1996; online edn, Oxford Academic, 3 Oct. 2011), https://doi.org/10.1093/acprof:oso/9780198117315.003.0004] * [https://www.academia.edu/123917862/Shakespeare_s_Richard_II_in_the_minds_of_the_Elizabethan_audiences Shakespeare’s Richard II in the minds of the Elizabethan audiences] * [https://journals.sagepub.com/doi/10.1177/03064220221084555 Plays, protests and the censor’s pencil] * [https://www.indexoncensorship.org/2022/03/plays-protests-and-the-censors-pencil/ Plays, protests and the censor’s pencil A brittle Gloriana: staging the deposition of Queen Elizabeth I]. * Saul, N., Richard II * [https://www.academia.edu/3236651/_A_god_on_earth_thou_art_Richard_II_and_the_nature_of_Kingship “A god on earth thou art”: Richard II and the nature of Kingship] * [https://www.jstor.org/stable/457398 Albright, Evelyn May. “Shakespeare’s Richard II and the Essex Conspiracy.” PMLA, vol. 42, no. 3, 1927, pp. 686–720. JSTOR, https://doi.org/10.2307/457398. Accessed 22 Apr. 2026] * [https://www.berghahnjournals.com/view/journals/critical-survey/37/3/cs370301.xml Holderness, G. (2025). Introduction: Shakespeare's Wars of the Roses. Critical Survey, 37(3), 1-4. Retrieved Apr 22, 2026, from https://doi.org/10.3167/cs.2025.370301] * [https://www.folger.edu/podcasts/shakespeare-unlimited/tragedy-richard-ii-henry-iv-helen-castor/ Folger: Shakespeare Unlimited: Episode 249] * [https://www.rachaeldickzen.com/blog/2021/1/10/charlesiii The Historical and Shakespearean References in King Charles III] * Kermode, The Age of Shakespeare, pg.53. * [https://heinonline.org/HOL/login-hol?redirect_url=https%3A%2F%2Fheinonline.org%2FHOL%2Fmojo%3F%252Fmojo%3D%26auth_token%3DyhcMOu8W7PCc5OrmTqVnr2qFPpoiOdhM5uQmkQyWufU%253D%26casa_token%3D%26div%3D5%26g_sent%3D1%26paywall_url%3D%26referrer%3Dhttps%253A%252F%252Fheinonline.org%252FHOL%252FPage%253Fhandle%253Dhein.journals%252Fneastlr1%26timestamp%3D1776855600%26url%3D%252FHOL%252FPage%253Fhandle%253Dhein.journals%252Fneastlr1%2526div%253D5%2526id%253D%2526page%253D Jones, H., ''I AM RICHARD II, KNOW YE NOT THAT?' - WORKS. OF WILLIAM SHAKESPEARE AND THE ART OF. GOVERNANCE'''] * Hattaway, M., William Shakespeare: Richard II (Literature Insights, 9781847600349) * [https://books.openedition.org/psn/8305 Findlay, A., “Good sometimes queen” (V.1.37). Richard II, Mary Stuart and the Politics of Queenship, 205-219] * [https://pure.manchester.ac.uk/ws/portalfiles/portal/54508829/FULL_TEXT.PDF Pritchard, K., 'Legitimacy, Illegitimacy and Sovereignty in Shakespeare’s British Plays' (2011 University of Manchester PhD] * CHATELANAT, Marine. Histrionic Future Kings: The Politics of Metadrama in Shakespeare’s Richard III, Richard II, and Henry IV Part 1. Master, 2023. * [https://www.persee.fr/doc/xvii_0291-3798_2004_num_59_1_1993 Mayer, J-C., 'The "Parliament Sceane" in Shakespeare's King Richard II', XVII-XVIII. Revue de la Société d'études anglo-américaines des XVIIe et XVIIIe siècles Année 2004 59 pp. 27-42] * [https://www.emerald.com/md/article-abstract/38/5/315/285132/Managers-as-kings-Shakespeare-on-modern-leadership?redirectedFrom=fulltext Egan M (2000), "Managers as kings: Shakespeare on modern leadership". Management Decision, Vol. 38 No. 5 pp. 315–327, doi: https://doi.org/10.1108/00251740010340490] * [https://d-nb.info/1279081511/34 Quabeck (2022): William Shakespeare, in: Thomas Gutmann, Eberhard Ortland, Klaus Stierstorfer, eds., Encyclopedia of Law and Literature (last edited 26 October 2022), doi: 10.17879/12009577118] * [https://lawliberty.org/book-review/was-the-bard-writing-about-the-donald/ Sullivan, V., 'Was the Bard Writing about The Donald?'] * [https://deveresociety.co.uk/wp-content/uploads/2022/11/FINAL_NL_Apr2022-Haste-July-Letterresponse.pdf Haste, I., 'THE PLAY ON THE EVE OF THE ESSEX REBELLION'] * [https://www.cherylmtaylor.com/2017/11/08/a-companion-to-shakespeares-richard-ii/ A Companion to Shakespeare’s Richard II] * [https://www.city-journal.org/article/shakespeares-richards Dalrymple, T., Shakespeare’s Richards: The playwright’s two historical dramas offer contrasts between political pathologies] * [https://hfriedberg.web.wesleyan.edu/engl205/wshakespeare/plotsandrebelions.htm ELIZABETHAN PLOTS AND REBELLIONS] * [https://www.jstor.org/stable/2869708 Rackin, Phyllis. “The Role of the Audience In Shakespeare’s Richard II.” Shakespeare Quarterly, vol. 36, no. 3, 1985, pp. 262–81. JSTOR, https://doi.org/10.2307/2869708. Accessed 22 Apr. 2026.] * [https://www.psychologytoday.com/ie/blog/whats-in-name/201603/group-think-and-academia-shocking-shakespeare-shenanigans Group Think and Academia: Shocking Shakespeare Shenanigans] * [https://journals.sagepub.com/doi/10.1177/01847678221099975 Hiscopck, A., '"Why I should welcome such a guest as grief [?": Lodging and dislodging in Shakespeare's Richard II'] * [https://academic.oup.com/book/51779/chapter-abstract/420487796?redirectedFrom=fulltext 'Epigraph', Imagining Time in the English Chronicle Play: Historical Futures, 1590-1660 (Oxford, 2023; online edn, Oxford Academic, 19 Oct. 2023), https://doi.org/10.1093/oso/9780198872658.002.0009] * [https://journals.openedition.org/interfaces/319?lang=en Blake & Shakespeare] * [https://www.cambridge.org/core/books/abs/shakespeare-and-the-visual-imagination/richard-iiand-the-politics-of-perspective/23A5D4615E2A8BF9380BAF1ECE99B46D Sillars S. Richard IIand the Politics of Perspective. In: Shakespeare and the Visual Imagination. Cambridge University Press; 2015:133-162.] * [https://www.persee.fr/doc/xvii_0291-3798_2004_num_59_1_1992 Le miroir des rois: de Richard II à Henry VIII] * [https://www.cambridge.org/core/books/abs/king-richard-ll/introduction/25419329A06BC1C2E4D3DDBF331B7F91 Shakespeare W, McEachern C. Introduction. In: Gurr A, ed. King Richard Ll. The New Cambridge Shakespeare. Cambridge University Press; 2018:1-66.] * [https://www.researchgate.net/publication/372862689_A_Critical_Discourse_Study_of_Shakespeare%27s_Theological_Conceptions_in_Acts_IV_and_V_of_Richard_II_The_%27Divine_Mandate%27_of_Richard_Kingship_Falls_Apart A Critical Discourse Study of Shakespeare’s Theological Conceptions in Acts IV and V of Richard II: The ‘Divine Mandate’ of Richard Kingship Falls Apart] * [https://www.jstor.org/stable/10.1525/lal.2004.16.3.313 Richard II, Bracton, and the End of Political Theology] * [https://dokumen.pub/the-art-of-law-in-shakespeare-9781509905478-9781509905508-9781509905492.html The Art of Law in Shakespeare] * [https://www.cambridge.org/core/books/abs/cambridge-companion-to-shakespeare-and-religion/providence-and-divine-right-in-the-english-histories/DBE9BFEBF5C199F9DD72D31ED9C3B041 Mayer J-C. Providence and Divine Right in the English Histories. In: Hamlin H, ed. The Cambridge Companion to Shakespeare and Religion. Cambridge Companions to Literature. Cambridge University Press; 2019:151-167] * [https://www.researchgate.net/publication/315698000_THE_CHOMSKY-FOUCAULT_DEBATE_WHERE_DOES_THE_BARD_STAND_IN_RICHARD_II Salami, Ali & Riahi Nouri, Amir. (2016). THE CHOMSKY-FOUCAULT DEBATE: WHERE DOES THE BARD STAND IN RICHARD II?] * [https://dokumen.pub/shakespeare-richard-ii-9781350363168-9780333416709.html Richard II] * [https://www.jstor.org/stable/457907?seq=1 Heffner, Ray. “Shakespeare, Hayward, and Essex.” PMLA, vol. 45, no. 3, 1930, pp. 754–80. JSTOR, https://doi.org/10.2307/457907. Accessed 22 Apr. 2026] * [https://www.cambridge.org/core/books/abs/cambridge-companion-to-shakespeare-and-war/shakespeare-on-civil-and-dynastic-wars/00D0269CFF998876DB961FB01C22954D Shakespeare on Civil and Dynastic Wars] * [https://brill.com/display/book/edcoll/9789004356900/B9789004356900_012.xml?srsltid=AfmBOopJMjwJAQnUGME9FmT6Gwg0ih9pArbXBdTJbN6YAqbdXqbkGyuo Is a Dramatic Theory of History Possible? Shakespeare’s Richard ii and ‘Historical Truth’] * [https://www.jstor.org/stable/24475607 THE CHILD'S TWO BODIES: SHAKESPEARE, SOVEREIGNTY, AND THE END OF SUCCESSION] * [https://dokumen.pub/english-renaissance-drama-an-introduction-to-theatre-and-theatres-in-shakespeares-time-9781847600202-9781847600714.html English Renaissance Drama : An Introduction to Theatre and Theatres in Shakespeare’s Time] * [https://www.cambridge.org/core/books/abs/believing-in-shakespeare/person-plot-place/BD4EF4CBA6E7D8E2ECE5519FA1D08C16 1. McEachern C. Person, Plot, Place. In: Believing in Shakespeare: Studies in Longing. Cambridge University Press; 2018:181-296] * [https://www.enotes.com/topics/richard-2/criticism/richard-ii-vol-91/criticism-production-reviews-1000228/nigel-saul-review-date-12-may-2000 With an Eye to the Present] * [https://www.academia.edu/128303633/Shakespeare_s_Other_Eden_Richard_II_by_Ouroboros_Theatre_Company_2013_ Shakespeare’s Other Eden Richard II by Ouroboros Theatre Company (2013)] * [https://dokumen.pub/shakespeare-upstart-crow-to-sweet-swan-1592-1623-9781472555496-9781408130148-9781408139189.html Shakespeare: Upstart Crow to Sweet Swan 1592-1623] * [https://www.jstor.org/stable/41917215 RACKIN, PHYLLIS. “Temporality, Anachronism, and Presence in Shakespeare’s English Histories.” Renaissance Drama, vol. 17, 1986, pp. 101–23. JSTOR, http://www.jstor.org/stable/41917215. Accessed 22 Apr. 2026.] * [https://www.cambridge.org/core/journals/greece-and-rome/article/abs/alexander-philotas-and-the-origins-of-modern-historiography/D90568C0A1131952625E0A3B44F43EC8 Stoneman R. Alexander, Philotas, and the origins of modern historiography. Greece and Rome. 2013;60(2):296-312. doi:10.1017/S0017383513000119] * [https://www.academia.edu/35800720/Shakespeares_Richard_II_and_Elizabethan_politics Shakespeare's Richard II and Elizabethan politics] * [https://www.cambridge.org/core/books/abs/rethinking-historicism-from-shakespeare-to-milton/old-new-now/2E8CBEFFEFB11D8B843BAF2C108E9768 Introduction - Old, New, Now] * [https://www.academia.edu/24392476/Richard_II_hamartia_and_climax Richard II hamartia and climax] * [https://www.cambridge.org/core/books/abs/babylonian-jews-and-sasanian-imperialism-in-late-antiquity/kings-and-religion-in-the-talmud-and-in-the-imagination-of-sasanian-communities/E2DCC2F3E0FBA74ABE543DDD2DD7F1A7 Kings and Religion in the Talmud and in the Imagination of Sasanian Communities] * Stephen Greenblatt, “Introduction,” in The Power of Forms in the English Renaissance, ed. Stephen Greenblatt (Norman, OK: 1982), 3-6] * [https://www.academia.edu/30472558/Americas_Donald_Trump_Shakespeares_Richard_II_15_Dec_2016 America's Donald Trump, Shakespeare's Richard II] * [https://www.academia.edu/5321346/Looking_Back_Shakespeare_s_Indebtedness_to_Chaucer_and_the_Representations_of_Chivalry_in_King_Richard_II_The_Two_Noble_Kinsmen_and_the_Knight_s_Tale Looking Back: Shakespeare’s Indebtedness to Chaucer and the Representations of Chivalry in King Richard II, The Two Noble Kinsmen and the Knight’s Tale] * [https://www.academia.edu/92423912/The_Good_Usurper_in_the_eyes_of_God_and_the_people_An_analysis_of_the_role_of_the_usurper_in_Shakespeares_Richard_II_and_Henry_IV_Part_1 The Good Usurper in the eyes of God and the people: An analysis of the role of the usurper in Shakespeare's Richard II and Henry IV Part 1] * [https://www.academia.edu/91789871/The_Bacon_Shakespeare_Manuscript_Hitherto_known_as_the_Northumberland_Manuscript_which_originally_Contained_Copies_of_his_Shakespeare_Plays_Richard_II_and_Richard_III THE BACON-SHAKESPEARE MANUSCRIPT (HITHERTO KNOWN AS THE NORTHUMBERLAND MANUSCRIPT) WHICH ORIGINALLY CONTAINED COPIES OF HIS SHAKESPEARE PLAYS RICHARD II AND RICHARD III] * [https://www.google.co.uk/books/edition/Richard_II/f4gGCAAAQBAJ?hl=en&gbpv=1&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&pg=PA88&printsec=frontcover Richard II: Critical Essays] * [https://www.google.co.uk/books/edition/A_Companion_to_Shakespeare_s_Works_Volum/JlDNEAAAQBAJ?hl=en&gbpv=1&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&pg=PA400&printsec=frontcover A Companion to Shakespeare's Works, Volume II] * [https://www.google.co.uk/books/edition/Shakespeare_Reread/Z6JhDwAAQBAJ?hl=en&gbpv=1&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&pg=PA239&printsec=frontcover Shakespeare Reread] * [https://www.google.co.uk/books/edition/Shakespeare_in_the_Present/yTKWEAAAQBAJ?hl=en&gbpv=1&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&pg=PT8&printsec=frontcover Shakespeare in the Present] * [https://www.google.co.uk/books/edition/Studying_Shakespeare/1N4FBAAAQBAJ?hl=en&gbpv=1&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&pg=PA129&printsec=frontcover Studying Shakespeare A Practical Introduction] * [https://www.google.co.uk/books/edition/Critical_Essays_on_William_Faulkner/kAGBEAAAQBAJ?hl=en&gbpv=1&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&pg=PT178&printsec=frontcover Critical Essays on William Faulkner] * [https://www.google.co.uk/books/edition/Shakespeare/QY6aEQAAQBAJ?hl=en&gbpv=1&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&pg=PA216&printsec=frontcover Shakespeare: An Anthology of Criticism and Theory 1945-2000] * [https://www.google.co.uk/books/edition/The_Reign_of_Richard_II/cv0WAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Dodd, G., R2] * [https://www.google.co.uk/books/edition/Shakespeare_and_the_Institution_of_Theat/Nkt9DAAAQBAJ?hl=en&gbpv=1&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&pg=PA108&printsec=frontcover Shakespeare and the Institution of Theatre] * [https://www.google.co.uk/books/edition/The_Theory_of_the_King_s_Two_Bodies_in_t/8SRXAAAAYAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Theory of the King's Two Bodies in the Age of Shakespeare] * [https://www.google.co.uk/books/edition/Stealing_the_Story/RaJlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Stealing the Story: Shakespeare's Self-Conscious Use of the Mimetic Tradition in the Tragedies] * [https://www.google.co.uk/books/edition/The_Greenwood_Companion_to_Shakespeare_O/JkcgAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Greenwood Companion to Shakespeare: Overviews and the history plays] * [https://www.google.co.uk/books/edition/British_and_Irish_Literature_and_Its_Tim/xH4jAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover British and Irish Literature and Its Times] * [https://www.google.co.uk/books/edition/Shakespeare_s_Philosophy_of_History_Reve/1jwgAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare's Philosophy of History Revealed in a Detailed Analysis of Henry V and Examined in Other History Plays] * [https://www.google.co.uk/books/edition/Shakespeare_and_His_Contemporaries/7em7coOthxQC?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare and His Contemporaries] * [https://www.google.co.uk/books/edition/The_Politics_of_the_Public_Sphere_in_Ear/0sGHAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Politics of the Public Sphere in Early Modern England] * [https://www.google.co.uk/books/edition/Shakespeare_and_the_Catholic_Religion/Fl4gAQAAIAAJ?hl=en&gbpv=0&bsq=%22I%20am%20Richard%20II,%20know%20ye%20not%20that?%22 Shakespeare and the Catholic religion] * [https://www.google.co.uk/books/edition/Proceedings_of_the_British_Academy_Volum/8CsoAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Lectures] * [https://www.google.co.uk/books/edition/Slander_and_Censorship_in_Late_Sixteenth/wb_oAvuJgbAC?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Slander and Censorship in Late Sixteenth Century Literature] * [https://www.google.co.uk/books/edition/Elizabethan_Literature_and_the_Law_of_Fr/vBFdAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Elizabethan Literature and the Law of Fraudulent Conveyance] * [https://www.google.co.uk/books/edition/Hamlet_History_and_commentary/yxgrAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Hamlet History etc] * [https://www.google.co.uk/books/edition/William_Shakespeare_A_Popular_Life/1BuaAAAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover William Shakespeare: a popular life] * [https://www.google.co.uk/books/edition/A_State_of_Mind/pCiIAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover A State of Mind?] * [https://www.google.co.uk/books/edition/Shakespeare_s_Dramatic_Genres/J5FlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare's Dramatic Genres] * [https://www.google.co.uk/books/edition/Literature_Criticism_from_1400_to_1800/O6VkAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Literature Criticism from 1400 to 1800 Volume 89] * [https://www.google.co.uk/books/edition/Shakespeare_the_Papist/LPwNAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare the Papist] * [https://www.google.co.uk/books/edition/Goslicius_Ideal_Senator_and_His_Cultural/1jc7AQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Goslicius' Ideal Senator and His Cultural Impact Over the Centuries] * [https://www.google.co.uk/books/edition/Shakespeare_by_Another_Name/FqllAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover "Shakespeare" by Another Name] * [https://www.google.co.uk/books/edition/Shakespeare_s_Friends/AlZlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare's Friends] * [https://www.google.co.uk/books/edition/Dr_Simon_Forman/qHceAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Dr Simon Forman] * [https://www.google.co.uk/books/edition/William_Shakespeare_the_Wars_of_the_Rose/dZFlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover William Shakespeare, the Wars of the Roses and the historians] * [https://www.google.co.uk/books/edition/Shakespearean_Criticism/2TdlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespearean Criticism] * [https://www.google.co.uk/books/edition/Poetry_History_and_Antiquarianism_in_Ref/jwIhAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Poetry, History, and Antiquarianism in Reformation England] * [https://www.google.co.uk/books/edition/Shakespeare_and_the_Prince_of_Love/D0QgAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare and the Prince of Love] * [https://www.google.co.uk/books/edition/Paper_Bullets_of_the_Brain/o0YgAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Paper Bullets of the Brain] * [https://www.google.co.uk/books/edition/As_You_Like_It/GiBaAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover As You Like It: Third Series] * [https://www.google.co.uk/books/edition/William_Shakespeare/WCqaAAAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover William Shakespeare, Dominic Shellard] * [https://www.google.co.uk/books/edition/Poets_and_God/0XZlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Poets and God] * [https://www.google.co.uk/books/edition/Law_and_Literature/Ax5MAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Law and Literature Volume 16] * [https://www.google.co.uk/books/edition/The_Embodied_Word/FV8sAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Embodied Word] * [https://www.google.co.uk/books/edition/The_Case_for_Shakespeare/WaRlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Case for Shakespeare: The End of the Authorship Question] * [https://www.google.co.uk/books/edition/Explorations_in_Renaissance_Culture/_SYrAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Explorations in Renaissance Culture Volumes 33-34] * [https://www.google.co.uk/books/edition/The_Touch_of_the_Real/ewdaAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Touch of the Real: Essays in Early Modern Culture in Honour of Stephen Greenblatt] * [https://www.google.co.uk/books/edition/Wotton_and_His_Worlds/ZfgNAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Wotton and his Worlds] * [https://www.google.co.uk/books/edition/Theatre_and_Religion/wo1lAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Theatre and Religion Lancastrian Shakespeare] * [https://www.google.co.uk/books/edition/Trying_Treason/TOKxAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Trying Treason] * [https://www.google.co.uk/books/edition/Willing_Subjects/IEX0sGwT1QQC?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Willing Subjects] * [https://www.google.co.uk/books/edition/Symbolism/Bt0ZAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Symbolism] * [https://www.google.co.uk/books/edition/Performing_Shakespeare/35pQAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Performing Shakespeare] * [https://www.google.co.uk/books/edition/Soul_of_the_Age/e0UgAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Soul of the Age] * [https://www.google.co.uk/books/edition/England/aD9nAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover England] * [https://www.google.co.uk/books/edition/Elizabeth_I/-GtnAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Elizabeth I] * [https://www.google.co.uk/books/edition/King_Richard_II/oGUMX4RntjgC?hl=en&gbpv=1&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&pg=PA25&printsec=frontcover King Richard II] * [https://www.google.co.uk/books/edition/Shakespeare_and_the_Legal_Imagination/OXPvBqQLw-4C?hl=en&gbpv=1&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&pg=PA38&printsec=frontcover Shakespeare and the legal imagination] * [https://www.google.co.uk/books/edition/Shakespeare_s_Theatre/GxN3ue9_r3oC?hl=en&gbpv=1&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&pg=PA69&printsec=frontcover Shakespeare's Theatre] * [https://www.google.co.uk/books/edition/Critical_Essays_on_Shakespeare_s_Richard/AaYoAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Critical Essays on Shakespeare's Richard II] * [https://www.google.co.uk/books/edition/The_Reign_of_Richard_II_Essays_in_Honour/y3xnAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Reign of Richard II: Essays in Honour of May McKisack] * [https://www.google.co.uk/books/edition/Poetry_and_the_Realm_of_Politics/oQFaAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Poetry and the Realm of Politics] * [https://www.google.co.uk/books/edition/Shakespeare/BM0mAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare] * [https://www.google.co.uk/books/edition/Shakespearean_Politics/oTdlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespearean Politics] * [https://www.google.co.uk/books/edition/Shakespeare_the_Theatrical_Dimension/wl4gAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare, the Theatrical Dimension] * [https://www.google.co.uk/books/edition/Who_was_Kit_Marlowe/zQ1aAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Kit Marlowe etc] * [https://www.google.co.uk/books/edition/From_Page_to_Performance/beQKAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover From Page to Performance] * [https://www.google.co.uk/books/edition/Exploring_Tudor_England/ax56AAAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Exploring Tudor England] * [https://www.google.co.uk/books/edition/The_Movement_Towards_Subversion/vyJaAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Movement Towards Subversion] * [https://www.google.co.uk/books/edition/Shakespeare_s_Typological_Satire/G5BlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare's Typological Satire] * [https://www.google.co.uk/books/edition/Shakespeare_Recycled/zzNlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare Recycled] * [https://www.google.co.uk/books/edition/Reinventing_the_Middle_Ages_the_Renaissa/fXFnAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Reinventing the Middle Ages & the Renaissance] * [https://www.google.co.uk/books/edition/The_Mysterious_William_Shakespeare/WnllAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Mysterious William Shakespeare] * [https://www.google.co.uk/books/edition/Richard_II/GHhlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Richard II Critical Essays] * [https://www.google.co.uk/books/edition/William_Shakespeare/WJvC6gu_I0gC?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover William Shakespeare: Records and Images] * [https://www.google.co.uk/books/edition/Shakespeare_and_the_Actors/HYtlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare and the Actors] * [https://www.google.co.uk/books/edition/Shakespeare_the_Man/BVdlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare the man] * [https://www.google.co.uk/books/edition/Henry_V/zXllAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Henry V: A Guide to the Play] * [https://www.google.co.uk/books/edition/Shakespearean_Contingencies/Cw1NAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespearean Contingencies] * [https://www.google.co.uk/books/edition/Renaissance_Drama/E60kAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Renaissance Drama 1990] * [https://www.google.co.uk/books/edition/Language_Discourse_Sign/uH4oAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Language, Discourse, Sign] * [https://www.google.co.uk/books/edition/Mock_Kings_in_Medieval_Society_and_Renai/T98KAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Mock kings in medieval society and Renaissance drama] * [https://www.google.co.uk/books/edition/Shakespeare_Invention_of_the_Human/ojHirImrtYoC?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare: Invention of the Human] * [https://www.google.co.uk/books/edition/Shakespeare/wn5lAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare] * [https://www.google.co.uk/books/edition/Persons_in_Groups/rQ24AAAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Persons in Groups] * [https://www.google.co.uk/books/edition/All_Semblative_a_Woman_s_Part/0DlaAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover All Semblative a Woman's Part?] * [https://www.google.co.uk/books/edition/Crossing_the_Mirror/qRZNAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Crossing the Mirror] * [https://www.google.co.uk/books/edition/De_Vere_is_Shakespeare/dKJlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover De Vere is Shakespeare] * [https://www.google.co.uk/books/edition/William_Lambarde_Elizabethan_Antiquary_1/x1RnAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover William Lambarde, Elizabethan Antiquary, 1536-1601] * [https://www.google.co.uk/books/edition/The_Power_of_Forms_in_the_English_Renais/cPtZAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Power of Forms in the English Renaissance] * [https://www.google.co.uk/books/edition/Ravishment_and_Rememberance/G31LAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Ravishment and Rememberance] * [https://www.google.co.uk/books/edition/Shakespeare_and_His_Theatre/8A5ZQq3uOVQC?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare and His Theatre] * [https://www.google.co.uk/books/edition/Critical_Hermeneutics_and_Shakespeare_s/O10gAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Critical Hermeneutics and Shakespeare's History Plays] * [https://www.google.co.uk/books/edition/Christian_England/K-WfAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Christian England] * [https://www.google.co.uk/books/edition/Shakespeare_s_Religious_Background/xDSaAAAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare's Religious Background] * [https://www.google.co.uk/books/edition/Shylock/N4RlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shylock] * [https://www.google.co.uk/books/edition/The_Shakespeare_Legacy/MM5XAAAAYAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Shakespeare Legacy] * [https://www.google.co.uk/books/edition/Renaissance_Genres/0uFZAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Renaissance Genres] * [https://www.google.co.uk/books/edition/Cannibals_Witches_and_Divorce/qZRpAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Cannibals, Witches, and Divorce] * [https://www.google.co.uk/books/edition/The_Problem_of_Religious_Knowledge/C29LAAAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Problem of Religious Knowledge] * [https://www.google.co.uk/books/edition/Essex_and_the_Great_Revolt_of_1381/J8RzAAAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Essex and the Great Revolt of 1381] * [https://www.google.co.uk/books/edition/Transactions_of_the_London_and_Middlesex/4dtJAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover TLMAS] * [https://www.google.co.uk/books/edition/Shakespeare_Politics_and_the_State/Mn9lAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare, Politics and the State] * [https://www.google.co.uk/books/edition/Allegories_of_Power_in_the_England_of_El/LIYgAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Allegories of Power in the England of Elizabeth] * [https://www.google.co.uk/books/edition/William_Shakespeare/rIVlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover William Shakespeare] * [https://www.google.co.uk/books/edition/Women_s_Matters/PDRlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Women's Matters] * [https://www.google.co.uk/books/edition/The_Weak_King_Dilemma_in_the_Shakespeare/0bJlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Weak King Dilemma in the Shakespearean History Play] * [https://www.google.co.uk/books/edition/The_Book_Known_as_Q/S2tlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Book Known as Q] * [https://www.google.co.uk/books/edition/Fields_of_Vision/OD0eAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Fields of Vision] * [https://www.google.co.uk/books/edition/Ungodly_Delights/RKgcAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Ungodly Delights] * [https://www.google.co.uk/books/edition/The_Shakespeare_Handbook/rLRlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Shakespeare HandboOK] * [https://www.google.co.uk/books/edition/Humanities/y5FZAAAAYAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Humanities] * [https://www.google.co.uk/books/edition/Richard_II_by_William_Shakespeare/Bb3yAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Richard II by William Shakespeare] * [https://www.google.co.uk/books/edition/King_Richard_II/50NnAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover King Richard II] * [https://www.google.co.uk/books/edition/Murder_Under_Trust_Or_The_Topical_Macbet/0oNlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Murder under trust] * [https://www.google.co.uk/books/edition/The_Shakespearean_Kings/tHBlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Shakespearean Kings] * [https://www.google.co.uk/books/edition/America_the_Mabr_e_y_Experience/mRQ3AAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover America, the Mabr(e)y Experience: Resistance, Revolution & Civil War] * [https://www.google.co.uk/books/edition/Richard_II/ZDEkAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Richard II: An Annotated Bibliography, Volume 2] * [https://www.google.co.uk/books/edition/The_Batsford_Companion_to_Medieval_Engla/ev78b9EJQy0C?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Batsford Companion to Medieval England] * [https://www.google.co.uk/books/edition/Shakespeare_s_Unruly_Women/FKFlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare's Unruly Women] * [https://www.google.co.uk/books/edition/Shakespeare_and_Others/iFEgAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare and Others] * [https://www.google.co.uk/books/edition/Kings_and_Chroniclers/L1wpAAAAYAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Kings and Chroniclers] * [https://www.google.co.uk/books/edition/A_Kingdom_for_a_Stage/UzxlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover A Kingdom for a Stage] * [https://www.google.co.uk/books/edition/The_House_of_Commons/Ezz4OZuYVFYC?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The History of Parliament: The House of Commons 1558-1603 (3 v.)] * [https://www.google.co.uk/books/edition/Shakespeare_Soul_of_the_Age/nMYCAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover shakespeare, Soul of the Age] * [https://www.google.co.uk/books/edition/After_Poststructuralism/TOaEAAAAIAAJ?hl=en&gbpv=0&bsq=%22I%20am%20Richard%20II,%20know%20ye%20not%20that?%22 After Poststructuralism: Interdisciplinarity and Literary Theory] * [https://www.google.co.uk/books/edition/The_Unschooled_Mind/C7WnYtt219IC?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Unschooled Mind] * [https://www.google.co.uk/books/edition/Elizabeth_I/hHZnAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Eliz I] * [https://www.google.co.uk/books/edition/Dramas_of_Christian_Time/mnIqAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Dramas of Christian Time] * [https://www.google.co.uk/books/edition/Elizabeth_I/XjQmAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Elizabeth I: The Shrewdness of Virtue] * [https://www.google.co.uk/books/edition/John_Dryden/9Q1aAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover John Dryden] * [https://www.google.co.uk/books/edition/Shakespeare_and_Early_Modern_Political_T/DUwhAwAAQBAJ?hl=en&gbpv=1&dq=%22shakespeare%22+%2B+%22political+propaganda%22&pg=PA259&printsec=frontcover Shakespeare and Early Modern Political Thought] * [https://www.google.co.uk/books/edition/The_English_History_Play_in_the_age_of_S/5TT-AQAAQBAJ?hl=en&gbpv=1&dq=%22shakespeare%22+%2B+%22political+propaganda%22&pg=PA158&printsec=frontcover The English History Play in the Age of Shakespeare] * [https://www.google.co.uk/books/edition/Shakespeare_and_the_Political/rEcREQAAQBAJ?hl=en&gbpv=1&dq=%22shakespeare%22+%2B+%22political+propaganda%22&pg=PA215&printsec=frontcover Shakespeare and the Political] * [https://www.google.co.uk/books/edition/William_Shakespeare_Subject_of_the_Crown/a7G6DAAAQBAJ?hl=en&gbpv=1&dq=%22shakespeare%22+%2B+%22political+propaganda%22&pg=PT18&printsec=frontcover William Shakespeare - Subject of the Crown?] * [https://www.google.com/search?q=%22shakespeare%22+%2B+%22political+propaganda%22&client=firefox-b-d&hs=4AQ&sca_esv=6d4ade7bd26771c9&udm=36&biw=2510&bih=1307&tbs=cdr%3A1%2Ccd_min%3A2000%2Ccd_max%3A2099&sxsrf=ANbL-n6I6Pkwl7mmdHK6N1xPQXLbGBIOSg%3A1776853062010&ei=RqDoaZUvztiFsg_I5bToDw&ved=0ahUKEwiV6tC8nYGUAxVObEEAHcgyDf0Q4dUDCBM&uact=5&oq=%22shakespeare%22+%2B+%22political+propaganda%22&gs_lp=EhBnd3Mtd2l6LW1vZGVsZXNzIiYic2hha2VzcGVhcmUiICsgInBvbGl0aWNhbCBwcm9wYWdhbmRhIjIIECEYoAEYwwRInQlQxgZYuwdwAXgAkAEAmAF_oAHPAaoBAzEuMbgBA8gBAPgBAZgCAqACVsICCxAAGIAEGKIEGLADmAMAiAYBkAYCkgcBMqAHowOyBwExuAdTwgcDMC4yyAcEgAgB&sclient=gws-wiz-modeless The Nazi Appropriation of Shakespeare: Cultural Politics in] {{refend}} hu3n31m9kxm64bt1voerl8ixmudbfzx 2818525 2818524 2026-07-19T11:55:46Z Fortuna imperatrix mundi 1058581 /* IaR2 */ date 2818525 wikitext text/x-wiki ==Background== ===Richard II=== ===Elizabeth I=== ====Essex's rebellion==== ==IaR2== Lambare's accoiunt of his intervieew with the queen in August 1601 is well known.{{sfn|Orgel|2011|pp=11–43}} ===Political anxiety and tension=== ===Censorship=== ==Consequences and aftermath== == Notes == {{reflist|group=note}} ==References== {{Reflist|20em}} ==Bibliography== {{refbegin|30em|indent=yes}} * {{Cite book|url=https://doi.org/10.1057/9780230307261_2|title=Prologue: I am Richard II|last=Orgel|first=Stephen|date=2011|publisher=Palgrave Macmillan UK|isbn=978-0-230-30726-1|editor-last=Petrina|editor-first=Alessandra|location=London|pages=11–43|language=en|doi=10.1057/9780230307261_2|editor-last2=Tosi|editor-first2=Laura}} * [https://broadlytextual.com/2017/12/15/i-am-richard-ii-know-ye-not-that-drama-and-political-anxiety-in-shakespeares-london/ Hixon, E., Syracuse Univ] * Bate, Jonathan (2008). Soul of the Age. London: Penguin. pp. 256–286. ISBN 978-0-670-91482-1. * [https://theconversation.com/richard-ii-by-william-shakespeare-why-the-divine-right-of-kings-still-matters-186648 McFarlane, K., Univ South Australia] * [https://muse.jhu.edu/article/31090/summary Lemon, Rebecca. "The Faulty Verdict in "The Crown v. John Hayward"." SEL Studies in English Literature 1500-1900, vol. 41 no. 1, 2001, p. 109-132. Project MUSE, https://dx.doi.org/10.1353/sel.2001.0009] * [https://brill.com/display/book/9789401211666/B9789401211666-s009.xml?language=en&srsltid=AfmBOoqTUaTQr0msNWRdtEZZ9qQ3rNcVkSoBvDPXuGv7nRvHrC8t5OzH Kizelbach, U. (2014), In The Pragmatics of Early Modern Politics: Power and Kingship in Shakespeare’s History Plays. Leiden, The Netherlands: Brill. https://doi.org/10.1163/9789401211666_009] * [https://www.google.co.uk/books/edition/Richard_II/z-kOAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Lopez, R2] * [https://www.manchesterhive.com/display/9781526130532/9781526130532.00008.xml?print rgel, S. (2017). "I am Richard II". In Spectacular Performances. Manchester, England: Manchester University Press. Retrieved Apr 22, 2026, from https://doi.org/10.7765/9781526130532.00008] * [https://www.researchgate.net/publication/271041996_Was_Elizabeth_I_Richard_II_The_Authenticity_of_Lambarde%27s_%27Conversation%27 Scott-Warren, Jason. (2012). Was Elizabeth I Richard II?: The Authenticity of Lambarde's 'Conversation'. The Review of English Studies. 64. 208-230. 10.1093/res/hgs062] * Stanley Wells, “Introduction” in Richard II, The New Penguin Shakespeare, ed. Stanley Wells (London: Penguin Books, 1969), 13. * [https://www.cambridge.org/core/books/abs/shakespeare-survey/shakespeare-and-history-divergencies-and-agreements/B065A23215FD86BEE8E1CFD51DC7C1FB Ives EW. Shakespeare and History: Divergencies and Agreements. In: Wells S, ed. Shakespeare Survey. Shakespeare Survey. Cambridge University Press; 1986:19-36] * [https://www.jstor.org/stable/457855 Albright, Evelyn May. “Shakespeare’s Richard II, Hayward’s History of Henry IV, and the Essex Conspiracy.” PMLA, vol. 46, no. 3, 1931, pp. 694–719. JSTOR, https://doi.org/10.2307/457855. Accessed 22 Apr. 2026] * [https://dsc.duq.edu/cgi/viewcontent.cgi?article=2800&context=etd Morris, A. (2019). "within the hollow crown": Performing Kingship in Richard II and Henry IV Part One (Master's thesis, Duquesne University). Retrieved from https://dsc.duq.edu/etd/1771] * [https://digitalcommons.lib.uconn.edu/cgi/viewcontent.cgi?article=1122&context=srhonors_theses Scannell, Sarah J., "Shakespeare's Richard II and Henry V and Political Rebellions in the Reign of Queen Elizabeth I" (2010). Honors Scholar Theses, 138] * [https://compass.onlinelibrary.wiley.com/doi/abs/10.1111/j.1741-4113.2011.00873.x Luecking Frost, L. (2012), “A Kyng That Ruled All By Lust”: Richard II in Elizabethan Literature. Literature Compass, 9: 183-198. https://doi.org/10.1111/j.1741-4113.2011.00873.x] * [https://www.academia.edu/8347491/A_Law_Case_in_Verse_Venus_and_Adonis_and_the_Authorship_Question Stritmatter, R., A Law Case in Verse: Venus and Adonis and the Authorship Question, 2004 University of Tennessee Law Review] * [https://www.researchgate.net/publication/304647742_Prologue_I_am_Richard_II Orgel, Stephen. (2011). Prologue: I am Richard II. 10.1057/9780230307261_2] * [https://go.gale.com/ps/i.do?id=GALE%7CA314252957&sid=googleScholar&v=2.1&it=r&linkaccess=abs&issn=15256863&sw=w&p=AONE&userGroupName=anon%7Ebf79dda0&aty=open-web-entry Egan, Michael. "The Essex Rebellion and Richard II: why wasn't Shakespeare arrested?" Shakespeare Oxford Newsletter, vol. 48, no. 3, summer-fall 2012, p. 20] * [https://www.cambridge.org/core/books/abs/richard-ii/stagehistory/D62D9CA061F03B2AE8D56E856288EEDA 1. Child H, Shakespeare W. THE STAGE-HISTORY. In: Dover Wilson J, ed. Richard II: The Cambridge Dover Wilson Shakespeare. Cambridge Library Collection - Literary Studies. Cambridge University Press; 2009:lxxvii-xcii] * [https://www.openstarts.units.it/entities/publication/16d8c447-62f4-48e8-8347-5d56af674aa3/details Snelling, D., 'Prospero on the Coast of Bohemia. "Nihil sed rara et miranda amabat". Diary of Melchior Goldast upon the late Emperor Rudolph II' 1994] * [https://www.jstor.org/stable/2902265 Gajowski, Evelyn. Shakespeare in Performance: King Richard IIby Margaret Shewring. Shakespeare Quarterly, vol. 49, no. 3, 1998, pp. 328–30. JSTOR, https://doi.org/10.2307/2902265. Accessed 22 Apr. 2026] * [https://muse.jhu.edu/article/217842/summary Mosse, Ramona Franziska. Review of Richard II. Shakespeare Bulletin, vol. 25 no. 2, 2007, p. 73-77. Project MUSE, https://dx.doi.org/10.1353/shb.2007.0036] * [https://www.journals.uchicago.edu/doi/abs/10.1086/665894?journalCode=rq Petrina and Tosi. Representations of Elizabeth I in Early Modern Culture Elizabeth Pentland Renaissance Quarterly 2012 65:1, 274-276] * [https://journals.co.za/doi/10.4314/sisa.v26i1.4 The construction of a king : waste, effeminacy and queerness in Shakespeare's Richard II Higginbotham, D.] * [https://academic.oup.com/book/2363/chapter-abstract/142533948?redirectedFrom=fulltext Erskine-Hill, Howard, 'The Second Tetralogy and After', Poetry and the Realm of Politics: Shakespeare to Dryden (Oxford, 1996; online edn, Oxford Academic, 3 Oct. 2011), https://doi.org/10.1093/acprof:oso/9780198117315.003.0004] * [https://www.academia.edu/123917862/Shakespeare_s_Richard_II_in_the_minds_of_the_Elizabethan_audiences Shakespeare’s Richard II in the minds of the Elizabethan audiences] * [https://journals.sagepub.com/doi/10.1177/03064220221084555 Plays, protests and the censor’s pencil] * [https://www.indexoncensorship.org/2022/03/plays-protests-and-the-censors-pencil/ Plays, protests and the censor’s pencil A brittle Gloriana: staging the deposition of Queen Elizabeth I]. * Saul, N., Richard II * [https://www.academia.edu/3236651/_A_god_on_earth_thou_art_Richard_II_and_the_nature_of_Kingship “A god on earth thou art”: Richard II and the nature of Kingship] * [https://www.jstor.org/stable/457398 Albright, Evelyn May. “Shakespeare’s Richard II and the Essex Conspiracy.” PMLA, vol. 42, no. 3, 1927, pp. 686–720. JSTOR, https://doi.org/10.2307/457398. Accessed 22 Apr. 2026] * [https://www.berghahnjournals.com/view/journals/critical-survey/37/3/cs370301.xml Holderness, G. (2025). Introduction: Shakespeare's Wars of the Roses. Critical Survey, 37(3), 1-4. Retrieved Apr 22, 2026, from https://doi.org/10.3167/cs.2025.370301] * [https://www.folger.edu/podcasts/shakespeare-unlimited/tragedy-richard-ii-henry-iv-helen-castor/ Folger: Shakespeare Unlimited: Episode 249] * [https://www.rachaeldickzen.com/blog/2021/1/10/charlesiii The Historical and Shakespearean References in King Charles III] * Kermode, The Age of Shakespeare, pg.53. * [https://heinonline.org/HOL/login-hol?redirect_url=https%3A%2F%2Fheinonline.org%2FHOL%2Fmojo%3F%252Fmojo%3D%26auth_token%3DyhcMOu8W7PCc5OrmTqVnr2qFPpoiOdhM5uQmkQyWufU%253D%26casa_token%3D%26div%3D5%26g_sent%3D1%26paywall_url%3D%26referrer%3Dhttps%253A%252F%252Fheinonline.org%252FHOL%252FPage%253Fhandle%253Dhein.journals%252Fneastlr1%26timestamp%3D1776855600%26url%3D%252FHOL%252FPage%253Fhandle%253Dhein.journals%252Fneastlr1%2526div%253D5%2526id%253D%2526page%253D Jones, H., ''I AM RICHARD II, KNOW YE NOT THAT?' - WORKS. OF WILLIAM SHAKESPEARE AND THE ART OF. GOVERNANCE'''] * Hattaway, M., William Shakespeare: Richard II (Literature Insights, 9781847600349) * [https://books.openedition.org/psn/8305 Findlay, A., “Good sometimes queen” (V.1.37). Richard II, Mary Stuart and the Politics of Queenship, 205-219] * [https://pure.manchester.ac.uk/ws/portalfiles/portal/54508829/FULL_TEXT.PDF Pritchard, K., 'Legitimacy, Illegitimacy and Sovereignty in Shakespeare’s British Plays' (2011 University of Manchester PhD] * CHATELANAT, Marine. Histrionic Future Kings: The Politics of Metadrama in Shakespeare’s Richard III, Richard II, and Henry IV Part 1. Master, 2023. * [https://www.persee.fr/doc/xvii_0291-3798_2004_num_59_1_1993 Mayer, J-C., 'The "Parliament Sceane" in Shakespeare's King Richard II', XVII-XVIII. Revue de la Société d'études anglo-américaines des XVIIe et XVIIIe siècles Année 2004 59 pp. 27-42] * [https://www.emerald.com/md/article-abstract/38/5/315/285132/Managers-as-kings-Shakespeare-on-modern-leadership?redirectedFrom=fulltext Egan M (2000), "Managers as kings: Shakespeare on modern leadership". Management Decision, Vol. 38 No. 5 pp. 315–327, doi: https://doi.org/10.1108/00251740010340490] * [https://d-nb.info/1279081511/34 Quabeck (2022): William Shakespeare, in: Thomas Gutmann, Eberhard Ortland, Klaus Stierstorfer, eds., Encyclopedia of Law and Literature (last edited 26 October 2022), doi: 10.17879/12009577118] * [https://lawliberty.org/book-review/was-the-bard-writing-about-the-donald/ Sullivan, V., 'Was the Bard Writing about The Donald?'] * [https://deveresociety.co.uk/wp-content/uploads/2022/11/FINAL_NL_Apr2022-Haste-July-Letterresponse.pdf Haste, I., 'THE PLAY ON THE EVE OF THE ESSEX REBELLION'] * [https://www.cherylmtaylor.com/2017/11/08/a-companion-to-shakespeares-richard-ii/ A Companion to Shakespeare’s Richard II] * [https://www.city-journal.org/article/shakespeares-richards Dalrymple, T., Shakespeare’s Richards: The playwright’s two historical dramas offer contrasts between political pathologies] * [https://hfriedberg.web.wesleyan.edu/engl205/wshakespeare/plotsandrebelions.htm ELIZABETHAN PLOTS AND REBELLIONS] * [https://www.jstor.org/stable/2869708 Rackin, Phyllis. “The Role of the Audience In Shakespeare’s Richard II.” Shakespeare Quarterly, vol. 36, no. 3, 1985, pp. 262–81. JSTOR, https://doi.org/10.2307/2869708. Accessed 22 Apr. 2026.] * [https://www.psychologytoday.com/ie/blog/whats-in-name/201603/group-think-and-academia-shocking-shakespeare-shenanigans Group Think and Academia: Shocking Shakespeare Shenanigans] * [https://journals.sagepub.com/doi/10.1177/01847678221099975 Hiscopck, A., '"Why I should welcome such a guest as grief [?": Lodging and dislodging in Shakespeare's Richard II'] * [https://academic.oup.com/book/51779/chapter-abstract/420487796?redirectedFrom=fulltext 'Epigraph', Imagining Time in the English Chronicle Play: Historical Futures, 1590-1660 (Oxford, 2023; online edn, Oxford Academic, 19 Oct. 2023), https://doi.org/10.1093/oso/9780198872658.002.0009] * [https://journals.openedition.org/interfaces/319?lang=en Blake & Shakespeare] * [https://www.cambridge.org/core/books/abs/shakespeare-and-the-visual-imagination/richard-iiand-the-politics-of-perspective/23A5D4615E2A8BF9380BAF1ECE99B46D Sillars S. Richard IIand the Politics of Perspective. In: Shakespeare and the Visual Imagination. Cambridge University Press; 2015:133-162.] * [https://www.persee.fr/doc/xvii_0291-3798_2004_num_59_1_1992 Le miroir des rois: de Richard II à Henry VIII] * [https://www.cambridge.org/core/books/abs/king-richard-ll/introduction/25419329A06BC1C2E4D3DDBF331B7F91 Shakespeare W, McEachern C. Introduction. In: Gurr A, ed. King Richard Ll. The New Cambridge Shakespeare. Cambridge University Press; 2018:1-66.] * [https://www.researchgate.net/publication/372862689_A_Critical_Discourse_Study_of_Shakespeare%27s_Theological_Conceptions_in_Acts_IV_and_V_of_Richard_II_The_%27Divine_Mandate%27_of_Richard_Kingship_Falls_Apart A Critical Discourse Study of Shakespeare’s Theological Conceptions in Acts IV and V of Richard II: The ‘Divine Mandate’ of Richard Kingship Falls Apart] * [https://www.jstor.org/stable/10.1525/lal.2004.16.3.313 Richard II, Bracton, and the End of Political Theology] * [https://dokumen.pub/the-art-of-law-in-shakespeare-9781509905478-9781509905508-9781509905492.html The Art of Law in Shakespeare] * [https://www.cambridge.org/core/books/abs/cambridge-companion-to-shakespeare-and-religion/providence-and-divine-right-in-the-english-histories/DBE9BFEBF5C199F9DD72D31ED9C3B041 Mayer J-C. Providence and Divine Right in the English Histories. In: Hamlin H, ed. The Cambridge Companion to Shakespeare and Religion. Cambridge Companions to Literature. Cambridge University Press; 2019:151-167] * [https://www.researchgate.net/publication/315698000_THE_CHOMSKY-FOUCAULT_DEBATE_WHERE_DOES_THE_BARD_STAND_IN_RICHARD_II Salami, Ali & Riahi Nouri, Amir. (2016). THE CHOMSKY-FOUCAULT DEBATE: WHERE DOES THE BARD STAND IN RICHARD II?] * [https://dokumen.pub/shakespeare-richard-ii-9781350363168-9780333416709.html Richard II] * [https://www.jstor.org/stable/457907?seq=1 Heffner, Ray. “Shakespeare, Hayward, and Essex.” PMLA, vol. 45, no. 3, 1930, pp. 754–80. JSTOR, https://doi.org/10.2307/457907. Accessed 22 Apr. 2026] * [https://www.cambridge.org/core/books/abs/cambridge-companion-to-shakespeare-and-war/shakespeare-on-civil-and-dynastic-wars/00D0269CFF998876DB961FB01C22954D Shakespeare on Civil and Dynastic Wars] * [https://brill.com/display/book/edcoll/9789004356900/B9789004356900_012.xml?srsltid=AfmBOopJMjwJAQnUGME9FmT6Gwg0ih9pArbXBdTJbN6YAqbdXqbkGyuo Is a Dramatic Theory of History Possible? Shakespeare’s Richard ii and ‘Historical Truth’] * [https://www.jstor.org/stable/24475607 THE CHILD'S TWO BODIES: SHAKESPEARE, SOVEREIGNTY, AND THE END OF SUCCESSION] * [https://dokumen.pub/english-renaissance-drama-an-introduction-to-theatre-and-theatres-in-shakespeares-time-9781847600202-9781847600714.html English Renaissance Drama : An Introduction to Theatre and Theatres in Shakespeare’s Time] * [https://www.cambridge.org/core/books/abs/believing-in-shakespeare/person-plot-place/BD4EF4CBA6E7D8E2ECE5519FA1D08C16 1. McEachern C. Person, Plot, Place. In: Believing in Shakespeare: Studies in Longing. Cambridge University Press; 2018:181-296] * [https://www.enotes.com/topics/richard-2/criticism/richard-ii-vol-91/criticism-production-reviews-1000228/nigel-saul-review-date-12-may-2000 With an Eye to the Present] * [https://www.academia.edu/128303633/Shakespeare_s_Other_Eden_Richard_II_by_Ouroboros_Theatre_Company_2013_ Shakespeare’s Other Eden Richard II by Ouroboros Theatre Company (2013)] * [https://dokumen.pub/shakespeare-upstart-crow-to-sweet-swan-1592-1623-9781472555496-9781408130148-9781408139189.html Shakespeare: Upstart Crow to Sweet Swan 1592-1623] * [https://www.jstor.org/stable/41917215 RACKIN, PHYLLIS. “Temporality, Anachronism, and Presence in Shakespeare’s English Histories.” Renaissance Drama, vol. 17, 1986, pp. 101–23. JSTOR, http://www.jstor.org/stable/41917215. Accessed 22 Apr. 2026.] * [https://www.cambridge.org/core/journals/greece-and-rome/article/abs/alexander-philotas-and-the-origins-of-modern-historiography/D90568C0A1131952625E0A3B44F43EC8 Stoneman R. Alexander, Philotas, and the origins of modern historiography. Greece and Rome. 2013;60(2):296-312. doi:10.1017/S0017383513000119] * [https://www.academia.edu/35800720/Shakespeares_Richard_II_and_Elizabethan_politics Shakespeare's Richard II and Elizabethan politics] * [https://www.cambridge.org/core/books/abs/rethinking-historicism-from-shakespeare-to-milton/old-new-now/2E8CBEFFEFB11D8B843BAF2C108E9768 Introduction - Old, New, Now] * [https://www.academia.edu/24392476/Richard_II_hamartia_and_climax Richard II hamartia and climax] * [https://www.cambridge.org/core/books/abs/babylonian-jews-and-sasanian-imperialism-in-late-antiquity/kings-and-religion-in-the-talmud-and-in-the-imagination-of-sasanian-communities/E2DCC2F3E0FBA74ABE543DDD2DD7F1A7 Kings and Religion in the Talmud and in the Imagination of Sasanian Communities] * Stephen Greenblatt, “Introduction,” in The Power of Forms in the English Renaissance, ed. Stephen Greenblatt (Norman, OK: 1982), 3-6] * [https://www.academia.edu/30472558/Americas_Donald_Trump_Shakespeares_Richard_II_15_Dec_2016 America's Donald Trump, Shakespeare's Richard II] * [https://www.academia.edu/5321346/Looking_Back_Shakespeare_s_Indebtedness_to_Chaucer_and_the_Representations_of_Chivalry_in_King_Richard_II_The_Two_Noble_Kinsmen_and_the_Knight_s_Tale Looking Back: Shakespeare’s Indebtedness to Chaucer and the Representations of Chivalry in King Richard II, The Two Noble Kinsmen and the Knight’s Tale] * [https://www.academia.edu/92423912/The_Good_Usurper_in_the_eyes_of_God_and_the_people_An_analysis_of_the_role_of_the_usurper_in_Shakespeares_Richard_II_and_Henry_IV_Part_1 The Good Usurper in the eyes of God and the people: An analysis of the role of the usurper in Shakespeare's Richard II and Henry IV Part 1] * [https://www.academia.edu/91789871/The_Bacon_Shakespeare_Manuscript_Hitherto_known_as_the_Northumberland_Manuscript_which_originally_Contained_Copies_of_his_Shakespeare_Plays_Richard_II_and_Richard_III THE BACON-SHAKESPEARE MANUSCRIPT (HITHERTO KNOWN AS THE NORTHUMBERLAND MANUSCRIPT) WHICH ORIGINALLY CONTAINED COPIES OF HIS SHAKESPEARE PLAYS RICHARD II AND RICHARD III] * [https://www.google.co.uk/books/edition/Richard_II/f4gGCAAAQBAJ?hl=en&gbpv=1&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&pg=PA88&printsec=frontcover Richard II: Critical Essays] * [https://www.google.co.uk/books/edition/A_Companion_to_Shakespeare_s_Works_Volum/JlDNEAAAQBAJ?hl=en&gbpv=1&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&pg=PA400&printsec=frontcover A Companion to Shakespeare's Works, Volume II] * [https://www.google.co.uk/books/edition/Shakespeare_Reread/Z6JhDwAAQBAJ?hl=en&gbpv=1&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&pg=PA239&printsec=frontcover Shakespeare Reread] * [https://www.google.co.uk/books/edition/Shakespeare_in_the_Present/yTKWEAAAQBAJ?hl=en&gbpv=1&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&pg=PT8&printsec=frontcover Shakespeare in the Present] * [https://www.google.co.uk/books/edition/Studying_Shakespeare/1N4FBAAAQBAJ?hl=en&gbpv=1&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&pg=PA129&printsec=frontcover Studying Shakespeare A Practical Introduction] * [https://www.google.co.uk/books/edition/Critical_Essays_on_William_Faulkner/kAGBEAAAQBAJ?hl=en&gbpv=1&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&pg=PT178&printsec=frontcover Critical Essays on William Faulkner] * [https://www.google.co.uk/books/edition/Shakespeare/QY6aEQAAQBAJ?hl=en&gbpv=1&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&pg=PA216&printsec=frontcover Shakespeare: An Anthology of Criticism and Theory 1945-2000] * [https://www.google.co.uk/books/edition/The_Reign_of_Richard_II/cv0WAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Dodd, G., R2] * [https://www.google.co.uk/books/edition/Shakespeare_and_the_Institution_of_Theat/Nkt9DAAAQBAJ?hl=en&gbpv=1&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&pg=PA108&printsec=frontcover Shakespeare and the Institution of Theatre] * [https://www.google.co.uk/books/edition/The_Theory_of_the_King_s_Two_Bodies_in_t/8SRXAAAAYAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Theory of the King's Two Bodies in the Age of Shakespeare] * [https://www.google.co.uk/books/edition/Stealing_the_Story/RaJlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Stealing the Story: Shakespeare's Self-Conscious Use of the Mimetic Tradition in the Tragedies] * [https://www.google.co.uk/books/edition/The_Greenwood_Companion_to_Shakespeare_O/JkcgAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Greenwood Companion to Shakespeare: Overviews and the history plays] * [https://www.google.co.uk/books/edition/British_and_Irish_Literature_and_Its_Tim/xH4jAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover British and Irish Literature and Its Times] * [https://www.google.co.uk/books/edition/Shakespeare_s_Philosophy_of_History_Reve/1jwgAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare's Philosophy of History Revealed in a Detailed Analysis of Henry V and Examined in Other History Plays] * [https://www.google.co.uk/books/edition/Shakespeare_and_His_Contemporaries/7em7coOthxQC?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare and His Contemporaries] * [https://www.google.co.uk/books/edition/The_Politics_of_the_Public_Sphere_in_Ear/0sGHAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Politics of the Public Sphere in Early Modern England] * [https://www.google.co.uk/books/edition/Shakespeare_and_the_Catholic_Religion/Fl4gAQAAIAAJ?hl=en&gbpv=0&bsq=%22I%20am%20Richard%20II,%20know%20ye%20not%20that?%22 Shakespeare and the Catholic religion] * [https://www.google.co.uk/books/edition/Proceedings_of_the_British_Academy_Volum/8CsoAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Lectures] * [https://www.google.co.uk/books/edition/Slander_and_Censorship_in_Late_Sixteenth/wb_oAvuJgbAC?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Slander and Censorship in Late Sixteenth Century Literature] * [https://www.google.co.uk/books/edition/Elizabethan_Literature_and_the_Law_of_Fr/vBFdAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Elizabethan Literature and the Law of Fraudulent Conveyance] * [https://www.google.co.uk/books/edition/Hamlet_History_and_commentary/yxgrAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Hamlet History etc] * [https://www.google.co.uk/books/edition/William_Shakespeare_A_Popular_Life/1BuaAAAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover William Shakespeare: a popular life] * [https://www.google.co.uk/books/edition/A_State_of_Mind/pCiIAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover A State of Mind?] * [https://www.google.co.uk/books/edition/Shakespeare_s_Dramatic_Genres/J5FlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare's Dramatic Genres] * [https://www.google.co.uk/books/edition/Literature_Criticism_from_1400_to_1800/O6VkAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Literature Criticism from 1400 to 1800 Volume 89] * [https://www.google.co.uk/books/edition/Shakespeare_the_Papist/LPwNAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare the Papist] * [https://www.google.co.uk/books/edition/Goslicius_Ideal_Senator_and_His_Cultural/1jc7AQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Goslicius' Ideal Senator and His Cultural Impact Over the Centuries] * [https://www.google.co.uk/books/edition/Shakespeare_by_Another_Name/FqllAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover "Shakespeare" by Another Name] * [https://www.google.co.uk/books/edition/Shakespeare_s_Friends/AlZlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare's Friends] * [https://www.google.co.uk/books/edition/Dr_Simon_Forman/qHceAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Dr Simon Forman] * [https://www.google.co.uk/books/edition/William_Shakespeare_the_Wars_of_the_Rose/dZFlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover William Shakespeare, the Wars of the Roses and the historians] * [https://www.google.co.uk/books/edition/Shakespearean_Criticism/2TdlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespearean Criticism] * [https://www.google.co.uk/books/edition/Poetry_History_and_Antiquarianism_in_Ref/jwIhAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Poetry, History, and Antiquarianism in Reformation England] * [https://www.google.co.uk/books/edition/Shakespeare_and_the_Prince_of_Love/D0QgAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare and the Prince of Love] * [https://www.google.co.uk/books/edition/Paper_Bullets_of_the_Brain/o0YgAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Paper Bullets of the Brain] * [https://www.google.co.uk/books/edition/As_You_Like_It/GiBaAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover As You Like It: Third Series] * [https://www.google.co.uk/books/edition/William_Shakespeare/WCqaAAAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover William Shakespeare, Dominic Shellard] * [https://www.google.co.uk/books/edition/Poets_and_God/0XZlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Poets and God] * [https://www.google.co.uk/books/edition/Law_and_Literature/Ax5MAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Law and Literature Volume 16] * [https://www.google.co.uk/books/edition/The_Embodied_Word/FV8sAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Embodied Word] * [https://www.google.co.uk/books/edition/The_Case_for_Shakespeare/WaRlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Case for Shakespeare: The End of the Authorship Question] * [https://www.google.co.uk/books/edition/Explorations_in_Renaissance_Culture/_SYrAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Explorations in Renaissance Culture Volumes 33-34] * [https://www.google.co.uk/books/edition/The_Touch_of_the_Real/ewdaAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Touch of the Real: Essays in Early Modern Culture in Honour of Stephen Greenblatt] * [https://www.google.co.uk/books/edition/Wotton_and_His_Worlds/ZfgNAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Wotton and his Worlds] * [https://www.google.co.uk/books/edition/Theatre_and_Religion/wo1lAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Theatre and Religion Lancastrian Shakespeare] * [https://www.google.co.uk/books/edition/Trying_Treason/TOKxAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Trying Treason] * [https://www.google.co.uk/books/edition/Willing_Subjects/IEX0sGwT1QQC?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Willing Subjects] * [https://www.google.co.uk/books/edition/Symbolism/Bt0ZAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Symbolism] * [https://www.google.co.uk/books/edition/Performing_Shakespeare/35pQAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Performing Shakespeare] * [https://www.google.co.uk/books/edition/Soul_of_the_Age/e0UgAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Soul of the Age] * [https://www.google.co.uk/books/edition/England/aD9nAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover England] * [https://www.google.co.uk/books/edition/Elizabeth_I/-GtnAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Elizabeth I] * [https://www.google.co.uk/books/edition/King_Richard_II/oGUMX4RntjgC?hl=en&gbpv=1&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&pg=PA25&printsec=frontcover King Richard II] * [https://www.google.co.uk/books/edition/Shakespeare_and_the_Legal_Imagination/OXPvBqQLw-4C?hl=en&gbpv=1&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&pg=PA38&printsec=frontcover Shakespeare and the legal imagination] * [https://www.google.co.uk/books/edition/Shakespeare_s_Theatre/GxN3ue9_r3oC?hl=en&gbpv=1&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&pg=PA69&printsec=frontcover Shakespeare's Theatre] * [https://www.google.co.uk/books/edition/Critical_Essays_on_Shakespeare_s_Richard/AaYoAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Critical Essays on Shakespeare's Richard II] * [https://www.google.co.uk/books/edition/The_Reign_of_Richard_II_Essays_in_Honour/y3xnAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Reign of Richard II: Essays in Honour of May McKisack] * [https://www.google.co.uk/books/edition/Poetry_and_the_Realm_of_Politics/oQFaAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Poetry and the Realm of Politics] * [https://www.google.co.uk/books/edition/Shakespeare/BM0mAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare] * [https://www.google.co.uk/books/edition/Shakespearean_Politics/oTdlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespearean Politics] * [https://www.google.co.uk/books/edition/Shakespeare_the_Theatrical_Dimension/wl4gAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare, the Theatrical Dimension] * [https://www.google.co.uk/books/edition/Who_was_Kit_Marlowe/zQ1aAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Kit Marlowe etc] * [https://www.google.co.uk/books/edition/From_Page_to_Performance/beQKAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover From Page to Performance] * [https://www.google.co.uk/books/edition/Exploring_Tudor_England/ax56AAAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Exploring Tudor England] * [https://www.google.co.uk/books/edition/The_Movement_Towards_Subversion/vyJaAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Movement Towards Subversion] * [https://www.google.co.uk/books/edition/Shakespeare_s_Typological_Satire/G5BlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare's Typological Satire] * [https://www.google.co.uk/books/edition/Shakespeare_Recycled/zzNlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare Recycled] * [https://www.google.co.uk/books/edition/Reinventing_the_Middle_Ages_the_Renaissa/fXFnAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Reinventing the Middle Ages & the Renaissance] * [https://www.google.co.uk/books/edition/The_Mysterious_William_Shakespeare/WnllAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Mysterious William Shakespeare] * [https://www.google.co.uk/books/edition/Richard_II/GHhlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Richard II Critical Essays] * [https://www.google.co.uk/books/edition/William_Shakespeare/WJvC6gu_I0gC?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover William Shakespeare: Records and Images] * [https://www.google.co.uk/books/edition/Shakespeare_and_the_Actors/HYtlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare and the Actors] * [https://www.google.co.uk/books/edition/Shakespeare_the_Man/BVdlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare the man] * [https://www.google.co.uk/books/edition/Henry_V/zXllAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Henry V: A Guide to the Play] * [https://www.google.co.uk/books/edition/Shakespearean_Contingencies/Cw1NAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespearean Contingencies] * [https://www.google.co.uk/books/edition/Renaissance_Drama/E60kAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Renaissance Drama 1990] * [https://www.google.co.uk/books/edition/Language_Discourse_Sign/uH4oAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Language, Discourse, Sign] * [https://www.google.co.uk/books/edition/Mock_Kings_in_Medieval_Society_and_Renai/T98KAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Mock kings in medieval society and Renaissance drama] * [https://www.google.co.uk/books/edition/Shakespeare_Invention_of_the_Human/ojHirImrtYoC?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare: Invention of the Human] * [https://www.google.co.uk/books/edition/Shakespeare/wn5lAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare] * [https://www.google.co.uk/books/edition/Persons_in_Groups/rQ24AAAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Persons in Groups] * [https://www.google.co.uk/books/edition/All_Semblative_a_Woman_s_Part/0DlaAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover All Semblative a Woman's Part?] * [https://www.google.co.uk/books/edition/Crossing_the_Mirror/qRZNAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Crossing the Mirror] * [https://www.google.co.uk/books/edition/De_Vere_is_Shakespeare/dKJlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover De Vere is Shakespeare] * [https://www.google.co.uk/books/edition/William_Lambarde_Elizabethan_Antiquary_1/x1RnAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover William Lambarde, Elizabethan Antiquary, 1536-1601] * [https://www.google.co.uk/books/edition/The_Power_of_Forms_in_the_English_Renais/cPtZAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Power of Forms in the English Renaissance] * [https://www.google.co.uk/books/edition/Ravishment_and_Rememberance/G31LAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Ravishment and Rememberance] * [https://www.google.co.uk/books/edition/Shakespeare_and_His_Theatre/8A5ZQq3uOVQC?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare and His Theatre] * [https://www.google.co.uk/books/edition/Critical_Hermeneutics_and_Shakespeare_s/O10gAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Critical Hermeneutics and Shakespeare's History Plays] * [https://www.google.co.uk/books/edition/Christian_England/K-WfAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Christian England] * [https://www.google.co.uk/books/edition/Shakespeare_s_Religious_Background/xDSaAAAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare's Religious Background] * [https://www.google.co.uk/books/edition/Shylock/N4RlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shylock] * [https://www.google.co.uk/books/edition/The_Shakespeare_Legacy/MM5XAAAAYAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Shakespeare Legacy] * [https://www.google.co.uk/books/edition/Renaissance_Genres/0uFZAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Renaissance Genres] * [https://www.google.co.uk/books/edition/Cannibals_Witches_and_Divorce/qZRpAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Cannibals, Witches, and Divorce] * [https://www.google.co.uk/books/edition/The_Problem_of_Religious_Knowledge/C29LAAAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Problem of Religious Knowledge] * [https://www.google.co.uk/books/edition/Essex_and_the_Great_Revolt_of_1381/J8RzAAAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Essex and the Great Revolt of 1381] * [https://www.google.co.uk/books/edition/Transactions_of_the_London_and_Middlesex/4dtJAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover TLMAS] * [https://www.google.co.uk/books/edition/Shakespeare_Politics_and_the_State/Mn9lAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare, Politics and the State] * [https://www.google.co.uk/books/edition/Allegories_of_Power_in_the_England_of_El/LIYgAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Allegories of Power in the England of Elizabeth] * [https://www.google.co.uk/books/edition/William_Shakespeare/rIVlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover William Shakespeare] * [https://www.google.co.uk/books/edition/Women_s_Matters/PDRlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Women's Matters] * [https://www.google.co.uk/books/edition/The_Weak_King_Dilemma_in_the_Shakespeare/0bJlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Weak King Dilemma in the Shakespearean History Play] * [https://www.google.co.uk/books/edition/The_Book_Known_as_Q/S2tlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Book Known as Q] * [https://www.google.co.uk/books/edition/Fields_of_Vision/OD0eAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Fields of Vision] * [https://www.google.co.uk/books/edition/Ungodly_Delights/RKgcAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Ungodly Delights] * [https://www.google.co.uk/books/edition/The_Shakespeare_Handbook/rLRlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Shakespeare HandboOK] * [https://www.google.co.uk/books/edition/Humanities/y5FZAAAAYAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Humanities] * [https://www.google.co.uk/books/edition/Richard_II_by_William_Shakespeare/Bb3yAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Richard II by William Shakespeare] * [https://www.google.co.uk/books/edition/King_Richard_II/50NnAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover King Richard II] * [https://www.google.co.uk/books/edition/Murder_Under_Trust_Or_The_Topical_Macbet/0oNlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Murder under trust] * [https://www.google.co.uk/books/edition/The_Shakespearean_Kings/tHBlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Shakespearean Kings] * [https://www.google.co.uk/books/edition/America_the_Mabr_e_y_Experience/mRQ3AAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover America, the Mabr(e)y Experience: Resistance, Revolution & Civil War] * [https://www.google.co.uk/books/edition/Richard_II/ZDEkAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Richard II: An Annotated Bibliography, Volume 2] * [https://www.google.co.uk/books/edition/The_Batsford_Companion_to_Medieval_Engla/ev78b9EJQy0C?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Batsford Companion to Medieval England] * [https://www.google.co.uk/books/edition/Shakespeare_s_Unruly_Women/FKFlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare's Unruly Women] * [https://www.google.co.uk/books/edition/Shakespeare_and_Others/iFEgAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare and Others] * [https://www.google.co.uk/books/edition/Kings_and_Chroniclers/L1wpAAAAYAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Kings and Chroniclers] * [https://www.google.co.uk/books/edition/A_Kingdom_for_a_Stage/UzxlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover A Kingdom for a Stage] * [https://www.google.co.uk/books/edition/The_House_of_Commons/Ezz4OZuYVFYC?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The History of Parliament: The House of Commons 1558-1603 (3 v.)] * [https://www.google.co.uk/books/edition/Shakespeare_Soul_of_the_Age/nMYCAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover shakespeare, Soul of the Age] * [https://www.google.co.uk/books/edition/After_Poststructuralism/TOaEAAAAIAAJ?hl=en&gbpv=0&bsq=%22I%20am%20Richard%20II,%20know%20ye%20not%20that?%22 After Poststructuralism: Interdisciplinarity and Literary Theory] * [https://www.google.co.uk/books/edition/The_Unschooled_Mind/C7WnYtt219IC?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Unschooled Mind] * [https://www.google.co.uk/books/edition/Elizabeth_I/hHZnAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Eliz I] * [https://www.google.co.uk/books/edition/Dramas_of_Christian_Time/mnIqAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Dramas of Christian Time] * [https://www.google.co.uk/books/edition/Elizabeth_I/XjQmAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Elizabeth I: The Shrewdness of Virtue] * [https://www.google.co.uk/books/edition/John_Dryden/9Q1aAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover John Dryden] * [https://www.google.co.uk/books/edition/Shakespeare_and_Early_Modern_Political_T/DUwhAwAAQBAJ?hl=en&gbpv=1&dq=%22shakespeare%22+%2B+%22political+propaganda%22&pg=PA259&printsec=frontcover Shakespeare and Early Modern Political Thought] * [https://www.google.co.uk/books/edition/The_English_History_Play_in_the_age_of_S/5TT-AQAAQBAJ?hl=en&gbpv=1&dq=%22shakespeare%22+%2B+%22political+propaganda%22&pg=PA158&printsec=frontcover The English History Play in the Age of Shakespeare] * [https://www.google.co.uk/books/edition/Shakespeare_and_the_Political/rEcREQAAQBAJ?hl=en&gbpv=1&dq=%22shakespeare%22+%2B+%22political+propaganda%22&pg=PA215&printsec=frontcover Shakespeare and the Political] * [https://www.google.co.uk/books/edition/William_Shakespeare_Subject_of_the_Crown/a7G6DAAAQBAJ?hl=en&gbpv=1&dq=%22shakespeare%22+%2B+%22political+propaganda%22&pg=PT18&printsec=frontcover William Shakespeare - Subject of the Crown?] * [https://www.google.com/search?q=%22shakespeare%22+%2B+%22political+propaganda%22&client=firefox-b-d&hs=4AQ&sca_esv=6d4ade7bd26771c9&udm=36&biw=2510&bih=1307&tbs=cdr%3A1%2Ccd_min%3A2000%2Ccd_max%3A2099&sxsrf=ANbL-n6I6Pkwl7mmdHK6N1xPQXLbGBIOSg%3A1776853062010&ei=RqDoaZUvztiFsg_I5bToDw&ved=0ahUKEwiV6tC8nYGUAxVObEEAHcgyDf0Q4dUDCBM&uact=5&oq=%22shakespeare%22+%2B+%22political+propaganda%22&gs_lp=EhBnd3Mtd2l6LW1vZGVsZXNzIiYic2hha2VzcGVhcmUiICsgInBvbGl0aWNhbCBwcm9wYWdhbmRhIjIIECEYoAEYwwRInQlQxgZYuwdwAXgAkAEAmAF_oAHPAaoBAzEuMbgBA8gBAPgBAZgCAqACVsICCxAAGIAEGKIEGLADmAMAiAYBkAYCkgcBMqAHowOyBwExuAdTwgcDMC4yyAcEgAgB&sclient=gws-wiz-modeless The Nazi Appropriation of Shakespeare: Cultural Politics in] {{refend}} hiw8zb6ecynrmrvj7wo3yh1v67iynqd 2818526 2818525 2026-07-19T11:56:06Z Fortuna imperatrix mundi 1058581 /* IaR2 */ sp 2818526 wikitext text/x-wiki ==Background== ===Richard II=== ===Elizabeth I=== ====Essex's rebellion==== ==IaR2== Lambarde's account of his interview with the queen in August 1601 is well known.{{sfn|Orgel|2011|pp=11–43}} ===Political anxiety and tension=== ===Censorship=== ==Consequences and aftermath== == Notes == {{reflist|group=note}} ==References== {{Reflist|20em}} ==Bibliography== {{refbegin|30em|indent=yes}} * {{Cite book|url=https://doi.org/10.1057/9780230307261_2|title=Prologue: I am Richard II|last=Orgel|first=Stephen|date=2011|publisher=Palgrave Macmillan UK|isbn=978-0-230-30726-1|editor-last=Petrina|editor-first=Alessandra|location=London|pages=11–43|language=en|doi=10.1057/9780230307261_2|editor-last2=Tosi|editor-first2=Laura}} * [https://broadlytextual.com/2017/12/15/i-am-richard-ii-know-ye-not-that-drama-and-political-anxiety-in-shakespeares-london/ Hixon, E., Syracuse Univ] * Bate, Jonathan (2008). Soul of the Age. London: Penguin. pp. 256–286. ISBN 978-0-670-91482-1. * [https://theconversation.com/richard-ii-by-william-shakespeare-why-the-divine-right-of-kings-still-matters-186648 McFarlane, K., Univ South Australia] * [https://muse.jhu.edu/article/31090/summary Lemon, Rebecca. "The Faulty Verdict in "The Crown v. John Hayward"." SEL Studies in English Literature 1500-1900, vol. 41 no. 1, 2001, p. 109-132. Project MUSE, https://dx.doi.org/10.1353/sel.2001.0009] * [https://brill.com/display/book/9789401211666/B9789401211666-s009.xml?language=en&srsltid=AfmBOoqTUaTQr0msNWRdtEZZ9qQ3rNcVkSoBvDPXuGv7nRvHrC8t5OzH Kizelbach, U. (2014), In The Pragmatics of Early Modern Politics: Power and Kingship in Shakespeare’s History Plays. Leiden, The Netherlands: Brill. https://doi.org/10.1163/9789401211666_009] * [https://www.google.co.uk/books/edition/Richard_II/z-kOAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Lopez, R2] * [https://www.manchesterhive.com/display/9781526130532/9781526130532.00008.xml?print rgel, S. (2017). "I am Richard II". In Spectacular Performances. Manchester, England: Manchester University Press. Retrieved Apr 22, 2026, from https://doi.org/10.7765/9781526130532.00008] * [https://www.researchgate.net/publication/271041996_Was_Elizabeth_I_Richard_II_The_Authenticity_of_Lambarde%27s_%27Conversation%27 Scott-Warren, Jason. (2012). Was Elizabeth I Richard II?: The Authenticity of Lambarde's 'Conversation'. The Review of English Studies. 64. 208-230. 10.1093/res/hgs062] * Stanley Wells, “Introduction” in Richard II, The New Penguin Shakespeare, ed. Stanley Wells (London: Penguin Books, 1969), 13. * [https://www.cambridge.org/core/books/abs/shakespeare-survey/shakespeare-and-history-divergencies-and-agreements/B065A23215FD86BEE8E1CFD51DC7C1FB Ives EW. Shakespeare and History: Divergencies and Agreements. In: Wells S, ed. Shakespeare Survey. Shakespeare Survey. Cambridge University Press; 1986:19-36] * [https://www.jstor.org/stable/457855 Albright, Evelyn May. “Shakespeare’s Richard II, Hayward’s History of Henry IV, and the Essex Conspiracy.” PMLA, vol. 46, no. 3, 1931, pp. 694–719. JSTOR, https://doi.org/10.2307/457855. Accessed 22 Apr. 2026] * [https://dsc.duq.edu/cgi/viewcontent.cgi?article=2800&context=etd Morris, A. (2019). "within the hollow crown": Performing Kingship in Richard II and Henry IV Part One (Master's thesis, Duquesne University). Retrieved from https://dsc.duq.edu/etd/1771] * [https://digitalcommons.lib.uconn.edu/cgi/viewcontent.cgi?article=1122&context=srhonors_theses Scannell, Sarah J., "Shakespeare's Richard II and Henry V and Political Rebellions in the Reign of Queen Elizabeth I" (2010). Honors Scholar Theses, 138] * [https://compass.onlinelibrary.wiley.com/doi/abs/10.1111/j.1741-4113.2011.00873.x Luecking Frost, L. (2012), “A Kyng That Ruled All By Lust”: Richard II in Elizabethan Literature. Literature Compass, 9: 183-198. https://doi.org/10.1111/j.1741-4113.2011.00873.x] * [https://www.academia.edu/8347491/A_Law_Case_in_Verse_Venus_and_Adonis_and_the_Authorship_Question Stritmatter, R., A Law Case in Verse: Venus and Adonis and the Authorship Question, 2004 University of Tennessee Law Review] * [https://www.researchgate.net/publication/304647742_Prologue_I_am_Richard_II Orgel, Stephen. (2011). Prologue: I am Richard II. 10.1057/9780230307261_2] * [https://go.gale.com/ps/i.do?id=GALE%7CA314252957&sid=googleScholar&v=2.1&it=r&linkaccess=abs&issn=15256863&sw=w&p=AONE&userGroupName=anon%7Ebf79dda0&aty=open-web-entry Egan, Michael. "The Essex Rebellion and Richard II: why wasn't Shakespeare arrested?" Shakespeare Oxford Newsletter, vol. 48, no. 3, summer-fall 2012, p. 20] * [https://www.cambridge.org/core/books/abs/richard-ii/stagehistory/D62D9CA061F03B2AE8D56E856288EEDA 1. Child H, Shakespeare W. THE STAGE-HISTORY. In: Dover Wilson J, ed. Richard II: The Cambridge Dover Wilson Shakespeare. Cambridge Library Collection - Literary Studies. Cambridge University Press; 2009:lxxvii-xcii] * [https://www.openstarts.units.it/entities/publication/16d8c447-62f4-48e8-8347-5d56af674aa3/details Snelling, D., 'Prospero on the Coast of Bohemia. "Nihil sed rara et miranda amabat". Diary of Melchior Goldast upon the late Emperor Rudolph II' 1994] * [https://www.jstor.org/stable/2902265 Gajowski, Evelyn. Shakespeare in Performance: King Richard IIby Margaret Shewring. Shakespeare Quarterly, vol. 49, no. 3, 1998, pp. 328–30. JSTOR, https://doi.org/10.2307/2902265. Accessed 22 Apr. 2026] * [https://muse.jhu.edu/article/217842/summary Mosse, Ramona Franziska. Review of Richard II. Shakespeare Bulletin, vol. 25 no. 2, 2007, p. 73-77. Project MUSE, https://dx.doi.org/10.1353/shb.2007.0036] * [https://www.journals.uchicago.edu/doi/abs/10.1086/665894?journalCode=rq Petrina and Tosi. Representations of Elizabeth I in Early Modern Culture Elizabeth Pentland Renaissance Quarterly 2012 65:1, 274-276] * [https://journals.co.za/doi/10.4314/sisa.v26i1.4 The construction of a king : waste, effeminacy and queerness in Shakespeare's Richard II Higginbotham, D.] * [https://academic.oup.com/book/2363/chapter-abstract/142533948?redirectedFrom=fulltext Erskine-Hill, Howard, 'The Second Tetralogy and After', Poetry and the Realm of Politics: Shakespeare to Dryden (Oxford, 1996; online edn, Oxford Academic, 3 Oct. 2011), https://doi.org/10.1093/acprof:oso/9780198117315.003.0004] * [https://www.academia.edu/123917862/Shakespeare_s_Richard_II_in_the_minds_of_the_Elizabethan_audiences Shakespeare’s Richard II in the minds of the Elizabethan audiences] * [https://journals.sagepub.com/doi/10.1177/03064220221084555 Plays, protests and the censor’s pencil] * [https://www.indexoncensorship.org/2022/03/plays-protests-and-the-censors-pencil/ Plays, protests and the censor’s pencil A brittle Gloriana: staging the deposition of Queen Elizabeth I]. * Saul, N., Richard II * [https://www.academia.edu/3236651/_A_god_on_earth_thou_art_Richard_II_and_the_nature_of_Kingship “A god on earth thou art”: Richard II and the nature of Kingship] * [https://www.jstor.org/stable/457398 Albright, Evelyn May. “Shakespeare’s Richard II and the Essex Conspiracy.” PMLA, vol. 42, no. 3, 1927, pp. 686–720. JSTOR, https://doi.org/10.2307/457398. Accessed 22 Apr. 2026] * [https://www.berghahnjournals.com/view/journals/critical-survey/37/3/cs370301.xml Holderness, G. (2025). Introduction: Shakespeare's Wars of the Roses. Critical Survey, 37(3), 1-4. Retrieved Apr 22, 2026, from https://doi.org/10.3167/cs.2025.370301] * [https://www.folger.edu/podcasts/shakespeare-unlimited/tragedy-richard-ii-henry-iv-helen-castor/ Folger: Shakespeare Unlimited: Episode 249] * [https://www.rachaeldickzen.com/blog/2021/1/10/charlesiii The Historical and Shakespearean References in King Charles III] * Kermode, The Age of Shakespeare, pg.53. * [https://heinonline.org/HOL/login-hol?redirect_url=https%3A%2F%2Fheinonline.org%2FHOL%2Fmojo%3F%252Fmojo%3D%26auth_token%3DyhcMOu8W7PCc5OrmTqVnr2qFPpoiOdhM5uQmkQyWufU%253D%26casa_token%3D%26div%3D5%26g_sent%3D1%26paywall_url%3D%26referrer%3Dhttps%253A%252F%252Fheinonline.org%252FHOL%252FPage%253Fhandle%253Dhein.journals%252Fneastlr1%26timestamp%3D1776855600%26url%3D%252FHOL%252FPage%253Fhandle%253Dhein.journals%252Fneastlr1%2526div%253D5%2526id%253D%2526page%253D Jones, H., ''I AM RICHARD II, KNOW YE NOT THAT?' - WORKS. OF WILLIAM SHAKESPEARE AND THE ART OF. GOVERNANCE'''] * Hattaway, M., William Shakespeare: Richard II (Literature Insights, 9781847600349) * [https://books.openedition.org/psn/8305 Findlay, A., “Good sometimes queen” (V.1.37). Richard II, Mary Stuart and the Politics of Queenship, 205-219] * [https://pure.manchester.ac.uk/ws/portalfiles/portal/54508829/FULL_TEXT.PDF Pritchard, K., 'Legitimacy, Illegitimacy and Sovereignty in Shakespeare’s British Plays' (2011 University of Manchester PhD] * CHATELANAT, Marine. Histrionic Future Kings: The Politics of Metadrama in Shakespeare’s Richard III, Richard II, and Henry IV Part 1. Master, 2023. * [https://www.persee.fr/doc/xvii_0291-3798_2004_num_59_1_1993 Mayer, J-C., 'The "Parliament Sceane" in Shakespeare's King Richard II', XVII-XVIII. Revue de la Société d'études anglo-américaines des XVIIe et XVIIIe siècles Année 2004 59 pp. 27-42] * [https://www.emerald.com/md/article-abstract/38/5/315/285132/Managers-as-kings-Shakespeare-on-modern-leadership?redirectedFrom=fulltext Egan M (2000), "Managers as kings: Shakespeare on modern leadership". Management Decision, Vol. 38 No. 5 pp. 315–327, doi: https://doi.org/10.1108/00251740010340490] * [https://d-nb.info/1279081511/34 Quabeck (2022): William Shakespeare, in: Thomas Gutmann, Eberhard Ortland, Klaus Stierstorfer, eds., Encyclopedia of Law and Literature (last edited 26 October 2022), doi: 10.17879/12009577118] * [https://lawliberty.org/book-review/was-the-bard-writing-about-the-donald/ Sullivan, V., 'Was the Bard Writing about The Donald?'] * [https://deveresociety.co.uk/wp-content/uploads/2022/11/FINAL_NL_Apr2022-Haste-July-Letterresponse.pdf Haste, I., 'THE PLAY ON THE EVE OF THE ESSEX REBELLION'] * [https://www.cherylmtaylor.com/2017/11/08/a-companion-to-shakespeares-richard-ii/ A Companion to Shakespeare’s Richard II] * [https://www.city-journal.org/article/shakespeares-richards Dalrymple, T., Shakespeare’s Richards: The playwright’s two historical dramas offer contrasts between political pathologies] * [https://hfriedberg.web.wesleyan.edu/engl205/wshakespeare/plotsandrebelions.htm ELIZABETHAN PLOTS AND REBELLIONS] * [https://www.jstor.org/stable/2869708 Rackin, Phyllis. “The Role of the Audience In Shakespeare’s Richard II.” Shakespeare Quarterly, vol. 36, no. 3, 1985, pp. 262–81. JSTOR, https://doi.org/10.2307/2869708. Accessed 22 Apr. 2026.] * [https://www.psychologytoday.com/ie/blog/whats-in-name/201603/group-think-and-academia-shocking-shakespeare-shenanigans Group Think and Academia: Shocking Shakespeare Shenanigans] * [https://journals.sagepub.com/doi/10.1177/01847678221099975 Hiscopck, A., '"Why I should welcome such a guest as grief [?": Lodging and dislodging in Shakespeare's Richard II'] * [https://academic.oup.com/book/51779/chapter-abstract/420487796?redirectedFrom=fulltext 'Epigraph', Imagining Time in the English Chronicle Play: Historical Futures, 1590-1660 (Oxford, 2023; online edn, Oxford Academic, 19 Oct. 2023), https://doi.org/10.1093/oso/9780198872658.002.0009] * [https://journals.openedition.org/interfaces/319?lang=en Blake & Shakespeare] * [https://www.cambridge.org/core/books/abs/shakespeare-and-the-visual-imagination/richard-iiand-the-politics-of-perspective/23A5D4615E2A8BF9380BAF1ECE99B46D Sillars S. Richard IIand the Politics of Perspective. In: Shakespeare and the Visual Imagination. Cambridge University Press; 2015:133-162.] * [https://www.persee.fr/doc/xvii_0291-3798_2004_num_59_1_1992 Le miroir des rois: de Richard II à Henry VIII] * [https://www.cambridge.org/core/books/abs/king-richard-ll/introduction/25419329A06BC1C2E4D3DDBF331B7F91 Shakespeare W, McEachern C. Introduction. In: Gurr A, ed. King Richard Ll. The New Cambridge Shakespeare. Cambridge University Press; 2018:1-66.] * [https://www.researchgate.net/publication/372862689_A_Critical_Discourse_Study_of_Shakespeare%27s_Theological_Conceptions_in_Acts_IV_and_V_of_Richard_II_The_%27Divine_Mandate%27_of_Richard_Kingship_Falls_Apart A Critical Discourse Study of Shakespeare’s Theological Conceptions in Acts IV and V of Richard II: The ‘Divine Mandate’ of Richard Kingship Falls Apart] * [https://www.jstor.org/stable/10.1525/lal.2004.16.3.313 Richard II, Bracton, and the End of Political Theology] * [https://dokumen.pub/the-art-of-law-in-shakespeare-9781509905478-9781509905508-9781509905492.html The Art of Law in Shakespeare] * [https://www.cambridge.org/core/books/abs/cambridge-companion-to-shakespeare-and-religion/providence-and-divine-right-in-the-english-histories/DBE9BFEBF5C199F9DD72D31ED9C3B041 Mayer J-C. Providence and Divine Right in the English Histories. In: Hamlin H, ed. The Cambridge Companion to Shakespeare and Religion. Cambridge Companions to Literature. Cambridge University Press; 2019:151-167] * [https://www.researchgate.net/publication/315698000_THE_CHOMSKY-FOUCAULT_DEBATE_WHERE_DOES_THE_BARD_STAND_IN_RICHARD_II Salami, Ali & Riahi Nouri, Amir. (2016). THE CHOMSKY-FOUCAULT DEBATE: WHERE DOES THE BARD STAND IN RICHARD II?] * [https://dokumen.pub/shakespeare-richard-ii-9781350363168-9780333416709.html Richard II] * [https://www.jstor.org/stable/457907?seq=1 Heffner, Ray. “Shakespeare, Hayward, and Essex.” PMLA, vol. 45, no. 3, 1930, pp. 754–80. JSTOR, https://doi.org/10.2307/457907. Accessed 22 Apr. 2026] * [https://www.cambridge.org/core/books/abs/cambridge-companion-to-shakespeare-and-war/shakespeare-on-civil-and-dynastic-wars/00D0269CFF998876DB961FB01C22954D Shakespeare on Civil and Dynastic Wars] * [https://brill.com/display/book/edcoll/9789004356900/B9789004356900_012.xml?srsltid=AfmBOopJMjwJAQnUGME9FmT6Gwg0ih9pArbXBdTJbN6YAqbdXqbkGyuo Is a Dramatic Theory of History Possible? Shakespeare’s Richard ii and ‘Historical Truth’] * [https://www.jstor.org/stable/24475607 THE CHILD'S TWO BODIES: SHAKESPEARE, SOVEREIGNTY, AND THE END OF SUCCESSION] * [https://dokumen.pub/english-renaissance-drama-an-introduction-to-theatre-and-theatres-in-shakespeares-time-9781847600202-9781847600714.html English Renaissance Drama : An Introduction to Theatre and Theatres in Shakespeare’s Time] * [https://www.cambridge.org/core/books/abs/believing-in-shakespeare/person-plot-place/BD4EF4CBA6E7D8E2ECE5519FA1D08C16 1. McEachern C. Person, Plot, Place. In: Believing in Shakespeare: Studies in Longing. Cambridge University Press; 2018:181-296] * [https://www.enotes.com/topics/richard-2/criticism/richard-ii-vol-91/criticism-production-reviews-1000228/nigel-saul-review-date-12-may-2000 With an Eye to the Present] * [https://www.academia.edu/128303633/Shakespeare_s_Other_Eden_Richard_II_by_Ouroboros_Theatre_Company_2013_ Shakespeare’s Other Eden Richard II by Ouroboros Theatre Company (2013)] * [https://dokumen.pub/shakespeare-upstart-crow-to-sweet-swan-1592-1623-9781472555496-9781408130148-9781408139189.html Shakespeare: Upstart Crow to Sweet Swan 1592-1623] * [https://www.jstor.org/stable/41917215 RACKIN, PHYLLIS. “Temporality, Anachronism, and Presence in Shakespeare’s English Histories.” Renaissance Drama, vol. 17, 1986, pp. 101–23. JSTOR, http://www.jstor.org/stable/41917215. Accessed 22 Apr. 2026.] * [https://www.cambridge.org/core/journals/greece-and-rome/article/abs/alexander-philotas-and-the-origins-of-modern-historiography/D90568C0A1131952625E0A3B44F43EC8 Stoneman R. Alexander, Philotas, and the origins of modern historiography. Greece and Rome. 2013;60(2):296-312. doi:10.1017/S0017383513000119] * [https://www.academia.edu/35800720/Shakespeares_Richard_II_and_Elizabethan_politics Shakespeare's Richard II and Elizabethan politics] * [https://www.cambridge.org/core/books/abs/rethinking-historicism-from-shakespeare-to-milton/old-new-now/2E8CBEFFEFB11D8B843BAF2C108E9768 Introduction - Old, New, Now] * [https://www.academia.edu/24392476/Richard_II_hamartia_and_climax Richard II hamartia and climax] * [https://www.cambridge.org/core/books/abs/babylonian-jews-and-sasanian-imperialism-in-late-antiquity/kings-and-religion-in-the-talmud-and-in-the-imagination-of-sasanian-communities/E2DCC2F3E0FBA74ABE543DDD2DD7F1A7 Kings and Religion in the Talmud and in the Imagination of Sasanian Communities] * Stephen Greenblatt, “Introduction,” in The Power of Forms in the English Renaissance, ed. Stephen Greenblatt (Norman, OK: 1982), 3-6] * [https://www.academia.edu/30472558/Americas_Donald_Trump_Shakespeares_Richard_II_15_Dec_2016 America's Donald Trump, Shakespeare's Richard II] * [https://www.academia.edu/5321346/Looking_Back_Shakespeare_s_Indebtedness_to_Chaucer_and_the_Representations_of_Chivalry_in_King_Richard_II_The_Two_Noble_Kinsmen_and_the_Knight_s_Tale Looking Back: Shakespeare’s Indebtedness to Chaucer and the Representations of Chivalry in King Richard II, The Two Noble Kinsmen and the Knight’s Tale] * [https://www.academia.edu/92423912/The_Good_Usurper_in_the_eyes_of_God_and_the_people_An_analysis_of_the_role_of_the_usurper_in_Shakespeares_Richard_II_and_Henry_IV_Part_1 The Good Usurper in the eyes of God and the people: An analysis of the role of the usurper in Shakespeare's Richard II and Henry IV Part 1] * [https://www.academia.edu/91789871/The_Bacon_Shakespeare_Manuscript_Hitherto_known_as_the_Northumberland_Manuscript_which_originally_Contained_Copies_of_his_Shakespeare_Plays_Richard_II_and_Richard_III THE BACON-SHAKESPEARE MANUSCRIPT (HITHERTO KNOWN AS THE NORTHUMBERLAND MANUSCRIPT) WHICH ORIGINALLY CONTAINED COPIES OF HIS SHAKESPEARE PLAYS RICHARD II AND RICHARD III] * [https://www.google.co.uk/books/edition/Richard_II/f4gGCAAAQBAJ?hl=en&gbpv=1&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&pg=PA88&printsec=frontcover Richard II: Critical Essays] * [https://www.google.co.uk/books/edition/A_Companion_to_Shakespeare_s_Works_Volum/JlDNEAAAQBAJ?hl=en&gbpv=1&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&pg=PA400&printsec=frontcover A Companion to Shakespeare's Works, Volume II] * [https://www.google.co.uk/books/edition/Shakespeare_Reread/Z6JhDwAAQBAJ?hl=en&gbpv=1&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&pg=PA239&printsec=frontcover Shakespeare Reread] * [https://www.google.co.uk/books/edition/Shakespeare_in_the_Present/yTKWEAAAQBAJ?hl=en&gbpv=1&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&pg=PT8&printsec=frontcover Shakespeare in the Present] * [https://www.google.co.uk/books/edition/Studying_Shakespeare/1N4FBAAAQBAJ?hl=en&gbpv=1&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&pg=PA129&printsec=frontcover Studying Shakespeare A Practical Introduction] * [https://www.google.co.uk/books/edition/Critical_Essays_on_William_Faulkner/kAGBEAAAQBAJ?hl=en&gbpv=1&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&pg=PT178&printsec=frontcover Critical Essays on William Faulkner] * [https://www.google.co.uk/books/edition/Shakespeare/QY6aEQAAQBAJ?hl=en&gbpv=1&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&pg=PA216&printsec=frontcover Shakespeare: An Anthology of Criticism and Theory 1945-2000] * [https://www.google.co.uk/books/edition/The_Reign_of_Richard_II/cv0WAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Dodd, G., R2] * [https://www.google.co.uk/books/edition/Shakespeare_and_the_Institution_of_Theat/Nkt9DAAAQBAJ?hl=en&gbpv=1&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&pg=PA108&printsec=frontcover Shakespeare and the Institution of Theatre] * [https://www.google.co.uk/books/edition/The_Theory_of_the_King_s_Two_Bodies_in_t/8SRXAAAAYAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Theory of the King's Two Bodies in the Age of Shakespeare] * [https://www.google.co.uk/books/edition/Stealing_the_Story/RaJlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Stealing the Story: Shakespeare's Self-Conscious Use of the Mimetic Tradition in the Tragedies] * [https://www.google.co.uk/books/edition/The_Greenwood_Companion_to_Shakespeare_O/JkcgAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Greenwood Companion to Shakespeare: Overviews and the history plays] * [https://www.google.co.uk/books/edition/British_and_Irish_Literature_and_Its_Tim/xH4jAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover British and Irish Literature and Its Times] * [https://www.google.co.uk/books/edition/Shakespeare_s_Philosophy_of_History_Reve/1jwgAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare's Philosophy of History Revealed in a Detailed Analysis of Henry V and Examined in Other History Plays] * [https://www.google.co.uk/books/edition/Shakespeare_and_His_Contemporaries/7em7coOthxQC?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare and His Contemporaries] * [https://www.google.co.uk/books/edition/The_Politics_of_the_Public_Sphere_in_Ear/0sGHAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Politics of the Public Sphere in Early Modern England] * [https://www.google.co.uk/books/edition/Shakespeare_and_the_Catholic_Religion/Fl4gAQAAIAAJ?hl=en&gbpv=0&bsq=%22I%20am%20Richard%20II,%20know%20ye%20not%20that?%22 Shakespeare and the Catholic religion] * [https://www.google.co.uk/books/edition/Proceedings_of_the_British_Academy_Volum/8CsoAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Lectures] * [https://www.google.co.uk/books/edition/Slander_and_Censorship_in_Late_Sixteenth/wb_oAvuJgbAC?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Slander and Censorship in Late Sixteenth Century Literature] * [https://www.google.co.uk/books/edition/Elizabethan_Literature_and_the_Law_of_Fr/vBFdAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Elizabethan Literature and the Law of Fraudulent Conveyance] * [https://www.google.co.uk/books/edition/Hamlet_History_and_commentary/yxgrAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Hamlet History etc] * [https://www.google.co.uk/books/edition/William_Shakespeare_A_Popular_Life/1BuaAAAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover William Shakespeare: a popular life] * [https://www.google.co.uk/books/edition/A_State_of_Mind/pCiIAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover A State of Mind?] * [https://www.google.co.uk/books/edition/Shakespeare_s_Dramatic_Genres/J5FlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare's Dramatic Genres] * [https://www.google.co.uk/books/edition/Literature_Criticism_from_1400_to_1800/O6VkAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Literature Criticism from 1400 to 1800 Volume 89] * [https://www.google.co.uk/books/edition/Shakespeare_the_Papist/LPwNAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare the Papist] * [https://www.google.co.uk/books/edition/Goslicius_Ideal_Senator_and_His_Cultural/1jc7AQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Goslicius' Ideal Senator and His Cultural Impact Over the Centuries] * [https://www.google.co.uk/books/edition/Shakespeare_by_Another_Name/FqllAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover "Shakespeare" by Another Name] * [https://www.google.co.uk/books/edition/Shakespeare_s_Friends/AlZlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare's Friends] * [https://www.google.co.uk/books/edition/Dr_Simon_Forman/qHceAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Dr Simon Forman] * [https://www.google.co.uk/books/edition/William_Shakespeare_the_Wars_of_the_Rose/dZFlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover William Shakespeare, the Wars of the Roses and the historians] * [https://www.google.co.uk/books/edition/Shakespearean_Criticism/2TdlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespearean Criticism] * [https://www.google.co.uk/books/edition/Poetry_History_and_Antiquarianism_in_Ref/jwIhAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Poetry, History, and Antiquarianism in Reformation England] * [https://www.google.co.uk/books/edition/Shakespeare_and_the_Prince_of_Love/D0QgAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare and the Prince of Love] * [https://www.google.co.uk/books/edition/Paper_Bullets_of_the_Brain/o0YgAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Paper Bullets of the Brain] * [https://www.google.co.uk/books/edition/As_You_Like_It/GiBaAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover As You Like It: Third Series] * [https://www.google.co.uk/books/edition/William_Shakespeare/WCqaAAAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover William Shakespeare, Dominic Shellard] * [https://www.google.co.uk/books/edition/Poets_and_God/0XZlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Poets and God] * [https://www.google.co.uk/books/edition/Law_and_Literature/Ax5MAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Law and Literature Volume 16] * [https://www.google.co.uk/books/edition/The_Embodied_Word/FV8sAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Embodied Word] * [https://www.google.co.uk/books/edition/The_Case_for_Shakespeare/WaRlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Case for Shakespeare: The End of the Authorship Question] * [https://www.google.co.uk/books/edition/Explorations_in_Renaissance_Culture/_SYrAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Explorations in Renaissance Culture Volumes 33-34] * [https://www.google.co.uk/books/edition/The_Touch_of_the_Real/ewdaAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Touch of the Real: Essays in Early Modern Culture in Honour of Stephen Greenblatt] * [https://www.google.co.uk/books/edition/Wotton_and_His_Worlds/ZfgNAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Wotton and his Worlds] * [https://www.google.co.uk/books/edition/Theatre_and_Religion/wo1lAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Theatre and Religion Lancastrian Shakespeare] * [https://www.google.co.uk/books/edition/Trying_Treason/TOKxAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Trying Treason] * [https://www.google.co.uk/books/edition/Willing_Subjects/IEX0sGwT1QQC?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Willing Subjects] * [https://www.google.co.uk/books/edition/Symbolism/Bt0ZAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Symbolism] * [https://www.google.co.uk/books/edition/Performing_Shakespeare/35pQAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Performing Shakespeare] * [https://www.google.co.uk/books/edition/Soul_of_the_Age/e0UgAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Soul of the Age] * [https://www.google.co.uk/books/edition/England/aD9nAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover England] * [https://www.google.co.uk/books/edition/Elizabeth_I/-GtnAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Elizabeth I] * [https://www.google.co.uk/books/edition/King_Richard_II/oGUMX4RntjgC?hl=en&gbpv=1&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&pg=PA25&printsec=frontcover King Richard II] * [https://www.google.co.uk/books/edition/Shakespeare_and_the_Legal_Imagination/OXPvBqQLw-4C?hl=en&gbpv=1&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&pg=PA38&printsec=frontcover Shakespeare and the legal imagination] * [https://www.google.co.uk/books/edition/Shakespeare_s_Theatre/GxN3ue9_r3oC?hl=en&gbpv=1&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&pg=PA69&printsec=frontcover Shakespeare's Theatre] * [https://www.google.co.uk/books/edition/Critical_Essays_on_Shakespeare_s_Richard/AaYoAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Critical Essays on Shakespeare's Richard II] * [https://www.google.co.uk/books/edition/The_Reign_of_Richard_II_Essays_in_Honour/y3xnAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Reign of Richard II: Essays in Honour of May McKisack] * [https://www.google.co.uk/books/edition/Poetry_and_the_Realm_of_Politics/oQFaAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Poetry and the Realm of Politics] * [https://www.google.co.uk/books/edition/Shakespeare/BM0mAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare] * [https://www.google.co.uk/books/edition/Shakespearean_Politics/oTdlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespearean Politics] * [https://www.google.co.uk/books/edition/Shakespeare_the_Theatrical_Dimension/wl4gAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare, the Theatrical Dimension] * [https://www.google.co.uk/books/edition/Who_was_Kit_Marlowe/zQ1aAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Kit Marlowe etc] * [https://www.google.co.uk/books/edition/From_Page_to_Performance/beQKAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover From Page to Performance] * [https://www.google.co.uk/books/edition/Exploring_Tudor_England/ax56AAAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Exploring Tudor England] * [https://www.google.co.uk/books/edition/The_Movement_Towards_Subversion/vyJaAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Movement Towards Subversion] * [https://www.google.co.uk/books/edition/Shakespeare_s_Typological_Satire/G5BlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare's Typological Satire] * [https://www.google.co.uk/books/edition/Shakespeare_Recycled/zzNlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare Recycled] * [https://www.google.co.uk/books/edition/Reinventing_the_Middle_Ages_the_Renaissa/fXFnAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Reinventing the Middle Ages & the Renaissance] * [https://www.google.co.uk/books/edition/The_Mysterious_William_Shakespeare/WnllAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Mysterious William Shakespeare] * [https://www.google.co.uk/books/edition/Richard_II/GHhlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Richard II Critical Essays] * [https://www.google.co.uk/books/edition/William_Shakespeare/WJvC6gu_I0gC?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover William Shakespeare: Records and Images] * [https://www.google.co.uk/books/edition/Shakespeare_and_the_Actors/HYtlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare and the Actors] * [https://www.google.co.uk/books/edition/Shakespeare_the_Man/BVdlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare the man] * [https://www.google.co.uk/books/edition/Henry_V/zXllAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Henry V: A Guide to the Play] * [https://www.google.co.uk/books/edition/Shakespearean_Contingencies/Cw1NAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespearean Contingencies] * [https://www.google.co.uk/books/edition/Renaissance_Drama/E60kAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Renaissance Drama 1990] * [https://www.google.co.uk/books/edition/Language_Discourse_Sign/uH4oAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Language, Discourse, Sign] * [https://www.google.co.uk/books/edition/Mock_Kings_in_Medieval_Society_and_Renai/T98KAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Mock kings in medieval society and Renaissance drama] * [https://www.google.co.uk/books/edition/Shakespeare_Invention_of_the_Human/ojHirImrtYoC?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare: Invention of the Human] * [https://www.google.co.uk/books/edition/Shakespeare/wn5lAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare] * [https://www.google.co.uk/books/edition/Persons_in_Groups/rQ24AAAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Persons in Groups] * [https://www.google.co.uk/books/edition/All_Semblative_a_Woman_s_Part/0DlaAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover All Semblative a Woman's Part?] * [https://www.google.co.uk/books/edition/Crossing_the_Mirror/qRZNAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Crossing the Mirror] * [https://www.google.co.uk/books/edition/De_Vere_is_Shakespeare/dKJlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover De Vere is Shakespeare] * [https://www.google.co.uk/books/edition/William_Lambarde_Elizabethan_Antiquary_1/x1RnAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover William Lambarde, Elizabethan Antiquary, 1536-1601] * [https://www.google.co.uk/books/edition/The_Power_of_Forms_in_the_English_Renais/cPtZAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Power of Forms in the English Renaissance] * [https://www.google.co.uk/books/edition/Ravishment_and_Rememberance/G31LAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Ravishment and Rememberance] * [https://www.google.co.uk/books/edition/Shakespeare_and_His_Theatre/8A5ZQq3uOVQC?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare and His Theatre] * [https://www.google.co.uk/books/edition/Critical_Hermeneutics_and_Shakespeare_s/O10gAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Critical Hermeneutics and Shakespeare's History Plays] * [https://www.google.co.uk/books/edition/Christian_England/K-WfAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Christian England] * [https://www.google.co.uk/books/edition/Shakespeare_s_Religious_Background/xDSaAAAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare's Religious Background] * [https://www.google.co.uk/books/edition/Shylock/N4RlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shylock] * [https://www.google.co.uk/books/edition/The_Shakespeare_Legacy/MM5XAAAAYAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Shakespeare Legacy] * [https://www.google.co.uk/books/edition/Renaissance_Genres/0uFZAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Renaissance Genres] * [https://www.google.co.uk/books/edition/Cannibals_Witches_and_Divorce/qZRpAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Cannibals, Witches, and Divorce] * [https://www.google.co.uk/books/edition/The_Problem_of_Religious_Knowledge/C29LAAAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Problem of Religious Knowledge] * [https://www.google.co.uk/books/edition/Essex_and_the_Great_Revolt_of_1381/J8RzAAAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Essex and the Great Revolt of 1381] * [https://www.google.co.uk/books/edition/Transactions_of_the_London_and_Middlesex/4dtJAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover TLMAS] * [https://www.google.co.uk/books/edition/Shakespeare_Politics_and_the_State/Mn9lAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare, Politics and the State] * [https://www.google.co.uk/books/edition/Allegories_of_Power_in_the_England_of_El/LIYgAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Allegories of Power in the England of Elizabeth] * [https://www.google.co.uk/books/edition/William_Shakespeare/rIVlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover William Shakespeare] * [https://www.google.co.uk/books/edition/Women_s_Matters/PDRlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Women's Matters] * [https://www.google.co.uk/books/edition/The_Weak_King_Dilemma_in_the_Shakespeare/0bJlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Weak King Dilemma in the Shakespearean History Play] * [https://www.google.co.uk/books/edition/The_Book_Known_as_Q/S2tlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Book Known as Q] * [https://www.google.co.uk/books/edition/Fields_of_Vision/OD0eAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Fields of Vision] * [https://www.google.co.uk/books/edition/Ungodly_Delights/RKgcAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Ungodly Delights] * [https://www.google.co.uk/books/edition/The_Shakespeare_Handbook/rLRlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Shakespeare HandboOK] * [https://www.google.co.uk/books/edition/Humanities/y5FZAAAAYAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Humanities] * [https://www.google.co.uk/books/edition/Richard_II_by_William_Shakespeare/Bb3yAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Richard II by William Shakespeare] * [https://www.google.co.uk/books/edition/King_Richard_II/50NnAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover King Richard II] * [https://www.google.co.uk/books/edition/Murder_Under_Trust_Or_The_Topical_Macbet/0oNlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Murder under trust] * [https://www.google.co.uk/books/edition/The_Shakespearean_Kings/tHBlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Shakespearean Kings] * [https://www.google.co.uk/books/edition/America_the_Mabr_e_y_Experience/mRQ3AAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover America, the Mabr(e)y Experience: Resistance, Revolution & Civil War] * [https://www.google.co.uk/books/edition/Richard_II/ZDEkAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Richard II: An Annotated Bibliography, Volume 2] * [https://www.google.co.uk/books/edition/The_Batsford_Companion_to_Medieval_Engla/ev78b9EJQy0C?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Batsford Companion to Medieval England] * [https://www.google.co.uk/books/edition/Shakespeare_s_Unruly_Women/FKFlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare's Unruly Women] * [https://www.google.co.uk/books/edition/Shakespeare_and_Others/iFEgAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare and Others] * [https://www.google.co.uk/books/edition/Kings_and_Chroniclers/L1wpAAAAYAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Kings and Chroniclers] * [https://www.google.co.uk/books/edition/A_Kingdom_for_a_Stage/UzxlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover A Kingdom for a Stage] * [https://www.google.co.uk/books/edition/The_House_of_Commons/Ezz4OZuYVFYC?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The History of Parliament: The House of Commons 1558-1603 (3 v.)] * [https://www.google.co.uk/books/edition/Shakespeare_Soul_of_the_Age/nMYCAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover shakespeare, Soul of the Age] * [https://www.google.co.uk/books/edition/After_Poststructuralism/TOaEAAAAIAAJ?hl=en&gbpv=0&bsq=%22I%20am%20Richard%20II,%20know%20ye%20not%20that?%22 After Poststructuralism: Interdisciplinarity and Literary Theory] * [https://www.google.co.uk/books/edition/The_Unschooled_Mind/C7WnYtt219IC?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Unschooled Mind] * [https://www.google.co.uk/books/edition/Elizabeth_I/hHZnAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Eliz I] * [https://www.google.co.uk/books/edition/Dramas_of_Christian_Time/mnIqAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Dramas of Christian Time] * [https://www.google.co.uk/books/edition/Elizabeth_I/XjQmAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Elizabeth I: The Shrewdness of Virtue] * [https://www.google.co.uk/books/edition/John_Dryden/9Q1aAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover John Dryden] * [https://www.google.co.uk/books/edition/Shakespeare_and_Early_Modern_Political_T/DUwhAwAAQBAJ?hl=en&gbpv=1&dq=%22shakespeare%22+%2B+%22political+propaganda%22&pg=PA259&printsec=frontcover Shakespeare and Early Modern Political Thought] * [https://www.google.co.uk/books/edition/The_English_History_Play_in_the_age_of_S/5TT-AQAAQBAJ?hl=en&gbpv=1&dq=%22shakespeare%22+%2B+%22political+propaganda%22&pg=PA158&printsec=frontcover The English History Play in the Age of Shakespeare] * [https://www.google.co.uk/books/edition/Shakespeare_and_the_Political/rEcREQAAQBAJ?hl=en&gbpv=1&dq=%22shakespeare%22+%2B+%22political+propaganda%22&pg=PA215&printsec=frontcover Shakespeare and the Political] * [https://www.google.co.uk/books/edition/William_Shakespeare_Subject_of_the_Crown/a7G6DAAAQBAJ?hl=en&gbpv=1&dq=%22shakespeare%22+%2B+%22political+propaganda%22&pg=PT18&printsec=frontcover William Shakespeare - Subject of the Crown?] * [https://www.google.com/search?q=%22shakespeare%22+%2B+%22political+propaganda%22&client=firefox-b-d&hs=4AQ&sca_esv=6d4ade7bd26771c9&udm=36&biw=2510&bih=1307&tbs=cdr%3A1%2Ccd_min%3A2000%2Ccd_max%3A2099&sxsrf=ANbL-n6I6Pkwl7mmdHK6N1xPQXLbGBIOSg%3A1776853062010&ei=RqDoaZUvztiFsg_I5bToDw&ved=0ahUKEwiV6tC8nYGUAxVObEEAHcgyDf0Q4dUDCBM&uact=5&oq=%22shakespeare%22+%2B+%22political+propaganda%22&gs_lp=EhBnd3Mtd2l6LW1vZGVsZXNzIiYic2hha2VzcGVhcmUiICsgInBvbGl0aWNhbCBwcm9wYWdhbmRhIjIIECEYoAEYwwRInQlQxgZYuwdwAXgAkAEAmAF_oAHPAaoBAzEuMbgBA8gBAPgBAZgCAqACVsICCxAAGIAEGKIEGLADmAMAiAYBkAYCkgcBMqAHowOyBwExuAdTwgcDMC4yyAcEgAgB&sclient=gws-wiz-modeless The Nazi Appropriation of Shakespeare: Cultural Politics in] {{refend}} ngydq95eohumyxrz2ybh3kyry7nepb6 2818527 2818526 2026-07-19T11:56:51Z Fortuna imperatrix mundi 1058581 /* Essex's rebellion */ + 2818527 wikitext text/x-wiki ==Background== ===Richard II=== ===Elizabeth I=== ====Essex's rebellion==== Seven months before Lambarde's discussion with Elizabeth. ==IaR2== Lambarde's account of his interview with the queen in August 1601 is well known.{{sfn|Orgel|2011|pp=11–43}} ===Political anxiety and tension=== ===Censorship=== ==Consequences and aftermath== == Notes == {{reflist|group=note}} ==References== {{Reflist|20em}} ==Bibliography== {{refbegin|30em|indent=yes}} * {{Cite book|url=https://doi.org/10.1057/9780230307261_2|title=Prologue: I am Richard II|last=Orgel|first=Stephen|date=2011|publisher=Palgrave Macmillan UK|isbn=978-0-230-30726-1|editor-last=Petrina|editor-first=Alessandra|location=London|pages=11–43|language=en|doi=10.1057/9780230307261_2|editor-last2=Tosi|editor-first2=Laura}} * [https://broadlytextual.com/2017/12/15/i-am-richard-ii-know-ye-not-that-drama-and-political-anxiety-in-shakespeares-london/ Hixon, E., Syracuse Univ] * Bate, Jonathan (2008). Soul of the Age. London: Penguin. pp. 256–286. ISBN 978-0-670-91482-1. * [https://theconversation.com/richard-ii-by-william-shakespeare-why-the-divine-right-of-kings-still-matters-186648 McFarlane, K., Univ South Australia] * [https://muse.jhu.edu/article/31090/summary Lemon, Rebecca. "The Faulty Verdict in "The Crown v. John Hayward"." SEL Studies in English Literature 1500-1900, vol. 41 no. 1, 2001, p. 109-132. Project MUSE, https://dx.doi.org/10.1353/sel.2001.0009] * [https://brill.com/display/book/9789401211666/B9789401211666-s009.xml?language=en&srsltid=AfmBOoqTUaTQr0msNWRdtEZZ9qQ3rNcVkSoBvDPXuGv7nRvHrC8t5OzH Kizelbach, U. (2014), In The Pragmatics of Early Modern Politics: Power and Kingship in Shakespeare’s History Plays. Leiden, The Netherlands: Brill. https://doi.org/10.1163/9789401211666_009] * [https://www.google.co.uk/books/edition/Richard_II/z-kOAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Lopez, R2] * [https://www.manchesterhive.com/display/9781526130532/9781526130532.00008.xml?print rgel, S. (2017). "I am Richard II". In Spectacular Performances. Manchester, England: Manchester University Press. Retrieved Apr 22, 2026, from https://doi.org/10.7765/9781526130532.00008] * [https://www.researchgate.net/publication/271041996_Was_Elizabeth_I_Richard_II_The_Authenticity_of_Lambarde%27s_%27Conversation%27 Scott-Warren, Jason. (2012). Was Elizabeth I Richard II?: The Authenticity of Lambarde's 'Conversation'. The Review of English Studies. 64. 208-230. 10.1093/res/hgs062] * Stanley Wells, “Introduction” in Richard II, The New Penguin Shakespeare, ed. Stanley Wells (London: Penguin Books, 1969), 13. * [https://www.cambridge.org/core/books/abs/shakespeare-survey/shakespeare-and-history-divergencies-and-agreements/B065A23215FD86BEE8E1CFD51DC7C1FB Ives EW. Shakespeare and History: Divergencies and Agreements. In: Wells S, ed. Shakespeare Survey. Shakespeare Survey. Cambridge University Press; 1986:19-36] * [https://www.jstor.org/stable/457855 Albright, Evelyn May. “Shakespeare’s Richard II, Hayward’s History of Henry IV, and the Essex Conspiracy.” PMLA, vol. 46, no. 3, 1931, pp. 694–719. JSTOR, https://doi.org/10.2307/457855. Accessed 22 Apr. 2026] * [https://dsc.duq.edu/cgi/viewcontent.cgi?article=2800&context=etd Morris, A. (2019). "within the hollow crown": Performing Kingship in Richard II and Henry IV Part One (Master's thesis, Duquesne University). Retrieved from https://dsc.duq.edu/etd/1771] * [https://digitalcommons.lib.uconn.edu/cgi/viewcontent.cgi?article=1122&context=srhonors_theses Scannell, Sarah J., "Shakespeare's Richard II and Henry V and Political Rebellions in the Reign of Queen Elizabeth I" (2010). Honors Scholar Theses, 138] * [https://compass.onlinelibrary.wiley.com/doi/abs/10.1111/j.1741-4113.2011.00873.x Luecking Frost, L. (2012), “A Kyng That Ruled All By Lust”: Richard II in Elizabethan Literature. Literature Compass, 9: 183-198. https://doi.org/10.1111/j.1741-4113.2011.00873.x] * [https://www.academia.edu/8347491/A_Law_Case_in_Verse_Venus_and_Adonis_and_the_Authorship_Question Stritmatter, R., A Law Case in Verse: Venus and Adonis and the Authorship Question, 2004 University of Tennessee Law Review] * [https://www.researchgate.net/publication/304647742_Prologue_I_am_Richard_II Orgel, Stephen. (2011). Prologue: I am Richard II. 10.1057/9780230307261_2] * [https://go.gale.com/ps/i.do?id=GALE%7CA314252957&sid=googleScholar&v=2.1&it=r&linkaccess=abs&issn=15256863&sw=w&p=AONE&userGroupName=anon%7Ebf79dda0&aty=open-web-entry Egan, Michael. "The Essex Rebellion and Richard II: why wasn't Shakespeare arrested?" Shakespeare Oxford Newsletter, vol. 48, no. 3, summer-fall 2012, p. 20] * [https://www.cambridge.org/core/books/abs/richard-ii/stagehistory/D62D9CA061F03B2AE8D56E856288EEDA 1. Child H, Shakespeare W. THE STAGE-HISTORY. In: Dover Wilson J, ed. Richard II: The Cambridge Dover Wilson Shakespeare. Cambridge Library Collection - Literary Studies. Cambridge University Press; 2009:lxxvii-xcii] * [https://www.openstarts.units.it/entities/publication/16d8c447-62f4-48e8-8347-5d56af674aa3/details Snelling, D., 'Prospero on the Coast of Bohemia. "Nihil sed rara et miranda amabat". Diary of Melchior Goldast upon the late Emperor Rudolph II' 1994] * [https://www.jstor.org/stable/2902265 Gajowski, Evelyn. Shakespeare in Performance: King Richard IIby Margaret Shewring. Shakespeare Quarterly, vol. 49, no. 3, 1998, pp. 328–30. JSTOR, https://doi.org/10.2307/2902265. Accessed 22 Apr. 2026] * [https://muse.jhu.edu/article/217842/summary Mosse, Ramona Franziska. Review of Richard II. Shakespeare Bulletin, vol. 25 no. 2, 2007, p. 73-77. Project MUSE, https://dx.doi.org/10.1353/shb.2007.0036] * [https://www.journals.uchicago.edu/doi/abs/10.1086/665894?journalCode=rq Petrina and Tosi. Representations of Elizabeth I in Early Modern Culture Elizabeth Pentland Renaissance Quarterly 2012 65:1, 274-276] * [https://journals.co.za/doi/10.4314/sisa.v26i1.4 The construction of a king : waste, effeminacy and queerness in Shakespeare's Richard II Higginbotham, D.] * [https://academic.oup.com/book/2363/chapter-abstract/142533948?redirectedFrom=fulltext Erskine-Hill, Howard, 'The Second Tetralogy and After', Poetry and the Realm of Politics: Shakespeare to Dryden (Oxford, 1996; online edn, Oxford Academic, 3 Oct. 2011), https://doi.org/10.1093/acprof:oso/9780198117315.003.0004] * [https://www.academia.edu/123917862/Shakespeare_s_Richard_II_in_the_minds_of_the_Elizabethan_audiences Shakespeare’s Richard II in the minds of the Elizabethan audiences] * [https://journals.sagepub.com/doi/10.1177/03064220221084555 Plays, protests and the censor’s pencil] * [https://www.indexoncensorship.org/2022/03/plays-protests-and-the-censors-pencil/ Plays, protests and the censor’s pencil A brittle Gloriana: staging the deposition of Queen Elizabeth I]. * Saul, N., Richard II * [https://www.academia.edu/3236651/_A_god_on_earth_thou_art_Richard_II_and_the_nature_of_Kingship “A god on earth thou art”: Richard II and the nature of Kingship] * [https://www.jstor.org/stable/457398 Albright, Evelyn May. “Shakespeare’s Richard II and the Essex Conspiracy.” PMLA, vol. 42, no. 3, 1927, pp. 686–720. JSTOR, https://doi.org/10.2307/457398. Accessed 22 Apr. 2026] * [https://www.berghahnjournals.com/view/journals/critical-survey/37/3/cs370301.xml Holderness, G. (2025). Introduction: Shakespeare's Wars of the Roses. Critical Survey, 37(3), 1-4. Retrieved Apr 22, 2026, from https://doi.org/10.3167/cs.2025.370301] * [https://www.folger.edu/podcasts/shakespeare-unlimited/tragedy-richard-ii-henry-iv-helen-castor/ Folger: Shakespeare Unlimited: Episode 249] * [https://www.rachaeldickzen.com/blog/2021/1/10/charlesiii The Historical and Shakespearean References in King Charles III] * Kermode, The Age of Shakespeare, pg.53. * [https://heinonline.org/HOL/login-hol?redirect_url=https%3A%2F%2Fheinonline.org%2FHOL%2Fmojo%3F%252Fmojo%3D%26auth_token%3DyhcMOu8W7PCc5OrmTqVnr2qFPpoiOdhM5uQmkQyWufU%253D%26casa_token%3D%26div%3D5%26g_sent%3D1%26paywall_url%3D%26referrer%3Dhttps%253A%252F%252Fheinonline.org%252FHOL%252FPage%253Fhandle%253Dhein.journals%252Fneastlr1%26timestamp%3D1776855600%26url%3D%252FHOL%252FPage%253Fhandle%253Dhein.journals%252Fneastlr1%2526div%253D5%2526id%253D%2526page%253D Jones, H., ''I AM RICHARD II, KNOW YE NOT THAT?' - WORKS. OF WILLIAM SHAKESPEARE AND THE ART OF. GOVERNANCE'''] * Hattaway, M., William Shakespeare: Richard II (Literature Insights, 9781847600349) * [https://books.openedition.org/psn/8305 Findlay, A., “Good sometimes queen” (V.1.37). Richard II, Mary Stuart and the Politics of Queenship, 205-219] * [https://pure.manchester.ac.uk/ws/portalfiles/portal/54508829/FULL_TEXT.PDF Pritchard, K., 'Legitimacy, Illegitimacy and Sovereignty in Shakespeare’s British Plays' (2011 University of Manchester PhD] * CHATELANAT, Marine. Histrionic Future Kings: The Politics of Metadrama in Shakespeare’s Richard III, Richard II, and Henry IV Part 1. Master, 2023. * [https://www.persee.fr/doc/xvii_0291-3798_2004_num_59_1_1993 Mayer, J-C., 'The "Parliament Sceane" in Shakespeare's King Richard II', XVII-XVIII. Revue de la Société d'études anglo-américaines des XVIIe et XVIIIe siècles Année 2004 59 pp. 27-42] * [https://www.emerald.com/md/article-abstract/38/5/315/285132/Managers-as-kings-Shakespeare-on-modern-leadership?redirectedFrom=fulltext Egan M (2000), "Managers as kings: Shakespeare on modern leadership". Management Decision, Vol. 38 No. 5 pp. 315–327, doi: https://doi.org/10.1108/00251740010340490] * [https://d-nb.info/1279081511/34 Quabeck (2022): William Shakespeare, in: Thomas Gutmann, Eberhard Ortland, Klaus Stierstorfer, eds., Encyclopedia of Law and Literature (last edited 26 October 2022), doi: 10.17879/12009577118] * [https://lawliberty.org/book-review/was-the-bard-writing-about-the-donald/ Sullivan, V., 'Was the Bard Writing about The Donald?'] * [https://deveresociety.co.uk/wp-content/uploads/2022/11/FINAL_NL_Apr2022-Haste-July-Letterresponse.pdf Haste, I., 'THE PLAY ON THE EVE OF THE ESSEX REBELLION'] * [https://www.cherylmtaylor.com/2017/11/08/a-companion-to-shakespeares-richard-ii/ A Companion to Shakespeare’s Richard II] * [https://www.city-journal.org/article/shakespeares-richards Dalrymple, T., Shakespeare’s Richards: The playwright’s two historical dramas offer contrasts between political pathologies] * [https://hfriedberg.web.wesleyan.edu/engl205/wshakespeare/plotsandrebelions.htm ELIZABETHAN PLOTS AND REBELLIONS] * [https://www.jstor.org/stable/2869708 Rackin, Phyllis. “The Role of the Audience In Shakespeare’s Richard II.” Shakespeare Quarterly, vol. 36, no. 3, 1985, pp. 262–81. JSTOR, https://doi.org/10.2307/2869708. Accessed 22 Apr. 2026.] * [https://www.psychologytoday.com/ie/blog/whats-in-name/201603/group-think-and-academia-shocking-shakespeare-shenanigans Group Think and Academia: Shocking Shakespeare Shenanigans] * [https://journals.sagepub.com/doi/10.1177/01847678221099975 Hiscopck, A., '"Why I should welcome such a guest as grief [?": Lodging and dislodging in Shakespeare's Richard II'] * [https://academic.oup.com/book/51779/chapter-abstract/420487796?redirectedFrom=fulltext 'Epigraph', Imagining Time in the English Chronicle Play: Historical Futures, 1590-1660 (Oxford, 2023; online edn, Oxford Academic, 19 Oct. 2023), https://doi.org/10.1093/oso/9780198872658.002.0009] * [https://journals.openedition.org/interfaces/319?lang=en Blake & Shakespeare] * [https://www.cambridge.org/core/books/abs/shakespeare-and-the-visual-imagination/richard-iiand-the-politics-of-perspective/23A5D4615E2A8BF9380BAF1ECE99B46D Sillars S. Richard IIand the Politics of Perspective. In: Shakespeare and the Visual Imagination. Cambridge University Press; 2015:133-162.] * [https://www.persee.fr/doc/xvii_0291-3798_2004_num_59_1_1992 Le miroir des rois: de Richard II à Henry VIII] * [https://www.cambridge.org/core/books/abs/king-richard-ll/introduction/25419329A06BC1C2E4D3DDBF331B7F91 Shakespeare W, McEachern C. Introduction. In: Gurr A, ed. King Richard Ll. The New Cambridge Shakespeare. Cambridge University Press; 2018:1-66.] * [https://www.researchgate.net/publication/372862689_A_Critical_Discourse_Study_of_Shakespeare%27s_Theological_Conceptions_in_Acts_IV_and_V_of_Richard_II_The_%27Divine_Mandate%27_of_Richard_Kingship_Falls_Apart A Critical Discourse Study of Shakespeare’s Theological Conceptions in Acts IV and V of Richard II: The ‘Divine Mandate’ of Richard Kingship Falls Apart] * [https://www.jstor.org/stable/10.1525/lal.2004.16.3.313 Richard II, Bracton, and the End of Political Theology] * [https://dokumen.pub/the-art-of-law-in-shakespeare-9781509905478-9781509905508-9781509905492.html The Art of Law in Shakespeare] * [https://www.cambridge.org/core/books/abs/cambridge-companion-to-shakespeare-and-religion/providence-and-divine-right-in-the-english-histories/DBE9BFEBF5C199F9DD72D31ED9C3B041 Mayer J-C. Providence and Divine Right in the English Histories. In: Hamlin H, ed. The Cambridge Companion to Shakespeare and Religion. Cambridge Companions to Literature. Cambridge University Press; 2019:151-167] * [https://www.researchgate.net/publication/315698000_THE_CHOMSKY-FOUCAULT_DEBATE_WHERE_DOES_THE_BARD_STAND_IN_RICHARD_II Salami, Ali & Riahi Nouri, Amir. (2016). THE CHOMSKY-FOUCAULT DEBATE: WHERE DOES THE BARD STAND IN RICHARD II?] * [https://dokumen.pub/shakespeare-richard-ii-9781350363168-9780333416709.html Richard II] * [https://www.jstor.org/stable/457907?seq=1 Heffner, Ray. “Shakespeare, Hayward, and Essex.” PMLA, vol. 45, no. 3, 1930, pp. 754–80. JSTOR, https://doi.org/10.2307/457907. Accessed 22 Apr. 2026] * [https://www.cambridge.org/core/books/abs/cambridge-companion-to-shakespeare-and-war/shakespeare-on-civil-and-dynastic-wars/00D0269CFF998876DB961FB01C22954D Shakespeare on Civil and Dynastic Wars] * [https://brill.com/display/book/edcoll/9789004356900/B9789004356900_012.xml?srsltid=AfmBOopJMjwJAQnUGME9FmT6Gwg0ih9pArbXBdTJbN6YAqbdXqbkGyuo Is a Dramatic Theory of History Possible? Shakespeare’s Richard ii and ‘Historical Truth’] * [https://www.jstor.org/stable/24475607 THE CHILD'S TWO BODIES: SHAKESPEARE, SOVEREIGNTY, AND THE END OF SUCCESSION] * [https://dokumen.pub/english-renaissance-drama-an-introduction-to-theatre-and-theatres-in-shakespeares-time-9781847600202-9781847600714.html English Renaissance Drama : An Introduction to Theatre and Theatres in Shakespeare’s Time] * [https://www.cambridge.org/core/books/abs/believing-in-shakespeare/person-plot-place/BD4EF4CBA6E7D8E2ECE5519FA1D08C16 1. McEachern C. Person, Plot, Place. In: Believing in Shakespeare: Studies in Longing. Cambridge University Press; 2018:181-296] * [https://www.enotes.com/topics/richard-2/criticism/richard-ii-vol-91/criticism-production-reviews-1000228/nigel-saul-review-date-12-may-2000 With an Eye to the Present] * [https://www.academia.edu/128303633/Shakespeare_s_Other_Eden_Richard_II_by_Ouroboros_Theatre_Company_2013_ Shakespeare’s Other Eden Richard II by Ouroboros Theatre Company (2013)] * [https://dokumen.pub/shakespeare-upstart-crow-to-sweet-swan-1592-1623-9781472555496-9781408130148-9781408139189.html Shakespeare: Upstart Crow to Sweet Swan 1592-1623] * [https://www.jstor.org/stable/41917215 RACKIN, PHYLLIS. “Temporality, Anachronism, and Presence in Shakespeare’s English Histories.” Renaissance Drama, vol. 17, 1986, pp. 101–23. JSTOR, http://www.jstor.org/stable/41917215. Accessed 22 Apr. 2026.] * [https://www.cambridge.org/core/journals/greece-and-rome/article/abs/alexander-philotas-and-the-origins-of-modern-historiography/D90568C0A1131952625E0A3B44F43EC8 Stoneman R. Alexander, Philotas, and the origins of modern historiography. Greece and Rome. 2013;60(2):296-312. doi:10.1017/S0017383513000119] * [https://www.academia.edu/35800720/Shakespeares_Richard_II_and_Elizabethan_politics Shakespeare's Richard II and Elizabethan politics] * [https://www.cambridge.org/core/books/abs/rethinking-historicism-from-shakespeare-to-milton/old-new-now/2E8CBEFFEFB11D8B843BAF2C108E9768 Introduction - Old, New, Now] * [https://www.academia.edu/24392476/Richard_II_hamartia_and_climax Richard II hamartia and climax] * [https://www.cambridge.org/core/books/abs/babylonian-jews-and-sasanian-imperialism-in-late-antiquity/kings-and-religion-in-the-talmud-and-in-the-imagination-of-sasanian-communities/E2DCC2F3E0FBA74ABE543DDD2DD7F1A7 Kings and Religion in the Talmud and in the Imagination of Sasanian Communities] * Stephen Greenblatt, “Introduction,” in The Power of Forms in the English Renaissance, ed. Stephen Greenblatt (Norman, OK: 1982), 3-6] * [https://www.academia.edu/30472558/Americas_Donald_Trump_Shakespeares_Richard_II_15_Dec_2016 America's Donald Trump, Shakespeare's Richard II] * [https://www.academia.edu/5321346/Looking_Back_Shakespeare_s_Indebtedness_to_Chaucer_and_the_Representations_of_Chivalry_in_King_Richard_II_The_Two_Noble_Kinsmen_and_the_Knight_s_Tale Looking Back: Shakespeare’s Indebtedness to Chaucer and the Representations of Chivalry in King Richard II, The Two Noble Kinsmen and the Knight’s Tale] * [https://www.academia.edu/92423912/The_Good_Usurper_in_the_eyes_of_God_and_the_people_An_analysis_of_the_role_of_the_usurper_in_Shakespeares_Richard_II_and_Henry_IV_Part_1 The Good Usurper in the eyes of God and the people: An analysis of the role of the usurper in Shakespeare's Richard II and Henry IV Part 1] * [https://www.academia.edu/91789871/The_Bacon_Shakespeare_Manuscript_Hitherto_known_as_the_Northumberland_Manuscript_which_originally_Contained_Copies_of_his_Shakespeare_Plays_Richard_II_and_Richard_III THE BACON-SHAKESPEARE MANUSCRIPT (HITHERTO KNOWN AS THE NORTHUMBERLAND MANUSCRIPT) WHICH ORIGINALLY CONTAINED COPIES OF HIS SHAKESPEARE PLAYS RICHARD II AND RICHARD III] * [https://www.google.co.uk/books/edition/Richard_II/f4gGCAAAQBAJ?hl=en&gbpv=1&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&pg=PA88&printsec=frontcover Richard II: Critical Essays] * [https://www.google.co.uk/books/edition/A_Companion_to_Shakespeare_s_Works_Volum/JlDNEAAAQBAJ?hl=en&gbpv=1&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&pg=PA400&printsec=frontcover A Companion to Shakespeare's Works, Volume II] * [https://www.google.co.uk/books/edition/Shakespeare_Reread/Z6JhDwAAQBAJ?hl=en&gbpv=1&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&pg=PA239&printsec=frontcover Shakespeare Reread] * [https://www.google.co.uk/books/edition/Shakespeare_in_the_Present/yTKWEAAAQBAJ?hl=en&gbpv=1&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&pg=PT8&printsec=frontcover Shakespeare in the Present] * [https://www.google.co.uk/books/edition/Studying_Shakespeare/1N4FBAAAQBAJ?hl=en&gbpv=1&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&pg=PA129&printsec=frontcover Studying Shakespeare A Practical Introduction] * [https://www.google.co.uk/books/edition/Critical_Essays_on_William_Faulkner/kAGBEAAAQBAJ?hl=en&gbpv=1&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&pg=PT178&printsec=frontcover Critical Essays on William Faulkner] * [https://www.google.co.uk/books/edition/Shakespeare/QY6aEQAAQBAJ?hl=en&gbpv=1&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&pg=PA216&printsec=frontcover Shakespeare: An Anthology of Criticism and Theory 1945-2000] * [https://www.google.co.uk/books/edition/The_Reign_of_Richard_II/cv0WAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Dodd, G., R2] * [https://www.google.co.uk/books/edition/Shakespeare_and_the_Institution_of_Theat/Nkt9DAAAQBAJ?hl=en&gbpv=1&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&pg=PA108&printsec=frontcover Shakespeare and the Institution of Theatre] * [https://www.google.co.uk/books/edition/The_Theory_of_the_King_s_Two_Bodies_in_t/8SRXAAAAYAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Theory of the King's Two Bodies in the Age of Shakespeare] * [https://www.google.co.uk/books/edition/Stealing_the_Story/RaJlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Stealing the Story: Shakespeare's Self-Conscious Use of the Mimetic Tradition in the Tragedies] * [https://www.google.co.uk/books/edition/The_Greenwood_Companion_to_Shakespeare_O/JkcgAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Greenwood Companion to Shakespeare: Overviews and the history plays] * [https://www.google.co.uk/books/edition/British_and_Irish_Literature_and_Its_Tim/xH4jAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover British and Irish Literature and Its Times] * [https://www.google.co.uk/books/edition/Shakespeare_s_Philosophy_of_History_Reve/1jwgAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare's Philosophy of History Revealed in a Detailed Analysis of Henry V and Examined in Other History Plays] * [https://www.google.co.uk/books/edition/Shakespeare_and_His_Contemporaries/7em7coOthxQC?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare and His Contemporaries] * [https://www.google.co.uk/books/edition/The_Politics_of_the_Public_Sphere_in_Ear/0sGHAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Politics of the Public Sphere in Early Modern England] * [https://www.google.co.uk/books/edition/Shakespeare_and_the_Catholic_Religion/Fl4gAQAAIAAJ?hl=en&gbpv=0&bsq=%22I%20am%20Richard%20II,%20know%20ye%20not%20that?%22 Shakespeare and the Catholic religion] * [https://www.google.co.uk/books/edition/Proceedings_of_the_British_Academy_Volum/8CsoAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Lectures] * [https://www.google.co.uk/books/edition/Slander_and_Censorship_in_Late_Sixteenth/wb_oAvuJgbAC?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Slander and Censorship in Late Sixteenth Century Literature] * [https://www.google.co.uk/books/edition/Elizabethan_Literature_and_the_Law_of_Fr/vBFdAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Elizabethan Literature and the Law of Fraudulent Conveyance] * [https://www.google.co.uk/books/edition/Hamlet_History_and_commentary/yxgrAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Hamlet History etc] * [https://www.google.co.uk/books/edition/William_Shakespeare_A_Popular_Life/1BuaAAAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover William Shakespeare: a popular life] * [https://www.google.co.uk/books/edition/A_State_of_Mind/pCiIAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover A State of Mind?] * [https://www.google.co.uk/books/edition/Shakespeare_s_Dramatic_Genres/J5FlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare's Dramatic Genres] * [https://www.google.co.uk/books/edition/Literature_Criticism_from_1400_to_1800/O6VkAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Literature Criticism from 1400 to 1800 Volume 89] * [https://www.google.co.uk/books/edition/Shakespeare_the_Papist/LPwNAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare the Papist] * [https://www.google.co.uk/books/edition/Goslicius_Ideal_Senator_and_His_Cultural/1jc7AQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Goslicius' Ideal Senator and His Cultural Impact Over the Centuries] * [https://www.google.co.uk/books/edition/Shakespeare_by_Another_Name/FqllAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover "Shakespeare" by Another Name] * [https://www.google.co.uk/books/edition/Shakespeare_s_Friends/AlZlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare's Friends] * [https://www.google.co.uk/books/edition/Dr_Simon_Forman/qHceAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Dr Simon Forman] * [https://www.google.co.uk/books/edition/William_Shakespeare_the_Wars_of_the_Rose/dZFlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover William Shakespeare, the Wars of the Roses and the historians] * [https://www.google.co.uk/books/edition/Shakespearean_Criticism/2TdlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespearean Criticism] * [https://www.google.co.uk/books/edition/Poetry_History_and_Antiquarianism_in_Ref/jwIhAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Poetry, History, and Antiquarianism in Reformation England] * [https://www.google.co.uk/books/edition/Shakespeare_and_the_Prince_of_Love/D0QgAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare and the Prince of Love] * [https://www.google.co.uk/books/edition/Paper_Bullets_of_the_Brain/o0YgAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Paper Bullets of the Brain] * [https://www.google.co.uk/books/edition/As_You_Like_It/GiBaAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover As You Like It: Third Series] * [https://www.google.co.uk/books/edition/William_Shakespeare/WCqaAAAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover William Shakespeare, Dominic Shellard] * [https://www.google.co.uk/books/edition/Poets_and_God/0XZlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Poets and God] * [https://www.google.co.uk/books/edition/Law_and_Literature/Ax5MAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Law and Literature Volume 16] * [https://www.google.co.uk/books/edition/The_Embodied_Word/FV8sAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Embodied Word] * [https://www.google.co.uk/books/edition/The_Case_for_Shakespeare/WaRlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Case for Shakespeare: The End of the Authorship Question] * [https://www.google.co.uk/books/edition/Explorations_in_Renaissance_Culture/_SYrAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Explorations in Renaissance Culture Volumes 33-34] * [https://www.google.co.uk/books/edition/The_Touch_of_the_Real/ewdaAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Touch of the Real: Essays in Early Modern Culture in Honour of Stephen Greenblatt] * [https://www.google.co.uk/books/edition/Wotton_and_His_Worlds/ZfgNAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Wotton and his Worlds] * [https://www.google.co.uk/books/edition/Theatre_and_Religion/wo1lAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Theatre and Religion Lancastrian Shakespeare] * [https://www.google.co.uk/books/edition/Trying_Treason/TOKxAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Trying Treason] * [https://www.google.co.uk/books/edition/Willing_Subjects/IEX0sGwT1QQC?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Willing Subjects] * [https://www.google.co.uk/books/edition/Symbolism/Bt0ZAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Symbolism] * [https://www.google.co.uk/books/edition/Performing_Shakespeare/35pQAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Performing Shakespeare] * [https://www.google.co.uk/books/edition/Soul_of_the_Age/e0UgAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Soul of the Age] * [https://www.google.co.uk/books/edition/England/aD9nAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover England] * [https://www.google.co.uk/books/edition/Elizabeth_I/-GtnAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Elizabeth I] * [https://www.google.co.uk/books/edition/King_Richard_II/oGUMX4RntjgC?hl=en&gbpv=1&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&pg=PA25&printsec=frontcover King Richard II] * [https://www.google.co.uk/books/edition/Shakespeare_and_the_Legal_Imagination/OXPvBqQLw-4C?hl=en&gbpv=1&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&pg=PA38&printsec=frontcover Shakespeare and the legal imagination] * [https://www.google.co.uk/books/edition/Shakespeare_s_Theatre/GxN3ue9_r3oC?hl=en&gbpv=1&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&pg=PA69&printsec=frontcover Shakespeare's Theatre] * [https://www.google.co.uk/books/edition/Critical_Essays_on_Shakespeare_s_Richard/AaYoAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Critical Essays on Shakespeare's Richard II] * [https://www.google.co.uk/books/edition/The_Reign_of_Richard_II_Essays_in_Honour/y3xnAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Reign of Richard II: Essays in Honour of May McKisack] * [https://www.google.co.uk/books/edition/Poetry_and_the_Realm_of_Politics/oQFaAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Poetry and the Realm of Politics] * [https://www.google.co.uk/books/edition/Shakespeare/BM0mAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare] * [https://www.google.co.uk/books/edition/Shakespearean_Politics/oTdlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespearean Politics] * [https://www.google.co.uk/books/edition/Shakespeare_the_Theatrical_Dimension/wl4gAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare, the Theatrical Dimension] * [https://www.google.co.uk/books/edition/Who_was_Kit_Marlowe/zQ1aAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Kit Marlowe etc] * [https://www.google.co.uk/books/edition/From_Page_to_Performance/beQKAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover From Page to Performance] * [https://www.google.co.uk/books/edition/Exploring_Tudor_England/ax56AAAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Exploring Tudor England] * [https://www.google.co.uk/books/edition/The_Movement_Towards_Subversion/vyJaAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Movement Towards Subversion] * [https://www.google.co.uk/books/edition/Shakespeare_s_Typological_Satire/G5BlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare's Typological Satire] * [https://www.google.co.uk/books/edition/Shakespeare_Recycled/zzNlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare Recycled] * [https://www.google.co.uk/books/edition/Reinventing_the_Middle_Ages_the_Renaissa/fXFnAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Reinventing the Middle Ages & the Renaissance] * [https://www.google.co.uk/books/edition/The_Mysterious_William_Shakespeare/WnllAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Mysterious William Shakespeare] * [https://www.google.co.uk/books/edition/Richard_II/GHhlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Richard II Critical Essays] * [https://www.google.co.uk/books/edition/William_Shakespeare/WJvC6gu_I0gC?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover William Shakespeare: Records and Images] * [https://www.google.co.uk/books/edition/Shakespeare_and_the_Actors/HYtlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare and the Actors] * [https://www.google.co.uk/books/edition/Shakespeare_the_Man/BVdlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare the man] * [https://www.google.co.uk/books/edition/Henry_V/zXllAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Henry V: A Guide to the Play] * [https://www.google.co.uk/books/edition/Shakespearean_Contingencies/Cw1NAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespearean Contingencies] * [https://www.google.co.uk/books/edition/Renaissance_Drama/E60kAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Renaissance Drama 1990] * [https://www.google.co.uk/books/edition/Language_Discourse_Sign/uH4oAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Language, Discourse, Sign] * [https://www.google.co.uk/books/edition/Mock_Kings_in_Medieval_Society_and_Renai/T98KAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Mock kings in medieval society and Renaissance drama] * [https://www.google.co.uk/books/edition/Shakespeare_Invention_of_the_Human/ojHirImrtYoC?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare: Invention of the Human] * [https://www.google.co.uk/books/edition/Shakespeare/wn5lAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare] * [https://www.google.co.uk/books/edition/Persons_in_Groups/rQ24AAAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Persons in Groups] * [https://www.google.co.uk/books/edition/All_Semblative_a_Woman_s_Part/0DlaAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover All Semblative a Woman's Part?] * [https://www.google.co.uk/books/edition/Crossing_the_Mirror/qRZNAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Crossing the Mirror] * [https://www.google.co.uk/books/edition/De_Vere_is_Shakespeare/dKJlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover De Vere is Shakespeare] * [https://www.google.co.uk/books/edition/William_Lambarde_Elizabethan_Antiquary_1/x1RnAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover William Lambarde, Elizabethan Antiquary, 1536-1601] * [https://www.google.co.uk/books/edition/The_Power_of_Forms_in_the_English_Renais/cPtZAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Power of Forms in the English Renaissance] * [https://www.google.co.uk/books/edition/Ravishment_and_Rememberance/G31LAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Ravishment and Rememberance] * [https://www.google.co.uk/books/edition/Shakespeare_and_His_Theatre/8A5ZQq3uOVQC?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare and His Theatre] * [https://www.google.co.uk/books/edition/Critical_Hermeneutics_and_Shakespeare_s/O10gAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Critical Hermeneutics and Shakespeare's History Plays] * [https://www.google.co.uk/books/edition/Christian_England/K-WfAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Christian England] * [https://www.google.co.uk/books/edition/Shakespeare_s_Religious_Background/xDSaAAAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare's Religious Background] * [https://www.google.co.uk/books/edition/Shylock/N4RlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shylock] * [https://www.google.co.uk/books/edition/The_Shakespeare_Legacy/MM5XAAAAYAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Shakespeare Legacy] * [https://www.google.co.uk/books/edition/Renaissance_Genres/0uFZAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Renaissance Genres] * [https://www.google.co.uk/books/edition/Cannibals_Witches_and_Divorce/qZRpAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Cannibals, Witches, and Divorce] * [https://www.google.co.uk/books/edition/The_Problem_of_Religious_Knowledge/C29LAAAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Problem of Religious Knowledge] * [https://www.google.co.uk/books/edition/Essex_and_the_Great_Revolt_of_1381/J8RzAAAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Essex and the Great Revolt of 1381] * [https://www.google.co.uk/books/edition/Transactions_of_the_London_and_Middlesex/4dtJAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover TLMAS] * [https://www.google.co.uk/books/edition/Shakespeare_Politics_and_the_State/Mn9lAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare, Politics and the State] * [https://www.google.co.uk/books/edition/Allegories_of_Power_in_the_England_of_El/LIYgAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Allegories of Power in the England of Elizabeth] * [https://www.google.co.uk/books/edition/William_Shakespeare/rIVlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover William Shakespeare] * [https://www.google.co.uk/books/edition/Women_s_Matters/PDRlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Women's Matters] * [https://www.google.co.uk/books/edition/The_Weak_King_Dilemma_in_the_Shakespeare/0bJlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Weak King Dilemma in the Shakespearean History Play] * [https://www.google.co.uk/books/edition/The_Book_Known_as_Q/S2tlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Book Known as Q] * [https://www.google.co.uk/books/edition/Fields_of_Vision/OD0eAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Fields of Vision] * [https://www.google.co.uk/books/edition/Ungodly_Delights/RKgcAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Ungodly Delights] * [https://www.google.co.uk/books/edition/The_Shakespeare_Handbook/rLRlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Shakespeare HandboOK] * [https://www.google.co.uk/books/edition/Humanities/y5FZAAAAYAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Humanities] * [https://www.google.co.uk/books/edition/Richard_II_by_William_Shakespeare/Bb3yAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Richard II by William Shakespeare] * [https://www.google.co.uk/books/edition/King_Richard_II/50NnAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover King Richard II] * [https://www.google.co.uk/books/edition/Murder_Under_Trust_Or_The_Topical_Macbet/0oNlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Murder under trust] * [https://www.google.co.uk/books/edition/The_Shakespearean_Kings/tHBlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Shakespearean Kings] * [https://www.google.co.uk/books/edition/America_the_Mabr_e_y_Experience/mRQ3AAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover America, the Mabr(e)y Experience: Resistance, Revolution & Civil War] * [https://www.google.co.uk/books/edition/Richard_II/ZDEkAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Richard II: An Annotated Bibliography, Volume 2] * [https://www.google.co.uk/books/edition/The_Batsford_Companion_to_Medieval_Engla/ev78b9EJQy0C?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Batsford Companion to Medieval England] * [https://www.google.co.uk/books/edition/Shakespeare_s_Unruly_Women/FKFlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare's Unruly Women] * [https://www.google.co.uk/books/edition/Shakespeare_and_Others/iFEgAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare and Others] * [https://www.google.co.uk/books/edition/Kings_and_Chroniclers/L1wpAAAAYAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Kings and Chroniclers] * [https://www.google.co.uk/books/edition/A_Kingdom_for_a_Stage/UzxlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover A Kingdom for a Stage] * [https://www.google.co.uk/books/edition/The_House_of_Commons/Ezz4OZuYVFYC?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The History of Parliament: The House of Commons 1558-1603 (3 v.)] * [https://www.google.co.uk/books/edition/Shakespeare_Soul_of_the_Age/nMYCAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover shakespeare, Soul of the Age] * [https://www.google.co.uk/books/edition/After_Poststructuralism/TOaEAAAAIAAJ?hl=en&gbpv=0&bsq=%22I%20am%20Richard%20II,%20know%20ye%20not%20that?%22 After Poststructuralism: Interdisciplinarity and Literary Theory] * [https://www.google.co.uk/books/edition/The_Unschooled_Mind/C7WnYtt219IC?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Unschooled Mind] * [https://www.google.co.uk/books/edition/Elizabeth_I/hHZnAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Eliz I] * [https://www.google.co.uk/books/edition/Dramas_of_Christian_Time/mnIqAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Dramas of Christian Time] * [https://www.google.co.uk/books/edition/Elizabeth_I/XjQmAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Elizabeth I: The Shrewdness of Virtue] * [https://www.google.co.uk/books/edition/John_Dryden/9Q1aAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover John Dryden] * [https://www.google.co.uk/books/edition/Shakespeare_and_Early_Modern_Political_T/DUwhAwAAQBAJ?hl=en&gbpv=1&dq=%22shakespeare%22+%2B+%22political+propaganda%22&pg=PA259&printsec=frontcover Shakespeare and Early Modern Political Thought] * [https://www.google.co.uk/books/edition/The_English_History_Play_in_the_age_of_S/5TT-AQAAQBAJ?hl=en&gbpv=1&dq=%22shakespeare%22+%2B+%22political+propaganda%22&pg=PA158&printsec=frontcover The English History Play in the Age of Shakespeare] * [https://www.google.co.uk/books/edition/Shakespeare_and_the_Political/rEcREQAAQBAJ?hl=en&gbpv=1&dq=%22shakespeare%22+%2B+%22political+propaganda%22&pg=PA215&printsec=frontcover Shakespeare and the Political] * [https://www.google.co.uk/books/edition/William_Shakespeare_Subject_of_the_Crown/a7G6DAAAQBAJ?hl=en&gbpv=1&dq=%22shakespeare%22+%2B+%22political+propaganda%22&pg=PT18&printsec=frontcover William Shakespeare - Subject of the Crown?] * [https://www.google.com/search?q=%22shakespeare%22+%2B+%22political+propaganda%22&client=firefox-b-d&hs=4AQ&sca_esv=6d4ade7bd26771c9&udm=36&biw=2510&bih=1307&tbs=cdr%3A1%2Ccd_min%3A2000%2Ccd_max%3A2099&sxsrf=ANbL-n6I6Pkwl7mmdHK6N1xPQXLbGBIOSg%3A1776853062010&ei=RqDoaZUvztiFsg_I5bToDw&ved=0ahUKEwiV6tC8nYGUAxVObEEAHcgyDf0Q4dUDCBM&uact=5&oq=%22shakespeare%22+%2B+%22political+propaganda%22&gs_lp=EhBnd3Mtd2l6LW1vZGVsZXNzIiYic2hha2VzcGVhcmUiICsgInBvbGl0aWNhbCBwcm9wYWdhbmRhIjIIECEYoAEYwwRInQlQxgZYuwdwAXgAkAEAmAF_oAHPAaoBAzEuMbgBA8gBAPgBAZgCAqACVsICCxAAGIAEGKIEGLADmAMAiAYBkAYCkgcBMqAHowOyBwExuAdTwgcDMC4yyAcEgAgB&sclient=gws-wiz-modeless The Nazi Appropriation of Shakespeare: Cultural Politics in] {{refend}} svb4k3sppvsyu4tll9xbclu2r69wrq5 2818528 2818527 2026-07-19T11:57:18Z Fortuna imperatrix mundi 1058581 /* Essex's rebellion */ sfn 2818528 wikitext text/x-wiki ==Background== ===Richard II=== ===Elizabeth I=== ====Essex's rebellion==== Seven months before Lambarde's discussion with Elizabeth.{{sfn|Orgel|2011|pp=11–43}} ==IaR2== Lambarde's account of his interview with the queen in August 1601 is well known.{{sfn|Orgel|2011|pp=11–43}} ===Political anxiety and tension=== ===Censorship=== ==Consequences and aftermath== == Notes == {{reflist|group=note}} ==References== {{Reflist|20em}} ==Bibliography== {{refbegin|30em|indent=yes}} * {{Cite book|url=https://doi.org/10.1057/9780230307261_2|title=Prologue: I am Richard II|last=Orgel|first=Stephen|date=2011|publisher=Palgrave Macmillan UK|isbn=978-0-230-30726-1|editor-last=Petrina|editor-first=Alessandra|location=London|pages=11–43|language=en|doi=10.1057/9780230307261_2|editor-last2=Tosi|editor-first2=Laura}} * [https://broadlytextual.com/2017/12/15/i-am-richard-ii-know-ye-not-that-drama-and-political-anxiety-in-shakespeares-london/ Hixon, E., Syracuse Univ] * Bate, Jonathan (2008). Soul of the Age. London: Penguin. pp. 256–286. ISBN 978-0-670-91482-1. * [https://theconversation.com/richard-ii-by-william-shakespeare-why-the-divine-right-of-kings-still-matters-186648 McFarlane, K., Univ South Australia] * [https://muse.jhu.edu/article/31090/summary Lemon, Rebecca. "The Faulty Verdict in "The Crown v. John Hayward"." SEL Studies in English Literature 1500-1900, vol. 41 no. 1, 2001, p. 109-132. Project MUSE, https://dx.doi.org/10.1353/sel.2001.0009] * [https://brill.com/display/book/9789401211666/B9789401211666-s009.xml?language=en&srsltid=AfmBOoqTUaTQr0msNWRdtEZZ9qQ3rNcVkSoBvDPXuGv7nRvHrC8t5OzH Kizelbach, U. (2014), In The Pragmatics of Early Modern Politics: Power and Kingship in Shakespeare’s History Plays. Leiden, The Netherlands: Brill. https://doi.org/10.1163/9789401211666_009] * [https://www.google.co.uk/books/edition/Richard_II/z-kOAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Lopez, R2] * [https://www.manchesterhive.com/display/9781526130532/9781526130532.00008.xml?print rgel, S. (2017). "I am Richard II". In Spectacular Performances. Manchester, England: Manchester University Press. Retrieved Apr 22, 2026, from https://doi.org/10.7765/9781526130532.00008] * [https://www.researchgate.net/publication/271041996_Was_Elizabeth_I_Richard_II_The_Authenticity_of_Lambarde%27s_%27Conversation%27 Scott-Warren, Jason. (2012). Was Elizabeth I Richard II?: The Authenticity of Lambarde's 'Conversation'. The Review of English Studies. 64. 208-230. 10.1093/res/hgs062] * Stanley Wells, “Introduction” in Richard II, The New Penguin Shakespeare, ed. Stanley Wells (London: Penguin Books, 1969), 13. * [https://www.cambridge.org/core/books/abs/shakespeare-survey/shakespeare-and-history-divergencies-and-agreements/B065A23215FD86BEE8E1CFD51DC7C1FB Ives EW. Shakespeare and History: Divergencies and Agreements. In: Wells S, ed. Shakespeare Survey. Shakespeare Survey. Cambridge University Press; 1986:19-36] * [https://www.jstor.org/stable/457855 Albright, Evelyn May. “Shakespeare’s Richard II, Hayward’s History of Henry IV, and the Essex Conspiracy.” PMLA, vol. 46, no. 3, 1931, pp. 694–719. JSTOR, https://doi.org/10.2307/457855. Accessed 22 Apr. 2026] * [https://dsc.duq.edu/cgi/viewcontent.cgi?article=2800&context=etd Morris, A. (2019). "within the hollow crown": Performing Kingship in Richard II and Henry IV Part One (Master's thesis, Duquesne University). Retrieved from https://dsc.duq.edu/etd/1771] * [https://digitalcommons.lib.uconn.edu/cgi/viewcontent.cgi?article=1122&context=srhonors_theses Scannell, Sarah J., "Shakespeare's Richard II and Henry V and Political Rebellions in the Reign of Queen Elizabeth I" (2010). Honors Scholar Theses, 138] * [https://compass.onlinelibrary.wiley.com/doi/abs/10.1111/j.1741-4113.2011.00873.x Luecking Frost, L. (2012), “A Kyng That Ruled All By Lust”: Richard II in Elizabethan Literature. Literature Compass, 9: 183-198. https://doi.org/10.1111/j.1741-4113.2011.00873.x] * [https://www.academia.edu/8347491/A_Law_Case_in_Verse_Venus_and_Adonis_and_the_Authorship_Question Stritmatter, R., A Law Case in Verse: Venus and Adonis and the Authorship Question, 2004 University of Tennessee Law Review] * [https://www.researchgate.net/publication/304647742_Prologue_I_am_Richard_II Orgel, Stephen. (2011). Prologue: I am Richard II. 10.1057/9780230307261_2] * [https://go.gale.com/ps/i.do?id=GALE%7CA314252957&sid=googleScholar&v=2.1&it=r&linkaccess=abs&issn=15256863&sw=w&p=AONE&userGroupName=anon%7Ebf79dda0&aty=open-web-entry Egan, Michael. "The Essex Rebellion and Richard II: why wasn't Shakespeare arrested?" Shakespeare Oxford Newsletter, vol. 48, no. 3, summer-fall 2012, p. 20] * [https://www.cambridge.org/core/books/abs/richard-ii/stagehistory/D62D9CA061F03B2AE8D56E856288EEDA 1. Child H, Shakespeare W. THE STAGE-HISTORY. In: Dover Wilson J, ed. Richard II: The Cambridge Dover Wilson Shakespeare. Cambridge Library Collection - Literary Studies. Cambridge University Press; 2009:lxxvii-xcii] * [https://www.openstarts.units.it/entities/publication/16d8c447-62f4-48e8-8347-5d56af674aa3/details Snelling, D., 'Prospero on the Coast of Bohemia. "Nihil sed rara et miranda amabat". Diary of Melchior Goldast upon the late Emperor Rudolph II' 1994] * [https://www.jstor.org/stable/2902265 Gajowski, Evelyn. Shakespeare in Performance: King Richard IIby Margaret Shewring. Shakespeare Quarterly, vol. 49, no. 3, 1998, pp. 328–30. JSTOR, https://doi.org/10.2307/2902265. Accessed 22 Apr. 2026] * [https://muse.jhu.edu/article/217842/summary Mosse, Ramona Franziska. Review of Richard II. Shakespeare Bulletin, vol. 25 no. 2, 2007, p. 73-77. Project MUSE, https://dx.doi.org/10.1353/shb.2007.0036] * [https://www.journals.uchicago.edu/doi/abs/10.1086/665894?journalCode=rq Petrina and Tosi. Representations of Elizabeth I in Early Modern Culture Elizabeth Pentland Renaissance Quarterly 2012 65:1, 274-276] * [https://journals.co.za/doi/10.4314/sisa.v26i1.4 The construction of a king : waste, effeminacy and queerness in Shakespeare's Richard II Higginbotham, D.] * [https://academic.oup.com/book/2363/chapter-abstract/142533948?redirectedFrom=fulltext Erskine-Hill, Howard, 'The Second Tetralogy and After', Poetry and the Realm of Politics: Shakespeare to Dryden (Oxford, 1996; online edn, Oxford Academic, 3 Oct. 2011), https://doi.org/10.1093/acprof:oso/9780198117315.003.0004] * [https://www.academia.edu/123917862/Shakespeare_s_Richard_II_in_the_minds_of_the_Elizabethan_audiences Shakespeare’s Richard II in the minds of the Elizabethan audiences] * [https://journals.sagepub.com/doi/10.1177/03064220221084555 Plays, protests and the censor’s pencil] * [https://www.indexoncensorship.org/2022/03/plays-protests-and-the-censors-pencil/ Plays, protests and the censor’s pencil A brittle Gloriana: staging the deposition of Queen Elizabeth I]. * Saul, N., Richard II * [https://www.academia.edu/3236651/_A_god_on_earth_thou_art_Richard_II_and_the_nature_of_Kingship “A god on earth thou art”: Richard II and the nature of Kingship] * [https://www.jstor.org/stable/457398 Albright, Evelyn May. “Shakespeare’s Richard II and the Essex Conspiracy.” PMLA, vol. 42, no. 3, 1927, pp. 686–720. JSTOR, https://doi.org/10.2307/457398. Accessed 22 Apr. 2026] * [https://www.berghahnjournals.com/view/journals/critical-survey/37/3/cs370301.xml Holderness, G. (2025). Introduction: Shakespeare's Wars of the Roses. Critical Survey, 37(3), 1-4. Retrieved Apr 22, 2026, from https://doi.org/10.3167/cs.2025.370301] * [https://www.folger.edu/podcasts/shakespeare-unlimited/tragedy-richard-ii-henry-iv-helen-castor/ Folger: Shakespeare Unlimited: Episode 249] * [https://www.rachaeldickzen.com/blog/2021/1/10/charlesiii The Historical and Shakespearean References in King Charles III] * Kermode, The Age of Shakespeare, pg.53. * [https://heinonline.org/HOL/login-hol?redirect_url=https%3A%2F%2Fheinonline.org%2FHOL%2Fmojo%3F%252Fmojo%3D%26auth_token%3DyhcMOu8W7PCc5OrmTqVnr2qFPpoiOdhM5uQmkQyWufU%253D%26casa_token%3D%26div%3D5%26g_sent%3D1%26paywall_url%3D%26referrer%3Dhttps%253A%252F%252Fheinonline.org%252FHOL%252FPage%253Fhandle%253Dhein.journals%252Fneastlr1%26timestamp%3D1776855600%26url%3D%252FHOL%252FPage%253Fhandle%253Dhein.journals%252Fneastlr1%2526div%253D5%2526id%253D%2526page%253D Jones, H., ''I AM RICHARD II, KNOW YE NOT THAT?' - WORKS. OF WILLIAM SHAKESPEARE AND THE ART OF. GOVERNANCE'''] * Hattaway, M., William Shakespeare: Richard II (Literature Insights, 9781847600349) * [https://books.openedition.org/psn/8305 Findlay, A., “Good sometimes queen” (V.1.37). Richard II, Mary Stuart and the Politics of Queenship, 205-219] * [https://pure.manchester.ac.uk/ws/portalfiles/portal/54508829/FULL_TEXT.PDF Pritchard, K., 'Legitimacy, Illegitimacy and Sovereignty in Shakespeare’s British Plays' (2011 University of Manchester PhD] * CHATELANAT, Marine. Histrionic Future Kings: The Politics of Metadrama in Shakespeare’s Richard III, Richard II, and Henry IV Part 1. Master, 2023. * [https://www.persee.fr/doc/xvii_0291-3798_2004_num_59_1_1993 Mayer, J-C., 'The "Parliament Sceane" in Shakespeare's King Richard II', XVII-XVIII. Revue de la Société d'études anglo-américaines des XVIIe et XVIIIe siècles Année 2004 59 pp. 27-42] * [https://www.emerald.com/md/article-abstract/38/5/315/285132/Managers-as-kings-Shakespeare-on-modern-leadership?redirectedFrom=fulltext Egan M (2000), "Managers as kings: Shakespeare on modern leadership". Management Decision, Vol. 38 No. 5 pp. 315–327, doi: https://doi.org/10.1108/00251740010340490] * [https://d-nb.info/1279081511/34 Quabeck (2022): William Shakespeare, in: Thomas Gutmann, Eberhard Ortland, Klaus Stierstorfer, eds., Encyclopedia of Law and Literature (last edited 26 October 2022), doi: 10.17879/12009577118] * [https://lawliberty.org/book-review/was-the-bard-writing-about-the-donald/ Sullivan, V., 'Was the Bard Writing about The Donald?'] * [https://deveresociety.co.uk/wp-content/uploads/2022/11/FINAL_NL_Apr2022-Haste-July-Letterresponse.pdf Haste, I., 'THE PLAY ON THE EVE OF THE ESSEX REBELLION'] * [https://www.cherylmtaylor.com/2017/11/08/a-companion-to-shakespeares-richard-ii/ A Companion to Shakespeare’s Richard II] * [https://www.city-journal.org/article/shakespeares-richards Dalrymple, T., Shakespeare’s Richards: The playwright’s two historical dramas offer contrasts between political pathologies] * [https://hfriedberg.web.wesleyan.edu/engl205/wshakespeare/plotsandrebelions.htm ELIZABETHAN PLOTS AND REBELLIONS] * [https://www.jstor.org/stable/2869708 Rackin, Phyllis. “The Role of the Audience In Shakespeare’s Richard II.” Shakespeare Quarterly, vol. 36, no. 3, 1985, pp. 262–81. JSTOR, https://doi.org/10.2307/2869708. Accessed 22 Apr. 2026.] * [https://www.psychologytoday.com/ie/blog/whats-in-name/201603/group-think-and-academia-shocking-shakespeare-shenanigans Group Think and Academia: Shocking Shakespeare Shenanigans] * [https://journals.sagepub.com/doi/10.1177/01847678221099975 Hiscopck, A., '"Why I should welcome such a guest as grief [?": Lodging and dislodging in Shakespeare's Richard II'] * [https://academic.oup.com/book/51779/chapter-abstract/420487796?redirectedFrom=fulltext 'Epigraph', Imagining Time in the English Chronicle Play: Historical Futures, 1590-1660 (Oxford, 2023; online edn, Oxford Academic, 19 Oct. 2023), https://doi.org/10.1093/oso/9780198872658.002.0009] * [https://journals.openedition.org/interfaces/319?lang=en Blake & Shakespeare] * [https://www.cambridge.org/core/books/abs/shakespeare-and-the-visual-imagination/richard-iiand-the-politics-of-perspective/23A5D4615E2A8BF9380BAF1ECE99B46D Sillars S. Richard IIand the Politics of Perspective. In: Shakespeare and the Visual Imagination. Cambridge University Press; 2015:133-162.] * [https://www.persee.fr/doc/xvii_0291-3798_2004_num_59_1_1992 Le miroir des rois: de Richard II à Henry VIII] * [https://www.cambridge.org/core/books/abs/king-richard-ll/introduction/25419329A06BC1C2E4D3DDBF331B7F91 Shakespeare W, McEachern C. Introduction. In: Gurr A, ed. King Richard Ll. The New Cambridge Shakespeare. Cambridge University Press; 2018:1-66.] * [https://www.researchgate.net/publication/372862689_A_Critical_Discourse_Study_of_Shakespeare%27s_Theological_Conceptions_in_Acts_IV_and_V_of_Richard_II_The_%27Divine_Mandate%27_of_Richard_Kingship_Falls_Apart A Critical Discourse Study of Shakespeare’s Theological Conceptions in Acts IV and V of Richard II: The ‘Divine Mandate’ of Richard Kingship Falls Apart] * [https://www.jstor.org/stable/10.1525/lal.2004.16.3.313 Richard II, Bracton, and the End of Political Theology] * [https://dokumen.pub/the-art-of-law-in-shakespeare-9781509905478-9781509905508-9781509905492.html The Art of Law in Shakespeare] * [https://www.cambridge.org/core/books/abs/cambridge-companion-to-shakespeare-and-religion/providence-and-divine-right-in-the-english-histories/DBE9BFEBF5C199F9DD72D31ED9C3B041 Mayer J-C. Providence and Divine Right in the English Histories. In: Hamlin H, ed. The Cambridge Companion to Shakespeare and Religion. Cambridge Companions to Literature. Cambridge University Press; 2019:151-167] * [https://www.researchgate.net/publication/315698000_THE_CHOMSKY-FOUCAULT_DEBATE_WHERE_DOES_THE_BARD_STAND_IN_RICHARD_II Salami, Ali & Riahi Nouri, Amir. (2016). THE CHOMSKY-FOUCAULT DEBATE: WHERE DOES THE BARD STAND IN RICHARD II?] * [https://dokumen.pub/shakespeare-richard-ii-9781350363168-9780333416709.html Richard II] * [https://www.jstor.org/stable/457907?seq=1 Heffner, Ray. “Shakespeare, Hayward, and Essex.” PMLA, vol. 45, no. 3, 1930, pp. 754–80. JSTOR, https://doi.org/10.2307/457907. Accessed 22 Apr. 2026] * [https://www.cambridge.org/core/books/abs/cambridge-companion-to-shakespeare-and-war/shakespeare-on-civil-and-dynastic-wars/00D0269CFF998876DB961FB01C22954D Shakespeare on Civil and Dynastic Wars] * [https://brill.com/display/book/edcoll/9789004356900/B9789004356900_012.xml?srsltid=AfmBOopJMjwJAQnUGME9FmT6Gwg0ih9pArbXBdTJbN6YAqbdXqbkGyuo Is a Dramatic Theory of History Possible? Shakespeare’s Richard ii and ‘Historical Truth’] * [https://www.jstor.org/stable/24475607 THE CHILD'S TWO BODIES: SHAKESPEARE, SOVEREIGNTY, AND THE END OF SUCCESSION] * [https://dokumen.pub/english-renaissance-drama-an-introduction-to-theatre-and-theatres-in-shakespeares-time-9781847600202-9781847600714.html English Renaissance Drama : An Introduction to Theatre and Theatres in Shakespeare’s Time] * [https://www.cambridge.org/core/books/abs/believing-in-shakespeare/person-plot-place/BD4EF4CBA6E7D8E2ECE5519FA1D08C16 1. McEachern C. Person, Plot, Place. In: Believing in Shakespeare: Studies in Longing. Cambridge University Press; 2018:181-296] * [https://www.enotes.com/topics/richard-2/criticism/richard-ii-vol-91/criticism-production-reviews-1000228/nigel-saul-review-date-12-may-2000 With an Eye to the Present] * [https://www.academia.edu/128303633/Shakespeare_s_Other_Eden_Richard_II_by_Ouroboros_Theatre_Company_2013_ Shakespeare’s Other Eden Richard II by Ouroboros Theatre Company (2013)] * [https://dokumen.pub/shakespeare-upstart-crow-to-sweet-swan-1592-1623-9781472555496-9781408130148-9781408139189.html Shakespeare: Upstart Crow to Sweet Swan 1592-1623] * [https://www.jstor.org/stable/41917215 RACKIN, PHYLLIS. “Temporality, Anachronism, and Presence in Shakespeare’s English Histories.” Renaissance Drama, vol. 17, 1986, pp. 101–23. JSTOR, http://www.jstor.org/stable/41917215. Accessed 22 Apr. 2026.] * [https://www.cambridge.org/core/journals/greece-and-rome/article/abs/alexander-philotas-and-the-origins-of-modern-historiography/D90568C0A1131952625E0A3B44F43EC8 Stoneman R. Alexander, Philotas, and the origins of modern historiography. Greece and Rome. 2013;60(2):296-312. doi:10.1017/S0017383513000119] * [https://www.academia.edu/35800720/Shakespeares_Richard_II_and_Elizabethan_politics Shakespeare's Richard II and Elizabethan politics] * [https://www.cambridge.org/core/books/abs/rethinking-historicism-from-shakespeare-to-milton/old-new-now/2E8CBEFFEFB11D8B843BAF2C108E9768 Introduction - Old, New, Now] * [https://www.academia.edu/24392476/Richard_II_hamartia_and_climax Richard II hamartia and climax] * [https://www.cambridge.org/core/books/abs/babylonian-jews-and-sasanian-imperialism-in-late-antiquity/kings-and-religion-in-the-talmud-and-in-the-imagination-of-sasanian-communities/E2DCC2F3E0FBA74ABE543DDD2DD7F1A7 Kings and Religion in the Talmud and in the Imagination of Sasanian Communities] * Stephen Greenblatt, “Introduction,” in The Power of Forms in the English Renaissance, ed. Stephen Greenblatt (Norman, OK: 1982), 3-6] * [https://www.academia.edu/30472558/Americas_Donald_Trump_Shakespeares_Richard_II_15_Dec_2016 America's Donald Trump, Shakespeare's Richard II] * [https://www.academia.edu/5321346/Looking_Back_Shakespeare_s_Indebtedness_to_Chaucer_and_the_Representations_of_Chivalry_in_King_Richard_II_The_Two_Noble_Kinsmen_and_the_Knight_s_Tale Looking Back: Shakespeare’s Indebtedness to Chaucer and the Representations of Chivalry in King Richard II, The Two Noble Kinsmen and the Knight’s Tale] * [https://www.academia.edu/92423912/The_Good_Usurper_in_the_eyes_of_God_and_the_people_An_analysis_of_the_role_of_the_usurper_in_Shakespeares_Richard_II_and_Henry_IV_Part_1 The Good Usurper in the eyes of God and the people: An analysis of the role of the usurper in Shakespeare's Richard II and Henry IV Part 1] * [https://www.academia.edu/91789871/The_Bacon_Shakespeare_Manuscript_Hitherto_known_as_the_Northumberland_Manuscript_which_originally_Contained_Copies_of_his_Shakespeare_Plays_Richard_II_and_Richard_III THE BACON-SHAKESPEARE MANUSCRIPT (HITHERTO KNOWN AS THE NORTHUMBERLAND MANUSCRIPT) WHICH ORIGINALLY CONTAINED COPIES OF HIS SHAKESPEARE PLAYS RICHARD II AND RICHARD III] * [https://www.google.co.uk/books/edition/Richard_II/f4gGCAAAQBAJ?hl=en&gbpv=1&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&pg=PA88&printsec=frontcover Richard II: Critical Essays] * [https://www.google.co.uk/books/edition/A_Companion_to_Shakespeare_s_Works_Volum/JlDNEAAAQBAJ?hl=en&gbpv=1&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&pg=PA400&printsec=frontcover A Companion to Shakespeare's Works, Volume II] * [https://www.google.co.uk/books/edition/Shakespeare_Reread/Z6JhDwAAQBAJ?hl=en&gbpv=1&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&pg=PA239&printsec=frontcover Shakespeare Reread] * [https://www.google.co.uk/books/edition/Shakespeare_in_the_Present/yTKWEAAAQBAJ?hl=en&gbpv=1&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&pg=PT8&printsec=frontcover Shakespeare in the Present] * [https://www.google.co.uk/books/edition/Studying_Shakespeare/1N4FBAAAQBAJ?hl=en&gbpv=1&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&pg=PA129&printsec=frontcover Studying Shakespeare A Practical Introduction] * [https://www.google.co.uk/books/edition/Critical_Essays_on_William_Faulkner/kAGBEAAAQBAJ?hl=en&gbpv=1&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&pg=PT178&printsec=frontcover Critical Essays on William Faulkner] * [https://www.google.co.uk/books/edition/Shakespeare/QY6aEQAAQBAJ?hl=en&gbpv=1&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&pg=PA216&printsec=frontcover Shakespeare: An Anthology of Criticism and Theory 1945-2000] * [https://www.google.co.uk/books/edition/The_Reign_of_Richard_II/cv0WAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Dodd, G., R2] * [https://www.google.co.uk/books/edition/Shakespeare_and_the_Institution_of_Theat/Nkt9DAAAQBAJ?hl=en&gbpv=1&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&pg=PA108&printsec=frontcover Shakespeare and the Institution of Theatre] * [https://www.google.co.uk/books/edition/The_Theory_of_the_King_s_Two_Bodies_in_t/8SRXAAAAYAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Theory of the King's Two Bodies in the Age of Shakespeare] * [https://www.google.co.uk/books/edition/Stealing_the_Story/RaJlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Stealing the Story: Shakespeare's Self-Conscious Use of the Mimetic Tradition in the Tragedies] * [https://www.google.co.uk/books/edition/The_Greenwood_Companion_to_Shakespeare_O/JkcgAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Greenwood Companion to Shakespeare: Overviews and the history plays] * [https://www.google.co.uk/books/edition/British_and_Irish_Literature_and_Its_Tim/xH4jAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover British and Irish Literature and Its Times] * [https://www.google.co.uk/books/edition/Shakespeare_s_Philosophy_of_History_Reve/1jwgAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare's Philosophy of History Revealed in a Detailed Analysis of Henry V and Examined in Other History Plays] * [https://www.google.co.uk/books/edition/Shakespeare_and_His_Contemporaries/7em7coOthxQC?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare and His Contemporaries] * [https://www.google.co.uk/books/edition/The_Politics_of_the_Public_Sphere_in_Ear/0sGHAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Politics of the Public Sphere in Early Modern England] * [https://www.google.co.uk/books/edition/Shakespeare_and_the_Catholic_Religion/Fl4gAQAAIAAJ?hl=en&gbpv=0&bsq=%22I%20am%20Richard%20II,%20know%20ye%20not%20that?%22 Shakespeare and the Catholic religion] * [https://www.google.co.uk/books/edition/Proceedings_of_the_British_Academy_Volum/8CsoAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Lectures] * [https://www.google.co.uk/books/edition/Slander_and_Censorship_in_Late_Sixteenth/wb_oAvuJgbAC?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Slander and Censorship in Late Sixteenth Century Literature] * [https://www.google.co.uk/books/edition/Elizabethan_Literature_and_the_Law_of_Fr/vBFdAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Elizabethan Literature and the Law of Fraudulent Conveyance] * [https://www.google.co.uk/books/edition/Hamlet_History_and_commentary/yxgrAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Hamlet History etc] * [https://www.google.co.uk/books/edition/William_Shakespeare_A_Popular_Life/1BuaAAAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover William Shakespeare: a popular life] * [https://www.google.co.uk/books/edition/A_State_of_Mind/pCiIAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover A State of Mind?] * [https://www.google.co.uk/books/edition/Shakespeare_s_Dramatic_Genres/J5FlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare's Dramatic Genres] * [https://www.google.co.uk/books/edition/Literature_Criticism_from_1400_to_1800/O6VkAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Literature Criticism from 1400 to 1800 Volume 89] * [https://www.google.co.uk/books/edition/Shakespeare_the_Papist/LPwNAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare the Papist] * [https://www.google.co.uk/books/edition/Goslicius_Ideal_Senator_and_His_Cultural/1jc7AQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Goslicius' Ideal Senator and His Cultural Impact Over the Centuries] * [https://www.google.co.uk/books/edition/Shakespeare_by_Another_Name/FqllAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover "Shakespeare" by Another Name] * [https://www.google.co.uk/books/edition/Shakespeare_s_Friends/AlZlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare's Friends] * [https://www.google.co.uk/books/edition/Dr_Simon_Forman/qHceAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Dr Simon Forman] * [https://www.google.co.uk/books/edition/William_Shakespeare_the_Wars_of_the_Rose/dZFlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover William Shakespeare, the Wars of the Roses and the historians] * [https://www.google.co.uk/books/edition/Shakespearean_Criticism/2TdlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespearean Criticism] * [https://www.google.co.uk/books/edition/Poetry_History_and_Antiquarianism_in_Ref/jwIhAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Poetry, History, and Antiquarianism in Reformation England] * [https://www.google.co.uk/books/edition/Shakespeare_and_the_Prince_of_Love/D0QgAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare and the Prince of Love] * [https://www.google.co.uk/books/edition/Paper_Bullets_of_the_Brain/o0YgAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Paper Bullets of the Brain] * [https://www.google.co.uk/books/edition/As_You_Like_It/GiBaAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover As You Like It: Third Series] * [https://www.google.co.uk/books/edition/William_Shakespeare/WCqaAAAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover William Shakespeare, Dominic Shellard] * [https://www.google.co.uk/books/edition/Poets_and_God/0XZlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Poets and God] * [https://www.google.co.uk/books/edition/Law_and_Literature/Ax5MAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Law and Literature Volume 16] * [https://www.google.co.uk/books/edition/The_Embodied_Word/FV8sAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Embodied Word] * [https://www.google.co.uk/books/edition/The_Case_for_Shakespeare/WaRlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Case for Shakespeare: The End of the Authorship Question] * [https://www.google.co.uk/books/edition/Explorations_in_Renaissance_Culture/_SYrAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Explorations in Renaissance Culture Volumes 33-34] * [https://www.google.co.uk/books/edition/The_Touch_of_the_Real/ewdaAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Touch of the Real: Essays in Early Modern Culture in Honour of Stephen Greenblatt] * [https://www.google.co.uk/books/edition/Wotton_and_His_Worlds/ZfgNAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Wotton and his Worlds] * [https://www.google.co.uk/books/edition/Theatre_and_Religion/wo1lAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Theatre and Religion Lancastrian Shakespeare] * [https://www.google.co.uk/books/edition/Trying_Treason/TOKxAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Trying Treason] * [https://www.google.co.uk/books/edition/Willing_Subjects/IEX0sGwT1QQC?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Willing Subjects] * [https://www.google.co.uk/books/edition/Symbolism/Bt0ZAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Symbolism] * [https://www.google.co.uk/books/edition/Performing_Shakespeare/35pQAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Performing Shakespeare] * [https://www.google.co.uk/books/edition/Soul_of_the_Age/e0UgAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Soul of the Age] * [https://www.google.co.uk/books/edition/England/aD9nAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover England] * [https://www.google.co.uk/books/edition/Elizabeth_I/-GtnAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Elizabeth I] * [https://www.google.co.uk/books/edition/King_Richard_II/oGUMX4RntjgC?hl=en&gbpv=1&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&pg=PA25&printsec=frontcover King Richard II] * [https://www.google.co.uk/books/edition/Shakespeare_and_the_Legal_Imagination/OXPvBqQLw-4C?hl=en&gbpv=1&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&pg=PA38&printsec=frontcover Shakespeare and the legal imagination] * [https://www.google.co.uk/books/edition/Shakespeare_s_Theatre/GxN3ue9_r3oC?hl=en&gbpv=1&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&pg=PA69&printsec=frontcover Shakespeare's Theatre] * [https://www.google.co.uk/books/edition/Critical_Essays_on_Shakespeare_s_Richard/AaYoAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Critical Essays on Shakespeare's Richard II] * [https://www.google.co.uk/books/edition/The_Reign_of_Richard_II_Essays_in_Honour/y3xnAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Reign of Richard II: Essays in Honour of May McKisack] * [https://www.google.co.uk/books/edition/Poetry_and_the_Realm_of_Politics/oQFaAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Poetry and the Realm of Politics] * [https://www.google.co.uk/books/edition/Shakespeare/BM0mAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare] * [https://www.google.co.uk/books/edition/Shakespearean_Politics/oTdlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespearean Politics] * [https://www.google.co.uk/books/edition/Shakespeare_the_Theatrical_Dimension/wl4gAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare, the Theatrical Dimension] * [https://www.google.co.uk/books/edition/Who_was_Kit_Marlowe/zQ1aAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Kit Marlowe etc] * [https://www.google.co.uk/books/edition/From_Page_to_Performance/beQKAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover From Page to Performance] * [https://www.google.co.uk/books/edition/Exploring_Tudor_England/ax56AAAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Exploring Tudor England] * [https://www.google.co.uk/books/edition/The_Movement_Towards_Subversion/vyJaAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Movement Towards Subversion] * [https://www.google.co.uk/books/edition/Shakespeare_s_Typological_Satire/G5BlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare's Typological Satire] * [https://www.google.co.uk/books/edition/Shakespeare_Recycled/zzNlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare Recycled] * [https://www.google.co.uk/books/edition/Reinventing_the_Middle_Ages_the_Renaissa/fXFnAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Reinventing the Middle Ages & the Renaissance] * [https://www.google.co.uk/books/edition/The_Mysterious_William_Shakespeare/WnllAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Mysterious William Shakespeare] * [https://www.google.co.uk/books/edition/Richard_II/GHhlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Richard II Critical Essays] * [https://www.google.co.uk/books/edition/William_Shakespeare/WJvC6gu_I0gC?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover William Shakespeare: Records and Images] * [https://www.google.co.uk/books/edition/Shakespeare_and_the_Actors/HYtlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare and the Actors] * [https://www.google.co.uk/books/edition/Shakespeare_the_Man/BVdlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare the man] * [https://www.google.co.uk/books/edition/Henry_V/zXllAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Henry V: A Guide to the Play] * [https://www.google.co.uk/books/edition/Shakespearean_Contingencies/Cw1NAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespearean Contingencies] * [https://www.google.co.uk/books/edition/Renaissance_Drama/E60kAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Renaissance Drama 1990] * [https://www.google.co.uk/books/edition/Language_Discourse_Sign/uH4oAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Language, Discourse, Sign] * [https://www.google.co.uk/books/edition/Mock_Kings_in_Medieval_Society_and_Renai/T98KAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Mock kings in medieval society and Renaissance drama] * [https://www.google.co.uk/books/edition/Shakespeare_Invention_of_the_Human/ojHirImrtYoC?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare: Invention of the Human] * [https://www.google.co.uk/books/edition/Shakespeare/wn5lAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare] * [https://www.google.co.uk/books/edition/Persons_in_Groups/rQ24AAAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Persons in Groups] * [https://www.google.co.uk/books/edition/All_Semblative_a_Woman_s_Part/0DlaAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover All Semblative a Woman's Part?] * [https://www.google.co.uk/books/edition/Crossing_the_Mirror/qRZNAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Crossing the Mirror] * [https://www.google.co.uk/books/edition/De_Vere_is_Shakespeare/dKJlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover De Vere is Shakespeare] * [https://www.google.co.uk/books/edition/William_Lambarde_Elizabethan_Antiquary_1/x1RnAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover William Lambarde, Elizabethan Antiquary, 1536-1601] * [https://www.google.co.uk/books/edition/The_Power_of_Forms_in_the_English_Renais/cPtZAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Power of Forms in the English Renaissance] * [https://www.google.co.uk/books/edition/Ravishment_and_Rememberance/G31LAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Ravishment and Rememberance] * [https://www.google.co.uk/books/edition/Shakespeare_and_His_Theatre/8A5ZQq3uOVQC?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare and His Theatre] * [https://www.google.co.uk/books/edition/Critical_Hermeneutics_and_Shakespeare_s/O10gAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Critical Hermeneutics and Shakespeare's History Plays] * [https://www.google.co.uk/books/edition/Christian_England/K-WfAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Christian England] * [https://www.google.co.uk/books/edition/Shakespeare_s_Religious_Background/xDSaAAAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare's Religious Background] * [https://www.google.co.uk/books/edition/Shylock/N4RlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shylock] * [https://www.google.co.uk/books/edition/The_Shakespeare_Legacy/MM5XAAAAYAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Shakespeare Legacy] * [https://www.google.co.uk/books/edition/Renaissance_Genres/0uFZAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Renaissance Genres] * [https://www.google.co.uk/books/edition/Cannibals_Witches_and_Divorce/qZRpAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Cannibals, Witches, and Divorce] * [https://www.google.co.uk/books/edition/The_Problem_of_Religious_Knowledge/C29LAAAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Problem of Religious Knowledge] * [https://www.google.co.uk/books/edition/Essex_and_the_Great_Revolt_of_1381/J8RzAAAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Essex and the Great Revolt of 1381] * [https://www.google.co.uk/books/edition/Transactions_of_the_London_and_Middlesex/4dtJAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover TLMAS] * [https://www.google.co.uk/books/edition/Shakespeare_Politics_and_the_State/Mn9lAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare, Politics and the State] * [https://www.google.co.uk/books/edition/Allegories_of_Power_in_the_England_of_El/LIYgAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Allegories of Power in the England of Elizabeth] * [https://www.google.co.uk/books/edition/William_Shakespeare/rIVlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover William Shakespeare] * [https://www.google.co.uk/books/edition/Women_s_Matters/PDRlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Women's Matters] * [https://www.google.co.uk/books/edition/The_Weak_King_Dilemma_in_the_Shakespeare/0bJlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Weak King Dilemma in the Shakespearean History Play] * [https://www.google.co.uk/books/edition/The_Book_Known_as_Q/S2tlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Book Known as Q] * [https://www.google.co.uk/books/edition/Fields_of_Vision/OD0eAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Fields of Vision] * [https://www.google.co.uk/books/edition/Ungodly_Delights/RKgcAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Ungodly Delights] * [https://www.google.co.uk/books/edition/The_Shakespeare_Handbook/rLRlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Shakespeare HandboOK] * [https://www.google.co.uk/books/edition/Humanities/y5FZAAAAYAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Humanities] * [https://www.google.co.uk/books/edition/Richard_II_by_William_Shakespeare/Bb3yAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Richard II by William Shakespeare] * [https://www.google.co.uk/books/edition/King_Richard_II/50NnAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover King Richard II] * [https://www.google.co.uk/books/edition/Murder_Under_Trust_Or_The_Topical_Macbet/0oNlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Murder under trust] * [https://www.google.co.uk/books/edition/The_Shakespearean_Kings/tHBlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Shakespearean Kings] * [https://www.google.co.uk/books/edition/America_the_Mabr_e_y_Experience/mRQ3AAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover America, the Mabr(e)y Experience: Resistance, Revolution & Civil War] * [https://www.google.co.uk/books/edition/Richard_II/ZDEkAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Richard II: An Annotated Bibliography, Volume 2] * [https://www.google.co.uk/books/edition/The_Batsford_Companion_to_Medieval_Engla/ev78b9EJQy0C?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Batsford Companion to Medieval England] * [https://www.google.co.uk/books/edition/Shakespeare_s_Unruly_Women/FKFlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare's Unruly Women] * [https://www.google.co.uk/books/edition/Shakespeare_and_Others/iFEgAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare and Others] * [https://www.google.co.uk/books/edition/Kings_and_Chroniclers/L1wpAAAAYAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Kings and Chroniclers] * [https://www.google.co.uk/books/edition/A_Kingdom_for_a_Stage/UzxlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover A Kingdom for a Stage] * [https://www.google.co.uk/books/edition/The_House_of_Commons/Ezz4OZuYVFYC?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The History of Parliament: The House of Commons 1558-1603 (3 v.)] * [https://www.google.co.uk/books/edition/Shakespeare_Soul_of_the_Age/nMYCAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover shakespeare, Soul of the Age] * [https://www.google.co.uk/books/edition/After_Poststructuralism/TOaEAAAAIAAJ?hl=en&gbpv=0&bsq=%22I%20am%20Richard%20II,%20know%20ye%20not%20that?%22 After Poststructuralism: Interdisciplinarity and Literary Theory] * [https://www.google.co.uk/books/edition/The_Unschooled_Mind/C7WnYtt219IC?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Unschooled Mind] * [https://www.google.co.uk/books/edition/Elizabeth_I/hHZnAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Eliz I] * [https://www.google.co.uk/books/edition/Dramas_of_Christian_Time/mnIqAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Dramas of Christian Time] * [https://www.google.co.uk/books/edition/Elizabeth_I/XjQmAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Elizabeth I: The Shrewdness of Virtue] * [https://www.google.co.uk/books/edition/John_Dryden/9Q1aAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover John Dryden] * [https://www.google.co.uk/books/edition/Shakespeare_and_Early_Modern_Political_T/DUwhAwAAQBAJ?hl=en&gbpv=1&dq=%22shakespeare%22+%2B+%22political+propaganda%22&pg=PA259&printsec=frontcover Shakespeare and Early Modern Political Thought] * [https://www.google.co.uk/books/edition/The_English_History_Play_in_the_age_of_S/5TT-AQAAQBAJ?hl=en&gbpv=1&dq=%22shakespeare%22+%2B+%22political+propaganda%22&pg=PA158&printsec=frontcover The English History Play in the Age of Shakespeare] * [https://www.google.co.uk/books/edition/Shakespeare_and_the_Political/rEcREQAAQBAJ?hl=en&gbpv=1&dq=%22shakespeare%22+%2B+%22political+propaganda%22&pg=PA215&printsec=frontcover Shakespeare and the Political] * [https://www.google.co.uk/books/edition/William_Shakespeare_Subject_of_the_Crown/a7G6DAAAQBAJ?hl=en&gbpv=1&dq=%22shakespeare%22+%2B+%22political+propaganda%22&pg=PT18&printsec=frontcover William Shakespeare - Subject of the Crown?] * [https://www.google.com/search?q=%22shakespeare%22+%2B+%22political+propaganda%22&client=firefox-b-d&hs=4AQ&sca_esv=6d4ade7bd26771c9&udm=36&biw=2510&bih=1307&tbs=cdr%3A1%2Ccd_min%3A2000%2Ccd_max%3A2099&sxsrf=ANbL-n6I6Pkwl7mmdHK6N1xPQXLbGBIOSg%3A1776853062010&ei=RqDoaZUvztiFsg_I5bToDw&ved=0ahUKEwiV6tC8nYGUAxVObEEAHcgyDf0Q4dUDCBM&uact=5&oq=%22shakespeare%22+%2B+%22political+propaganda%22&gs_lp=EhBnd3Mtd2l6LW1vZGVsZXNzIiYic2hha2VzcGVhcmUiICsgInBvbGl0aWNhbCBwcm9wYWdhbmRhIjIIECEYoAEYwwRInQlQxgZYuwdwAXgAkAEAmAF_oAHPAaoBAzEuMbgBA8gBAPgBAZgCAqACVsICCxAAGIAEGKIEGLADmAMAiAYBkAYCkgcBMqAHowOyBwExuAdTwgcDMC4yyAcEgAgB&sclient=gws-wiz-modeless The Nazi Appropriation of Shakespeare: Cultural Politics in] {{refend}} 3uaftxdvf5c2ouraono0od5lftla2p8 2818529 2818528 2026-07-19T11:58:23Z Fortuna imperatrix mundi 1058581 /* Richard II */ L3 2818529 wikitext text/x-wiki ==Background== ===Richard II=== ===William Lambarde=== ===Elizabeth I=== ====Essex's rebellion==== Seven months before Lambarde's discussion with Elizabeth.{{sfn|Orgel|2011|pp=11–43}} ==IaR2== Lambarde's account of his interview with the queen in August 1601 is well known.{{sfn|Orgel|2011|pp=11–43}} ===Political anxiety and tension=== ===Censorship=== ==Consequences and aftermath== == Notes == {{reflist|group=note}} ==References== {{Reflist|20em}} ==Bibliography== {{refbegin|30em|indent=yes}} * {{Cite book|url=https://doi.org/10.1057/9780230307261_2|title=Prologue: I am Richard II|last=Orgel|first=Stephen|date=2011|publisher=Palgrave Macmillan UK|isbn=978-0-230-30726-1|editor-last=Petrina|editor-first=Alessandra|location=London|pages=11–43|language=en|doi=10.1057/9780230307261_2|editor-last2=Tosi|editor-first2=Laura}} * [https://broadlytextual.com/2017/12/15/i-am-richard-ii-know-ye-not-that-drama-and-political-anxiety-in-shakespeares-london/ Hixon, E., Syracuse Univ] * Bate, Jonathan (2008). Soul of the Age. London: Penguin. pp. 256–286. ISBN 978-0-670-91482-1. * [https://theconversation.com/richard-ii-by-william-shakespeare-why-the-divine-right-of-kings-still-matters-186648 McFarlane, K., Univ South Australia] * [https://muse.jhu.edu/article/31090/summary Lemon, Rebecca. "The Faulty Verdict in "The Crown v. John Hayward"." SEL Studies in English Literature 1500-1900, vol. 41 no. 1, 2001, p. 109-132. Project MUSE, https://dx.doi.org/10.1353/sel.2001.0009] * [https://brill.com/display/book/9789401211666/B9789401211666-s009.xml?language=en&srsltid=AfmBOoqTUaTQr0msNWRdtEZZ9qQ3rNcVkSoBvDPXuGv7nRvHrC8t5OzH Kizelbach, U. (2014), In The Pragmatics of Early Modern Politics: Power and Kingship in Shakespeare’s History Plays. Leiden, The Netherlands: Brill. https://doi.org/10.1163/9789401211666_009] * [https://www.google.co.uk/books/edition/Richard_II/z-kOAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Lopez, R2] * [https://www.manchesterhive.com/display/9781526130532/9781526130532.00008.xml?print rgel, S. (2017). "I am Richard II". In Spectacular Performances. Manchester, England: Manchester University Press. Retrieved Apr 22, 2026, from https://doi.org/10.7765/9781526130532.00008] * [https://www.researchgate.net/publication/271041996_Was_Elizabeth_I_Richard_II_The_Authenticity_of_Lambarde%27s_%27Conversation%27 Scott-Warren, Jason. (2012). Was Elizabeth I Richard II?: The Authenticity of Lambarde's 'Conversation'. The Review of English Studies. 64. 208-230. 10.1093/res/hgs062] * Stanley Wells, “Introduction” in Richard II, The New Penguin Shakespeare, ed. Stanley Wells (London: Penguin Books, 1969), 13. * [https://www.cambridge.org/core/books/abs/shakespeare-survey/shakespeare-and-history-divergencies-and-agreements/B065A23215FD86BEE8E1CFD51DC7C1FB Ives EW. Shakespeare and History: Divergencies and Agreements. In: Wells S, ed. Shakespeare Survey. Shakespeare Survey. Cambridge University Press; 1986:19-36] * [https://www.jstor.org/stable/457855 Albright, Evelyn May. “Shakespeare’s Richard II, Hayward’s History of Henry IV, and the Essex Conspiracy.” PMLA, vol. 46, no. 3, 1931, pp. 694–719. JSTOR, https://doi.org/10.2307/457855. Accessed 22 Apr. 2026] * [https://dsc.duq.edu/cgi/viewcontent.cgi?article=2800&context=etd Morris, A. (2019). "within the hollow crown": Performing Kingship in Richard II and Henry IV Part One (Master's thesis, Duquesne University). Retrieved from https://dsc.duq.edu/etd/1771] * [https://digitalcommons.lib.uconn.edu/cgi/viewcontent.cgi?article=1122&context=srhonors_theses Scannell, Sarah J., "Shakespeare's Richard II and Henry V and Political Rebellions in the Reign of Queen Elizabeth I" (2010). Honors Scholar Theses, 138] * [https://compass.onlinelibrary.wiley.com/doi/abs/10.1111/j.1741-4113.2011.00873.x Luecking Frost, L. (2012), “A Kyng That Ruled All By Lust”: Richard II in Elizabethan Literature. Literature Compass, 9: 183-198. https://doi.org/10.1111/j.1741-4113.2011.00873.x] * [https://www.academia.edu/8347491/A_Law_Case_in_Verse_Venus_and_Adonis_and_the_Authorship_Question Stritmatter, R., A Law Case in Verse: Venus and Adonis and the Authorship Question, 2004 University of Tennessee Law Review] * [https://www.researchgate.net/publication/304647742_Prologue_I_am_Richard_II Orgel, Stephen. (2011). Prologue: I am Richard II. 10.1057/9780230307261_2] * [https://go.gale.com/ps/i.do?id=GALE%7CA314252957&sid=googleScholar&v=2.1&it=r&linkaccess=abs&issn=15256863&sw=w&p=AONE&userGroupName=anon%7Ebf79dda0&aty=open-web-entry Egan, Michael. "The Essex Rebellion and Richard II: why wasn't Shakespeare arrested?" Shakespeare Oxford Newsletter, vol. 48, no. 3, summer-fall 2012, p. 20] * [https://www.cambridge.org/core/books/abs/richard-ii/stagehistory/D62D9CA061F03B2AE8D56E856288EEDA 1. Child H, Shakespeare W. THE STAGE-HISTORY. In: Dover Wilson J, ed. Richard II: The Cambridge Dover Wilson Shakespeare. Cambridge Library Collection - Literary Studies. Cambridge University Press; 2009:lxxvii-xcii] * [https://www.openstarts.units.it/entities/publication/16d8c447-62f4-48e8-8347-5d56af674aa3/details Snelling, D., 'Prospero on the Coast of Bohemia. "Nihil sed rara et miranda amabat". Diary of Melchior Goldast upon the late Emperor Rudolph II' 1994] * [https://www.jstor.org/stable/2902265 Gajowski, Evelyn. Shakespeare in Performance: King Richard IIby Margaret Shewring. Shakespeare Quarterly, vol. 49, no. 3, 1998, pp. 328–30. JSTOR, https://doi.org/10.2307/2902265. Accessed 22 Apr. 2026] * [https://muse.jhu.edu/article/217842/summary Mosse, Ramona Franziska. Review of Richard II. Shakespeare Bulletin, vol. 25 no. 2, 2007, p. 73-77. Project MUSE, https://dx.doi.org/10.1353/shb.2007.0036] * [https://www.journals.uchicago.edu/doi/abs/10.1086/665894?journalCode=rq Petrina and Tosi. Representations of Elizabeth I in Early Modern Culture Elizabeth Pentland Renaissance Quarterly 2012 65:1, 274-276] * [https://journals.co.za/doi/10.4314/sisa.v26i1.4 The construction of a king : waste, effeminacy and queerness in Shakespeare's Richard II Higginbotham, D.] * [https://academic.oup.com/book/2363/chapter-abstract/142533948?redirectedFrom=fulltext Erskine-Hill, Howard, 'The Second Tetralogy and After', Poetry and the Realm of Politics: Shakespeare to Dryden (Oxford, 1996; online edn, Oxford Academic, 3 Oct. 2011), https://doi.org/10.1093/acprof:oso/9780198117315.003.0004] * [https://www.academia.edu/123917862/Shakespeare_s_Richard_II_in_the_minds_of_the_Elizabethan_audiences Shakespeare’s Richard II in the minds of the Elizabethan audiences] * [https://journals.sagepub.com/doi/10.1177/03064220221084555 Plays, protests and the censor’s pencil] * [https://www.indexoncensorship.org/2022/03/plays-protests-and-the-censors-pencil/ Plays, protests and the censor’s pencil A brittle Gloriana: staging the deposition of Queen Elizabeth I]. * Saul, N., Richard II * [https://www.academia.edu/3236651/_A_god_on_earth_thou_art_Richard_II_and_the_nature_of_Kingship “A god on earth thou art”: Richard II and the nature of Kingship] * [https://www.jstor.org/stable/457398 Albright, Evelyn May. “Shakespeare’s Richard II and the Essex Conspiracy.” PMLA, vol. 42, no. 3, 1927, pp. 686–720. JSTOR, https://doi.org/10.2307/457398. Accessed 22 Apr. 2026] * [https://www.berghahnjournals.com/view/journals/critical-survey/37/3/cs370301.xml Holderness, G. (2025). Introduction: Shakespeare's Wars of the Roses. Critical Survey, 37(3), 1-4. Retrieved Apr 22, 2026, from https://doi.org/10.3167/cs.2025.370301] * [https://www.folger.edu/podcasts/shakespeare-unlimited/tragedy-richard-ii-henry-iv-helen-castor/ Folger: Shakespeare Unlimited: Episode 249] * [https://www.rachaeldickzen.com/blog/2021/1/10/charlesiii The Historical and Shakespearean References in King Charles III] * Kermode, The Age of Shakespeare, pg.53. * [https://heinonline.org/HOL/login-hol?redirect_url=https%3A%2F%2Fheinonline.org%2FHOL%2Fmojo%3F%252Fmojo%3D%26auth_token%3DyhcMOu8W7PCc5OrmTqVnr2qFPpoiOdhM5uQmkQyWufU%253D%26casa_token%3D%26div%3D5%26g_sent%3D1%26paywall_url%3D%26referrer%3Dhttps%253A%252F%252Fheinonline.org%252FHOL%252FPage%253Fhandle%253Dhein.journals%252Fneastlr1%26timestamp%3D1776855600%26url%3D%252FHOL%252FPage%253Fhandle%253Dhein.journals%252Fneastlr1%2526div%253D5%2526id%253D%2526page%253D Jones, H., ''I AM RICHARD II, KNOW YE NOT THAT?' - WORKS. OF WILLIAM SHAKESPEARE AND THE ART OF. GOVERNANCE'''] * Hattaway, M., William Shakespeare: Richard II (Literature Insights, 9781847600349) * [https://books.openedition.org/psn/8305 Findlay, A., “Good sometimes queen” (V.1.37). Richard II, Mary Stuart and the Politics of Queenship, 205-219] * [https://pure.manchester.ac.uk/ws/portalfiles/portal/54508829/FULL_TEXT.PDF Pritchard, K., 'Legitimacy, Illegitimacy and Sovereignty in Shakespeare’s British Plays' (2011 University of Manchester PhD] * CHATELANAT, Marine. Histrionic Future Kings: The Politics of Metadrama in Shakespeare’s Richard III, Richard II, and Henry IV Part 1. Master, 2023. * [https://www.persee.fr/doc/xvii_0291-3798_2004_num_59_1_1993 Mayer, J-C., 'The "Parliament Sceane" in Shakespeare's King Richard II', XVII-XVIII. Revue de la Société d'études anglo-américaines des XVIIe et XVIIIe siècles Année 2004 59 pp. 27-42] * [https://www.emerald.com/md/article-abstract/38/5/315/285132/Managers-as-kings-Shakespeare-on-modern-leadership?redirectedFrom=fulltext Egan M (2000), "Managers as kings: Shakespeare on modern leadership". Management Decision, Vol. 38 No. 5 pp. 315–327, doi: https://doi.org/10.1108/00251740010340490] * [https://d-nb.info/1279081511/34 Quabeck (2022): William Shakespeare, in: Thomas Gutmann, Eberhard Ortland, Klaus Stierstorfer, eds., Encyclopedia of Law and Literature (last edited 26 October 2022), doi: 10.17879/12009577118] * [https://lawliberty.org/book-review/was-the-bard-writing-about-the-donald/ Sullivan, V., 'Was the Bard Writing about The Donald?'] * [https://deveresociety.co.uk/wp-content/uploads/2022/11/FINAL_NL_Apr2022-Haste-July-Letterresponse.pdf Haste, I., 'THE PLAY ON THE EVE OF THE ESSEX REBELLION'] * [https://www.cherylmtaylor.com/2017/11/08/a-companion-to-shakespeares-richard-ii/ A Companion to Shakespeare’s Richard II] * [https://www.city-journal.org/article/shakespeares-richards Dalrymple, T., Shakespeare’s Richards: The playwright’s two historical dramas offer contrasts between political pathologies] * [https://hfriedberg.web.wesleyan.edu/engl205/wshakespeare/plotsandrebelions.htm ELIZABETHAN PLOTS AND REBELLIONS] * [https://www.jstor.org/stable/2869708 Rackin, Phyllis. “The Role of the Audience In Shakespeare’s Richard II.” Shakespeare Quarterly, vol. 36, no. 3, 1985, pp. 262–81. JSTOR, https://doi.org/10.2307/2869708. Accessed 22 Apr. 2026.] * [https://www.psychologytoday.com/ie/blog/whats-in-name/201603/group-think-and-academia-shocking-shakespeare-shenanigans Group Think and Academia: Shocking Shakespeare Shenanigans] * [https://journals.sagepub.com/doi/10.1177/01847678221099975 Hiscopck, A., '"Why I should welcome such a guest as grief [?": Lodging and dislodging in Shakespeare's Richard II'] * [https://academic.oup.com/book/51779/chapter-abstract/420487796?redirectedFrom=fulltext 'Epigraph', Imagining Time in the English Chronicle Play: Historical Futures, 1590-1660 (Oxford, 2023; online edn, Oxford Academic, 19 Oct. 2023), https://doi.org/10.1093/oso/9780198872658.002.0009] * [https://journals.openedition.org/interfaces/319?lang=en Blake & Shakespeare] * [https://www.cambridge.org/core/books/abs/shakespeare-and-the-visual-imagination/richard-iiand-the-politics-of-perspective/23A5D4615E2A8BF9380BAF1ECE99B46D Sillars S. Richard IIand the Politics of Perspective. In: Shakespeare and the Visual Imagination. Cambridge University Press; 2015:133-162.] * [https://www.persee.fr/doc/xvii_0291-3798_2004_num_59_1_1992 Le miroir des rois: de Richard II à Henry VIII] * [https://www.cambridge.org/core/books/abs/king-richard-ll/introduction/25419329A06BC1C2E4D3DDBF331B7F91 Shakespeare W, McEachern C. Introduction. In: Gurr A, ed. King Richard Ll. The New Cambridge Shakespeare. Cambridge University Press; 2018:1-66.] * [https://www.researchgate.net/publication/372862689_A_Critical_Discourse_Study_of_Shakespeare%27s_Theological_Conceptions_in_Acts_IV_and_V_of_Richard_II_The_%27Divine_Mandate%27_of_Richard_Kingship_Falls_Apart A Critical Discourse Study of Shakespeare’s Theological Conceptions in Acts IV and V of Richard II: The ‘Divine Mandate’ of Richard Kingship Falls Apart] * [https://www.jstor.org/stable/10.1525/lal.2004.16.3.313 Richard II, Bracton, and the End of Political Theology] * [https://dokumen.pub/the-art-of-law-in-shakespeare-9781509905478-9781509905508-9781509905492.html The Art of Law in Shakespeare] * [https://www.cambridge.org/core/books/abs/cambridge-companion-to-shakespeare-and-religion/providence-and-divine-right-in-the-english-histories/DBE9BFEBF5C199F9DD72D31ED9C3B041 Mayer J-C. Providence and Divine Right in the English Histories. In: Hamlin H, ed. The Cambridge Companion to Shakespeare and Religion. Cambridge Companions to Literature. Cambridge University Press; 2019:151-167] * [https://www.researchgate.net/publication/315698000_THE_CHOMSKY-FOUCAULT_DEBATE_WHERE_DOES_THE_BARD_STAND_IN_RICHARD_II Salami, Ali & Riahi Nouri, Amir. (2016). THE CHOMSKY-FOUCAULT DEBATE: WHERE DOES THE BARD STAND IN RICHARD II?] * [https://dokumen.pub/shakespeare-richard-ii-9781350363168-9780333416709.html Richard II] * [https://www.jstor.org/stable/457907?seq=1 Heffner, Ray. “Shakespeare, Hayward, and Essex.” PMLA, vol. 45, no. 3, 1930, pp. 754–80. JSTOR, https://doi.org/10.2307/457907. Accessed 22 Apr. 2026] * [https://www.cambridge.org/core/books/abs/cambridge-companion-to-shakespeare-and-war/shakespeare-on-civil-and-dynastic-wars/00D0269CFF998876DB961FB01C22954D Shakespeare on Civil and Dynastic Wars] * [https://brill.com/display/book/edcoll/9789004356900/B9789004356900_012.xml?srsltid=AfmBOopJMjwJAQnUGME9FmT6Gwg0ih9pArbXBdTJbN6YAqbdXqbkGyuo Is a Dramatic Theory of History Possible? Shakespeare’s Richard ii and ‘Historical Truth’] * [https://www.jstor.org/stable/24475607 THE CHILD'S TWO BODIES: SHAKESPEARE, SOVEREIGNTY, AND THE END OF SUCCESSION] * [https://dokumen.pub/english-renaissance-drama-an-introduction-to-theatre-and-theatres-in-shakespeares-time-9781847600202-9781847600714.html English Renaissance Drama : An Introduction to Theatre and Theatres in Shakespeare’s Time] * [https://www.cambridge.org/core/books/abs/believing-in-shakespeare/person-plot-place/BD4EF4CBA6E7D8E2ECE5519FA1D08C16 1. McEachern C. Person, Plot, Place. In: Believing in Shakespeare: Studies in Longing. Cambridge University Press; 2018:181-296] * [https://www.enotes.com/topics/richard-2/criticism/richard-ii-vol-91/criticism-production-reviews-1000228/nigel-saul-review-date-12-may-2000 With an Eye to the Present] * [https://www.academia.edu/128303633/Shakespeare_s_Other_Eden_Richard_II_by_Ouroboros_Theatre_Company_2013_ Shakespeare’s Other Eden Richard II by Ouroboros Theatre Company (2013)] * [https://dokumen.pub/shakespeare-upstart-crow-to-sweet-swan-1592-1623-9781472555496-9781408130148-9781408139189.html Shakespeare: Upstart Crow to Sweet Swan 1592-1623] * [https://www.jstor.org/stable/41917215 RACKIN, PHYLLIS. “Temporality, Anachronism, and Presence in Shakespeare’s English Histories.” Renaissance Drama, vol. 17, 1986, pp. 101–23. JSTOR, http://www.jstor.org/stable/41917215. Accessed 22 Apr. 2026.] * [https://www.cambridge.org/core/journals/greece-and-rome/article/abs/alexander-philotas-and-the-origins-of-modern-historiography/D90568C0A1131952625E0A3B44F43EC8 Stoneman R. Alexander, Philotas, and the origins of modern historiography. Greece and Rome. 2013;60(2):296-312. doi:10.1017/S0017383513000119] * [https://www.academia.edu/35800720/Shakespeares_Richard_II_and_Elizabethan_politics Shakespeare's Richard II and Elizabethan politics] * [https://www.cambridge.org/core/books/abs/rethinking-historicism-from-shakespeare-to-milton/old-new-now/2E8CBEFFEFB11D8B843BAF2C108E9768 Introduction - Old, New, Now] * [https://www.academia.edu/24392476/Richard_II_hamartia_and_climax Richard II hamartia and climax] * [https://www.cambridge.org/core/books/abs/babylonian-jews-and-sasanian-imperialism-in-late-antiquity/kings-and-religion-in-the-talmud-and-in-the-imagination-of-sasanian-communities/E2DCC2F3E0FBA74ABE543DDD2DD7F1A7 Kings and Religion in the Talmud and in the Imagination of Sasanian Communities] * Stephen Greenblatt, “Introduction,” in The Power of Forms in the English Renaissance, ed. Stephen Greenblatt (Norman, OK: 1982), 3-6] * [https://www.academia.edu/30472558/Americas_Donald_Trump_Shakespeares_Richard_II_15_Dec_2016 America's Donald Trump, Shakespeare's Richard II] * [https://www.academia.edu/5321346/Looking_Back_Shakespeare_s_Indebtedness_to_Chaucer_and_the_Representations_of_Chivalry_in_King_Richard_II_The_Two_Noble_Kinsmen_and_the_Knight_s_Tale Looking Back: Shakespeare’s Indebtedness to Chaucer and the Representations of Chivalry in King Richard II, The Two Noble Kinsmen and the Knight’s Tale] * [https://www.academia.edu/92423912/The_Good_Usurper_in_the_eyes_of_God_and_the_people_An_analysis_of_the_role_of_the_usurper_in_Shakespeares_Richard_II_and_Henry_IV_Part_1 The Good Usurper in the eyes of God and the people: An analysis of the role of the usurper in Shakespeare's Richard II and Henry IV Part 1] * [https://www.academia.edu/91789871/The_Bacon_Shakespeare_Manuscript_Hitherto_known_as_the_Northumberland_Manuscript_which_originally_Contained_Copies_of_his_Shakespeare_Plays_Richard_II_and_Richard_III THE BACON-SHAKESPEARE MANUSCRIPT (HITHERTO KNOWN AS THE NORTHUMBERLAND MANUSCRIPT) WHICH ORIGINALLY CONTAINED COPIES OF HIS SHAKESPEARE PLAYS RICHARD II AND RICHARD III] * [https://www.google.co.uk/books/edition/Richard_II/f4gGCAAAQBAJ?hl=en&gbpv=1&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&pg=PA88&printsec=frontcover Richard II: Critical Essays] * [https://www.google.co.uk/books/edition/A_Companion_to_Shakespeare_s_Works_Volum/JlDNEAAAQBAJ?hl=en&gbpv=1&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&pg=PA400&printsec=frontcover A Companion to Shakespeare's Works, Volume II] * [https://www.google.co.uk/books/edition/Shakespeare_Reread/Z6JhDwAAQBAJ?hl=en&gbpv=1&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&pg=PA239&printsec=frontcover Shakespeare Reread] * [https://www.google.co.uk/books/edition/Shakespeare_in_the_Present/yTKWEAAAQBAJ?hl=en&gbpv=1&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&pg=PT8&printsec=frontcover Shakespeare in the Present] * [https://www.google.co.uk/books/edition/Studying_Shakespeare/1N4FBAAAQBAJ?hl=en&gbpv=1&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&pg=PA129&printsec=frontcover Studying Shakespeare A Practical Introduction] * [https://www.google.co.uk/books/edition/Critical_Essays_on_William_Faulkner/kAGBEAAAQBAJ?hl=en&gbpv=1&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&pg=PT178&printsec=frontcover Critical Essays on William Faulkner] * [https://www.google.co.uk/books/edition/Shakespeare/QY6aEQAAQBAJ?hl=en&gbpv=1&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&pg=PA216&printsec=frontcover Shakespeare: An Anthology of Criticism and Theory 1945-2000] * [https://www.google.co.uk/books/edition/The_Reign_of_Richard_II/cv0WAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Dodd, G., R2] * [https://www.google.co.uk/books/edition/Shakespeare_and_the_Institution_of_Theat/Nkt9DAAAQBAJ?hl=en&gbpv=1&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&pg=PA108&printsec=frontcover Shakespeare and the Institution of Theatre] * [https://www.google.co.uk/books/edition/The_Theory_of_the_King_s_Two_Bodies_in_t/8SRXAAAAYAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Theory of the King's Two Bodies in the Age of Shakespeare] * [https://www.google.co.uk/books/edition/Stealing_the_Story/RaJlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Stealing the Story: Shakespeare's Self-Conscious Use of the Mimetic Tradition in the Tragedies] * [https://www.google.co.uk/books/edition/The_Greenwood_Companion_to_Shakespeare_O/JkcgAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Greenwood Companion to Shakespeare: Overviews and the history plays] * [https://www.google.co.uk/books/edition/British_and_Irish_Literature_and_Its_Tim/xH4jAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover British and Irish Literature and Its Times] * [https://www.google.co.uk/books/edition/Shakespeare_s_Philosophy_of_History_Reve/1jwgAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare's Philosophy of History Revealed in a Detailed Analysis of Henry V and Examined in Other History Plays] * [https://www.google.co.uk/books/edition/Shakespeare_and_His_Contemporaries/7em7coOthxQC?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare and His Contemporaries] * [https://www.google.co.uk/books/edition/The_Politics_of_the_Public_Sphere_in_Ear/0sGHAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Politics of the Public Sphere in Early Modern England] * [https://www.google.co.uk/books/edition/Shakespeare_and_the_Catholic_Religion/Fl4gAQAAIAAJ?hl=en&gbpv=0&bsq=%22I%20am%20Richard%20II,%20know%20ye%20not%20that?%22 Shakespeare and the Catholic religion] * [https://www.google.co.uk/books/edition/Proceedings_of_the_British_Academy_Volum/8CsoAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Lectures] * [https://www.google.co.uk/books/edition/Slander_and_Censorship_in_Late_Sixteenth/wb_oAvuJgbAC?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Slander and Censorship in Late Sixteenth Century Literature] * [https://www.google.co.uk/books/edition/Elizabethan_Literature_and_the_Law_of_Fr/vBFdAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Elizabethan Literature and the Law of Fraudulent Conveyance] * [https://www.google.co.uk/books/edition/Hamlet_History_and_commentary/yxgrAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Hamlet History etc] * [https://www.google.co.uk/books/edition/William_Shakespeare_A_Popular_Life/1BuaAAAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover William Shakespeare: a popular life] * [https://www.google.co.uk/books/edition/A_State_of_Mind/pCiIAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover A State of Mind?] * [https://www.google.co.uk/books/edition/Shakespeare_s_Dramatic_Genres/J5FlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare's Dramatic Genres] * [https://www.google.co.uk/books/edition/Literature_Criticism_from_1400_to_1800/O6VkAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Literature Criticism from 1400 to 1800 Volume 89] * [https://www.google.co.uk/books/edition/Shakespeare_the_Papist/LPwNAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare the Papist] * [https://www.google.co.uk/books/edition/Goslicius_Ideal_Senator_and_His_Cultural/1jc7AQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Goslicius' Ideal Senator and His Cultural Impact Over the Centuries] * [https://www.google.co.uk/books/edition/Shakespeare_by_Another_Name/FqllAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover "Shakespeare" by Another Name] * [https://www.google.co.uk/books/edition/Shakespeare_s_Friends/AlZlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare's Friends] * [https://www.google.co.uk/books/edition/Dr_Simon_Forman/qHceAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Dr Simon Forman] * [https://www.google.co.uk/books/edition/William_Shakespeare_the_Wars_of_the_Rose/dZFlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover William Shakespeare, the Wars of the Roses and the historians] * [https://www.google.co.uk/books/edition/Shakespearean_Criticism/2TdlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespearean Criticism] * [https://www.google.co.uk/books/edition/Poetry_History_and_Antiquarianism_in_Ref/jwIhAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Poetry, History, and Antiquarianism in Reformation England] * [https://www.google.co.uk/books/edition/Shakespeare_and_the_Prince_of_Love/D0QgAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare and the Prince of Love] * [https://www.google.co.uk/books/edition/Paper_Bullets_of_the_Brain/o0YgAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Paper Bullets of the Brain] * [https://www.google.co.uk/books/edition/As_You_Like_It/GiBaAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover As You Like It: Third Series] * [https://www.google.co.uk/books/edition/William_Shakespeare/WCqaAAAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover William Shakespeare, Dominic Shellard] * [https://www.google.co.uk/books/edition/Poets_and_God/0XZlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Poets and God] * [https://www.google.co.uk/books/edition/Law_and_Literature/Ax5MAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Law and Literature Volume 16] * [https://www.google.co.uk/books/edition/The_Embodied_Word/FV8sAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Embodied Word] * [https://www.google.co.uk/books/edition/The_Case_for_Shakespeare/WaRlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Case for Shakespeare: The End of the Authorship Question] * [https://www.google.co.uk/books/edition/Explorations_in_Renaissance_Culture/_SYrAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Explorations in Renaissance Culture Volumes 33-34] * [https://www.google.co.uk/books/edition/The_Touch_of_the_Real/ewdaAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Touch of the Real: Essays in Early Modern Culture in Honour of Stephen Greenblatt] * [https://www.google.co.uk/books/edition/Wotton_and_His_Worlds/ZfgNAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Wotton and his Worlds] * [https://www.google.co.uk/books/edition/Theatre_and_Religion/wo1lAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Theatre and Religion Lancastrian Shakespeare] * [https://www.google.co.uk/books/edition/Trying_Treason/TOKxAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Trying Treason] * [https://www.google.co.uk/books/edition/Willing_Subjects/IEX0sGwT1QQC?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Willing Subjects] * [https://www.google.co.uk/books/edition/Symbolism/Bt0ZAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Symbolism] * [https://www.google.co.uk/books/edition/Performing_Shakespeare/35pQAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Performing Shakespeare] * [https://www.google.co.uk/books/edition/Soul_of_the_Age/e0UgAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Soul of the Age] * [https://www.google.co.uk/books/edition/England/aD9nAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover England] * [https://www.google.co.uk/books/edition/Elizabeth_I/-GtnAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Elizabeth I] * [https://www.google.co.uk/books/edition/King_Richard_II/oGUMX4RntjgC?hl=en&gbpv=1&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&pg=PA25&printsec=frontcover King Richard II] * [https://www.google.co.uk/books/edition/Shakespeare_and_the_Legal_Imagination/OXPvBqQLw-4C?hl=en&gbpv=1&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&pg=PA38&printsec=frontcover Shakespeare and the legal imagination] * [https://www.google.co.uk/books/edition/Shakespeare_s_Theatre/GxN3ue9_r3oC?hl=en&gbpv=1&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&pg=PA69&printsec=frontcover Shakespeare's Theatre] * [https://www.google.co.uk/books/edition/Critical_Essays_on_Shakespeare_s_Richard/AaYoAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Critical Essays on Shakespeare's Richard II] * [https://www.google.co.uk/books/edition/The_Reign_of_Richard_II_Essays_in_Honour/y3xnAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Reign of Richard II: Essays in Honour of May McKisack] * [https://www.google.co.uk/books/edition/Poetry_and_the_Realm_of_Politics/oQFaAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Poetry and the Realm of Politics] * [https://www.google.co.uk/books/edition/Shakespeare/BM0mAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare] * [https://www.google.co.uk/books/edition/Shakespearean_Politics/oTdlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespearean Politics] * [https://www.google.co.uk/books/edition/Shakespeare_the_Theatrical_Dimension/wl4gAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare, the Theatrical Dimension] * [https://www.google.co.uk/books/edition/Who_was_Kit_Marlowe/zQ1aAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Kit Marlowe etc] * [https://www.google.co.uk/books/edition/From_Page_to_Performance/beQKAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover From Page to Performance] * [https://www.google.co.uk/books/edition/Exploring_Tudor_England/ax56AAAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Exploring Tudor England] * [https://www.google.co.uk/books/edition/The_Movement_Towards_Subversion/vyJaAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Movement Towards Subversion] * [https://www.google.co.uk/books/edition/Shakespeare_s_Typological_Satire/G5BlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare's Typological Satire] * [https://www.google.co.uk/books/edition/Shakespeare_Recycled/zzNlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare Recycled] * [https://www.google.co.uk/books/edition/Reinventing_the_Middle_Ages_the_Renaissa/fXFnAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Reinventing the Middle Ages & the Renaissance] * [https://www.google.co.uk/books/edition/The_Mysterious_William_Shakespeare/WnllAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Mysterious William Shakespeare] * [https://www.google.co.uk/books/edition/Richard_II/GHhlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Richard II Critical Essays] * [https://www.google.co.uk/books/edition/William_Shakespeare/WJvC6gu_I0gC?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover William Shakespeare: Records and Images] * [https://www.google.co.uk/books/edition/Shakespeare_and_the_Actors/HYtlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare and the Actors] * [https://www.google.co.uk/books/edition/Shakespeare_the_Man/BVdlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare the man] * [https://www.google.co.uk/books/edition/Henry_V/zXllAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Henry V: A Guide to the Play] * [https://www.google.co.uk/books/edition/Shakespearean_Contingencies/Cw1NAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespearean Contingencies] * [https://www.google.co.uk/books/edition/Renaissance_Drama/E60kAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Renaissance Drama 1990] * [https://www.google.co.uk/books/edition/Language_Discourse_Sign/uH4oAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Language, Discourse, Sign] * [https://www.google.co.uk/books/edition/Mock_Kings_in_Medieval_Society_and_Renai/T98KAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Mock kings in medieval society and Renaissance drama] * [https://www.google.co.uk/books/edition/Shakespeare_Invention_of_the_Human/ojHirImrtYoC?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare: Invention of the Human] * [https://www.google.co.uk/books/edition/Shakespeare/wn5lAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare] * [https://www.google.co.uk/books/edition/Persons_in_Groups/rQ24AAAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Persons in Groups] * [https://www.google.co.uk/books/edition/All_Semblative_a_Woman_s_Part/0DlaAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover All Semblative a Woman's Part?] * [https://www.google.co.uk/books/edition/Crossing_the_Mirror/qRZNAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Crossing the Mirror] * [https://www.google.co.uk/books/edition/De_Vere_is_Shakespeare/dKJlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover De Vere is Shakespeare] * [https://www.google.co.uk/books/edition/William_Lambarde_Elizabethan_Antiquary_1/x1RnAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover William Lambarde, Elizabethan Antiquary, 1536-1601] * [https://www.google.co.uk/books/edition/The_Power_of_Forms_in_the_English_Renais/cPtZAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Power of Forms in the English Renaissance] * [https://www.google.co.uk/books/edition/Ravishment_and_Rememberance/G31LAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Ravishment and Rememberance] * [https://www.google.co.uk/books/edition/Shakespeare_and_His_Theatre/8A5ZQq3uOVQC?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare and His Theatre] * [https://www.google.co.uk/books/edition/Critical_Hermeneutics_and_Shakespeare_s/O10gAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Critical Hermeneutics and Shakespeare's History Plays] * [https://www.google.co.uk/books/edition/Christian_England/K-WfAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Christian England] * [https://www.google.co.uk/books/edition/Shakespeare_s_Religious_Background/xDSaAAAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare's Religious Background] * [https://www.google.co.uk/books/edition/Shylock/N4RlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shylock] * [https://www.google.co.uk/books/edition/The_Shakespeare_Legacy/MM5XAAAAYAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Shakespeare Legacy] * [https://www.google.co.uk/books/edition/Renaissance_Genres/0uFZAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Renaissance Genres] * [https://www.google.co.uk/books/edition/Cannibals_Witches_and_Divorce/qZRpAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Cannibals, Witches, and Divorce] * [https://www.google.co.uk/books/edition/The_Problem_of_Religious_Knowledge/C29LAAAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Problem of Religious Knowledge] * [https://www.google.co.uk/books/edition/Essex_and_the_Great_Revolt_of_1381/J8RzAAAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Essex and the Great Revolt of 1381] * [https://www.google.co.uk/books/edition/Transactions_of_the_London_and_Middlesex/4dtJAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover TLMAS] * [https://www.google.co.uk/books/edition/Shakespeare_Politics_and_the_State/Mn9lAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare, Politics and the State] * [https://www.google.co.uk/books/edition/Allegories_of_Power_in_the_England_of_El/LIYgAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Allegories of Power in the England of Elizabeth] * [https://www.google.co.uk/books/edition/William_Shakespeare/rIVlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover William Shakespeare] * [https://www.google.co.uk/books/edition/Women_s_Matters/PDRlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Women's Matters] * [https://www.google.co.uk/books/edition/The_Weak_King_Dilemma_in_the_Shakespeare/0bJlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Weak King Dilemma in the Shakespearean History Play] * [https://www.google.co.uk/books/edition/The_Book_Known_as_Q/S2tlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Book Known as Q] * [https://www.google.co.uk/books/edition/Fields_of_Vision/OD0eAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Fields of Vision] * [https://www.google.co.uk/books/edition/Ungodly_Delights/RKgcAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Ungodly Delights] * [https://www.google.co.uk/books/edition/The_Shakespeare_Handbook/rLRlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Shakespeare HandboOK] * [https://www.google.co.uk/books/edition/Humanities/y5FZAAAAYAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Humanities] * [https://www.google.co.uk/books/edition/Richard_II_by_William_Shakespeare/Bb3yAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Richard II by William Shakespeare] * [https://www.google.co.uk/books/edition/King_Richard_II/50NnAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover King Richard II] * [https://www.google.co.uk/books/edition/Murder_Under_Trust_Or_The_Topical_Macbet/0oNlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Murder under trust] * [https://www.google.co.uk/books/edition/The_Shakespearean_Kings/tHBlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Shakespearean Kings] * [https://www.google.co.uk/books/edition/America_the_Mabr_e_y_Experience/mRQ3AAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover America, the Mabr(e)y Experience: Resistance, Revolution & Civil War] * [https://www.google.co.uk/books/edition/Richard_II/ZDEkAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Richard II: An Annotated Bibliography, Volume 2] * [https://www.google.co.uk/books/edition/The_Batsford_Companion_to_Medieval_Engla/ev78b9EJQy0C?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Batsford Companion to Medieval England] * [https://www.google.co.uk/books/edition/Shakespeare_s_Unruly_Women/FKFlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare's Unruly Women] * [https://www.google.co.uk/books/edition/Shakespeare_and_Others/iFEgAQAAIAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Shakespeare and Others] * [https://www.google.co.uk/books/edition/Kings_and_Chroniclers/L1wpAAAAYAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Kings and Chroniclers] * [https://www.google.co.uk/books/edition/A_Kingdom_for_a_Stage/UzxlAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover A Kingdom for a Stage] * [https://www.google.co.uk/books/edition/The_House_of_Commons/Ezz4OZuYVFYC?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The History of Parliament: The House of Commons 1558-1603 (3 v.)] * [https://www.google.co.uk/books/edition/Shakespeare_Soul_of_the_Age/nMYCAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover shakespeare, Soul of the Age] * [https://www.google.co.uk/books/edition/After_Poststructuralism/TOaEAAAAIAAJ?hl=en&gbpv=0&bsq=%22I%20am%20Richard%20II,%20know%20ye%20not%20that?%22 After Poststructuralism: Interdisciplinarity and Literary Theory] * [https://www.google.co.uk/books/edition/The_Unschooled_Mind/C7WnYtt219IC?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover The Unschooled Mind] * [https://www.google.co.uk/books/edition/Elizabeth_I/hHZnAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Eliz I] * [https://www.google.co.uk/books/edition/Dramas_of_Christian_Time/mnIqAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Dramas of Christian Time] * [https://www.google.co.uk/books/edition/Elizabeth_I/XjQmAQAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover Elizabeth I: The Shrewdness of Virtue] * [https://www.google.co.uk/books/edition/John_Dryden/9Q1aAAAAMAAJ?hl=en&gbpv=1&bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&printsec=frontcover John Dryden] * [https://www.google.co.uk/books/edition/Shakespeare_and_Early_Modern_Political_T/DUwhAwAAQBAJ?hl=en&gbpv=1&dq=%22shakespeare%22+%2B+%22political+propaganda%22&pg=PA259&printsec=frontcover Shakespeare and Early Modern Political Thought] * [https://www.google.co.uk/books/edition/The_English_History_Play_in_the_age_of_S/5TT-AQAAQBAJ?hl=en&gbpv=1&dq=%22shakespeare%22+%2B+%22political+propaganda%22&pg=PA158&printsec=frontcover The English History Play in the Age of Shakespeare] * [https://www.google.co.uk/books/edition/Shakespeare_and_the_Political/rEcREQAAQBAJ?hl=en&gbpv=1&dq=%22shakespeare%22+%2B+%22political+propaganda%22&pg=PA215&printsec=frontcover Shakespeare and the Political] * [https://www.google.co.uk/books/edition/William_Shakespeare_Subject_of_the_Crown/a7G6DAAAQBAJ?hl=en&gbpv=1&dq=%22shakespeare%22+%2B+%22political+propaganda%22&pg=PT18&printsec=frontcover William Shakespeare - Subject of the Crown?] * [https://www.google.com/search?q=%22shakespeare%22+%2B+%22political+propaganda%22&client=firefox-b-d&hs=4AQ&sca_esv=6d4ade7bd26771c9&udm=36&biw=2510&bih=1307&tbs=cdr%3A1%2Ccd_min%3A2000%2Ccd_max%3A2099&sxsrf=ANbL-n6I6Pkwl7mmdHK6N1xPQXLbGBIOSg%3A1776853062010&ei=RqDoaZUvztiFsg_I5bToDw&ved=0ahUKEwiV6tC8nYGUAxVObEEAHcgyDf0Q4dUDCBM&uact=5&oq=%22shakespeare%22+%2B+%22political+propaganda%22&gs_lp=EhBnd3Mtd2l6LW1vZGVsZXNzIiYic2hha2VzcGVhcmUiICsgInBvbGl0aWNhbCBwcm9wYWdhbmRhIjIIECEYoAEYwwRInQlQxgZYuwdwAXgAkAEAmAF_oAHPAaoBAzEuMbgBA8gBAPgBAZgCAqACVsICCxAAGIAEGKIEGLADmAMAiAYBkAYCkgcBMqAHowOyBwExuAdTwgcDMC4yyAcEgAgB&sclient=gws-wiz-modeless The Nazi Appropriation of Shakespeare: Cultural Politics in] {{refend}} 96ax4lh0ukin57l9qa9tga6enrz48si 0 330007 2818510 2818371 2026-07-19T08:35:09Z Rabbi Mendl 772449 ([[c:GR|GR]]) [[c:COM:FR|File renamed]]: [[File:Igbo Woman wearing gele.jpg]] → [[File:Igbo woman wearing Akwete obiakwa(double wrapper) with uweobi(blouse) and Ichafu headdress.jpg]] [[c:COM:FR#FR2|Criterion 2]] (meaningless or ambiguous name) · The current filename incorrectly identifies the headwear as "Ichafu". Reliable published sources identify this style as gele. 2818510 wikitext text/x-wiki {{contrib-creator|Wmbata}} {{launch}} {{course}} {{humanities}} == Introduction == '''Igbo culture''' are the customs, practices and traditions of the Igbo people of southeastern Nigeria. It consists of ancient practices known as ''Odinala'' ''ndi'' ''igbo'' as well as new concepts added into the Igbo culture either by cultural evolution or by outside influence. These customs and traditions includes the Igbo people's visual art, music and dance forms, as well as their attire, Food, cuisine and language dialects. Because of their various subgroups, the variety of their culture is heightened further. == Learning Objectives == By reviewing this material, you should be able to: * '''Identify the major traditional musical instruments and art forms of Ndị Igbo.''' * '''Contrast the components of traditional Ndị Igbo cosmology.''' * '''Detail the socio-economic functions of historical practices like traditional marriage, dressing, architecture, and the apprenticeship system.''' == Module 1: Creative Arts and Expressive Traditions == === Music === [[File:Udu.jpg|thumb|right|95px|Udu, an Igbo instrument]] The Igbo people have a melodic and symphonic musical style. Instruments include Ọ̀pì otherwise known as '''Oja''' a wind instrument similar to the flute, '''igba''', and '''ichaka'''. Another popular musical form among Igbo people is highlife, which is a fusion of jazz and traditional music and widely popular in West Africa. The modern Igbo highlife is seen in the works of Prince Nico Mbarga, Dr Sir Warrior, Oliver De Coque, Bright Chimezie, Celestine Ukwu,Chief Osita Osadebe, And many others who are some of the greatest Igbo highlife musicians of the twentieth century. There are also other notable Igbo highlife artists, like the Mike Ejeagha, Paulson Kalu, Ali Chukwuma, Ozoemena Nwa Nsugbe. === Art === Igbo art is known for various types of masquerades, masks, outfits (symbolizing people), animals and abstract conceptions. Igbo art is also known for its bronze castings found in the town of Igbo Ukwu from the 9th century. <gallery widths="200" heights="200" mode="packed"> File:Nigeria, igbo, maschera-elmo della società mmuo, xx secolo.jpg|Helmet-mask; 20th century; Indianapolis Museum of Art (USA) File:Nigeria, igbo, figura femminile per un tempietto, xx secolo.jpg|Female figure for a small temple, 20th century; Indianapolis Museum of Art File:Igbo brass anklet.jpg|Anklet beaten from a solid brass bar of the type worn by Igbo women. Now in the collection of Wolverhampton Art Gallery. The leg-tube extends approximately 7&nbsp;cm each side of the 35&nbsp;cm disc. File:Bronze ceremonial vessel in form of a snail shell, 9th century, Igbo-Ukwu, Nigeria.JPG|Bronze ceremonial vessel in form of a snail shell; 9th century; from Igbo-Ukwu; Nigerian National Museum (Lagos, Nigeria) File:Eze Onyiudo (2).jpg|Eze Onyiudo Masquerade Awka-Etiti </gallery> === Igbo masks and masquerades === There are two basic types of masquerades, visible and invisible. The visible masquerades are meant for the public. They often are more entertaining. Masks used offer a visual appeal for their shapes and forms. In these visible masquerades, performances of harassment, music, dance, and parodies are acted out (Oyeneke 25). The invisible masquerades take place at night. Sound is the main tool for them. The masquerader uses his voice to scream so it may be heard throughout the village. The masks used are usually fierce looking and their interpretation is only fully understood by the society's members. These invisible masquerades call upon a silent village to strike fear in the hearts of those not initiated into their society. == Module 2: Spiritual Beliefs and Cosmological Frameworks == === Mythology === While today many Igbo people are Christian, the traditional ancient Igbo religion is known as Odinani. In the Igbo mythology, which is part of their ancient religion, the supreme God is called Chineke ("the God of creation"); Chineke created the world and everything in it and is associated with all things on Earth. To the ancient Igbo, the cosmos is divided into four complex parts: * OKIKE (Creation) * ALUSI (Supernatural Forces or Deities) * MMUO (Spirit) * UWA (World) ==== Alusi ==== [[File:Complex sculpture Nigeria BM Af1954 23 522 img02.jpg|thumb|alt=A photo of a complex wooden carving of animals, people and spirits laid on each other to about 2 meters in height|Complex wooden carving depicting images of power and daily life, such as horsemen, imported goods, military insignia, Europeans, rifles, wild beasts and masqueraders.]]'''Alusi''', also known as '''Arusi''' or '''Arushi''', are minor deities that are worshiped and served in Igbo mythology. There are a list of many different Alusi that exists within each community and each has its own purpose. When there is no longer need for the deity, it is returned to its source, through the help of a Chief Priest or Dibia, who is aware of the procedure and ensures that its done properly. ==== Mmuo ==== Mmuo simply means spirit. It is either a good and godly spirit (mmuo oma) or it is an evil spirit (mmuo ojo). For example, the Ogbanje spirit is seen as an evil spirit (mmuo ojo) and anyone possessed by this spirit is given spiritual attention. (Spiritual attention means a way of casting out the evil spirit through deliverance (Christian way) or through African Traditional Religion&nbsp; (i.e. digging out his/her '''“iyi uwa”'''. the ATR way)). Ogbanje is an Igbo (Nigeria) term that means a repeater or someone who comes and departs. Ogbanje is not a bad spirit in Igbo Cosmology. It is a word widely used to describe a kid or teenager who is claimed to die and be born repeatedly by the same person. === Osu caste system === Osu are a group of people whose ancestors were dedicated to serving in shrines and temples for the deities of the Igbo, and therefore were deemed property of the gods. Relationships and sometimes interactions with Osu were (and to this day, still are) in many cases, forbidden. To this day being called an ''Osu'' remains a stigma that prevents people's progress and lifestyles. == Module 3: Social Milestones and Economic Structures == === Umuada === The married and unmarried daughters of a particular clan or village in [[wikipedia: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. === Yam === The yam is very important to the Igbo as it is their staple crop. There are celebrations such as the New yam festival which are held Every August of Every year for the harvesting of the yam. The New Yam festival is celebrated annually to secure a good harvest of the staple crop. The festival is practiced primarily in Nigeria and other countries in West Africa. === Traditional marriage === Marriages in Igbo community follow a multi-step process before the bride and groom are proclaimed husband and wife in accordance with local law and tradition. [[File:Igba nkwu ceremony 04.jpg|alt=Igbo Traditional Marriage|thumb|Traditional Igbo Marriage Attire]] The traditional marriage is known as "Igbankwu Alumdi" in Igbo land, or wine carrying, since it involves the bride serving up a cup of palm wine to her fiancé. Prior to the wedding, the groom must go to the bride's compound with his father before the Igbankwu day to get the bride's father's consent to marry his daughter. If the bride's father is late, in this case, the bride's brother, uncle or male relative fills in for the bride's late father, as applies to the groom. On the second visit, when kola nuts (oji Igbo) are offered, the two fathers must arrange a price for the bride. In most cases, the bride's price is just symbolic, in addition to other requirements like kola nuts, goats, wine, fowl and so on. Normally, it takes more than one evening until the bride price is agreed upon, after which a feast is served to both parents. When the bride price is paid, another evening is set aside for the ceremony. During the ceremony, the bride's father fills a cup with palm wine and hands it over to the daughter. Accompanied by her brides maids known as umuagbo nwunye, she then searches for the groom among the crowd of wedding guests to offer him the drink. Once the drink is offered, the bride and groom dance to the bride's father. They kneel before him and he will give them his blessings. After that, the couple dances for a while before taking their seats, then refreshment takes place followed by presentation of gifts, at times a speech from the MC, and then closing prayer and departure. === Apprenticeship === The Igbo have a unique form of apprenticeship in which either a male family member or a community member will spend time (usually in their teens to their adulthood) with another family, when they work for them. After the time spent with the family, the head of the host household, who is usually the older man who brought the apprentice into his household, will establish the apprentice by either setting up a business for him or giving money or tools by which to make a living. This practice was exploited by Europeans, who used this practice as a way of trading in enslaved people. Olaudah Equiano, although stolen from his home, was an Igbo person who was forced into service to an African family. He said that he felt part of the family, unlike later, when he was shipped to North America and enslaved in the Thirteen Colonies. The Igbo apprenticeship system is called Imu Ahia or Igba Boy in Igboland. It became more prominent among the Igbos after the Nigerian civil war, in a quest to survive the £20 policy which was proposed by Obafemi Awolowo that only £20 be given to every Biafran citizen to survive on regardless of what they had in the bank before the war and the rest of the money were held by the Nigerian government. Petty trade was one of the only ways to build back destroyed communities as well as farming, but then, farming required time that was not readily available at that moment. Essentially, most people went into trading. This Imu-Ahia/Igba Boy model was simple, it works in such a way that business owners would take in younger boys which can be relative, sibling or non-relative from same region, house them and have them work as apprentices in business while learning how it works and the secrets of the business. After the allotted time for the training was reached, 5–8 years’ time, a little graduation ceremony would be held for the '''Nwa Boy''' (the person that learnt the trade). He would also be paid a lump sum for their services over the years, and the money will be used to start a business for the '''Nwa Boy'''. === Chieftaincy Title === [[File:Igbo ichi marks.jpg|thumb|An Igbo man with ''Ichi'' marks, a sign of rank as an Ozo]] Highly accomplished men and women are admitted into their noble orders for people of title such as Ndi Ozo or Ndi Nze. These people receive insignia to show their stature. Membership is highly exclusive, and to qualify an individual need to be highly regarded and well-spoken of in the community. === Kola nut (Ọjị) === [[File:Kola nut.jpg|alt=Kola nut|thumb|Kola nut]] Kola nut occupies a unique position in the cultural life of Igbo people. Ọjị is the first thing served to any visitor in an Igbo home. Ọjị is served before an important function begins, be it marriage ceremony, settlement of family disputes or entering into any type of agreement. Ọjị is traditionally broken into pieces by hand, and if the Kola nut breaks into 3 pieces a special celebration is arranged. == Module 4: Material, Material Culture, Architecture, and Systems of Time == === Igbo Architecture === Igbo architecture refers to the architectural styles and building traditions of the Igbo people. The architectural style is closely tied to the Igbo society's culture, beliefs, and social structure. While the architectural style has evolved, traditional Igbo architecture shares some common characteristics such as: '''Compound layout'''- Igbo architectural traditions often revolve around the concept of a compound which is characterized by an enclosed area encompassing multiple family residences, open central courtyards, verandas, and auxiliary structures. These compounds are meticulously planned and sometimes paved with flat stones to foster communal living and facilitate familial engagements. Additionally, certain compounds feature unique elements like Impluvium houses, Gardens, Moats, and water wells demonstrating the diversity within Igbo architectural practices. '''Ventilation''' - Igbo architecture integrates strategic placement of openings in buildings to promote cross-ventilation, aiding in regulating indoor temperatures. Employing expansive openings facilitates air circulation, ensuring occupant comfort. Depending on the area with high temperatures and humidity, evaporation of sweat becomes challenging; however, airflow aids this process, enhancing comfort. Moreover, construction practices involve thick walls, thatched roofs, and raised foundations to mitigate environmental challenges. The thick walls maintain cooler interiors in hot weather and warmth during rainy seasons. Thatched roofs provide insulation from direct sunlight, offering shade and contributing to thermal comfort. '''Shrines and Sacred Spaces'''- Igbo architecture often includes designated spaces in compounds or community areas for ancestral shrines/temples and secret society meeting houses. These spaces are considered sacred and are an essential part of Igbo cultural and religious practices. These sacred structures may vary in design, ranging from simple open-air spaces to more elaborate structures with specific architectural features. '''Decorative Elements -''' Traditional Igbo architecture often incorporates decorative elements, including painted designs on walls such as [[Uli (design)|uli]], carved wooden door frames, and intricate patterns on ceilings. These decorations may have symbolic or religious significance. === Traditional attire === Igbo traditional attire varies across regions of Southeastern and south south Nigeria with various cultural significance. '''<big>Men</big>''' For men, common garments include ''uwe mwuda'' or ''afe ntutu'' ( robe) or ''efe elu'', a basic shirt paired with underneath wrappers or skirts complemented by the ''okpu ozo'' (the feathered red cap), or ''Okpu aji'' (woolen cap), ''ofo'', ''mkpara'' (staff) and ''Akupe'' (handfans) for ceremonial or titled occasions while loin clothes or waist wrappers were usually worn as casual wears or basic activities like hunting or farming. <big>'''Women'''</big> [[File:Igbo woman wearing Akwete obiakwa(double wrapper) with uweobi(blouse) and Ichafu headdress.jpg|thumb|Igbo women’s traditional attire showing Obiakwa(matching double wrappers) made of Akwete George, uweobi (blouse with puffed sleeves) and stiff Ichafu (headdress)]] Traditional Igbo women's attire comprises many regional and age-based (''Ụmụagbọ'') variant, including the Obiakwa pair of matching wrappers, Uweobi (blouse), and Ịchafụ̀ (head-tie), an elaborate and voluminous headdress traditionally worn by mature women. [[File:Igbo_woman_wearing_Isiagu_obiakwa_maiden_attire,_aka_olu(coral_beads)_ngala(head_beads)_and_nza(horsetail).jpg|thumb|A short Obiakwa (wrapper-style) ensemble paired with a fitted blouse, complemented by nza (horsewhisk), ngala (head beads), and other beaded accessories. The textile features Isiagu motif]] Younger women may wear shorter Obiakwa wrappers paired with a tubular Uweobi blouse. Traditional adornments include ''aka olu'' (coral beads), ''ngala'' (head beads), and ''mgbaji'' (waist beads), often complemented by other ceremonial accessories like the ''akupe'' (hand fans) and ''nza'' (horsetail whisks). Traditional attire and adornment form many parts of Igbo cultural expressions associated with age, status, ceremony, and identity. '''<u>Obiakwa</u>''' The ''Obiakwa'' is a traditional women's double- wrapper attire unique to Igbo weaving traditions such as Akwete cloth. It consists of a pair of matching wrappers Descriptions of Akwete weaving note that such wrapper sets were engineered during the weaving process to be worn together. It is therefore sold in matching pairs. These wrappers are standardly paired with a blouse called uweobi and the Ichafu headdress. Younger women usually wear shorter ''obiakwa'' waist wrapper sets combined with fitted or tubular blouses or wrappers . These clothings are also complemented with ''ngala'', ''mbaji'' and ''aka'' (beaded accessories) as well ''uli'' body arts. In some regions, The Uli body art was also used to decorate both men and women in the form of lines forming patterns and shapes on the body. '''<u>Blouse</u>''' To complete the silhouette, double wrappers(obiakwa) are paired with a Blouse (or ''uweobi''), a traditional fitted blouse. Short ''obiakwa'' styles are usually paired with tubular blouses. '''<u>Ichafu</u>''' [[File:Igbo_woman_wearing_Joojii_and_Ichafu._Igbo_regality.jpg|thumb|An Igbo woman dressed in traditional attire consisting of a white puff-sleeve Uweobi blouse, a red and gold double George wrapper, and a stiff, elaborately structured Ichafu headdress, accessorized with pearl jewelry.]] ''Ichafu'' is an elaborate head-tie or headdress worn by Igbo women, especially for church services, ceremonies and other social occasions. It forms part of a broader clothing ensemble that may include wrappers, blouses and jewellery, and is typically tied in volumnious elevated layered styles with large folds and pleats rising above the head. Ichafu is tied with various textiles including synthetic damask, brocade, Akwete and George fabric, which gives it the stiff and highly elaborated look. In Ogadinma, published by Granta, women were described wearing colourful blouses with “expensive ichafu” tied “in layers and pleats until the scarves were piled atop their heads like large plants”. Other dialectical variations for Ichafu is ''Akwaisi'', ''ulari'', ''unari'', ''nsu n'isi'', ''ufu isi'', ''asusu isi'', ''nchafu isi,'' ''Akishi''. '''<u>Textiles</u>''' Textiles commonly used across Igbo land include ''Isiagu'' (often patterned with the tiger or Lion head motifs), ''Akwete'' and ''Akwaocha'' handwoven clothes, and richly patterned George wrappers. [[File:Little world, Aichi prefecture - African plaza - Hat of a vassal - Ìgbo people in Nigeria - Collected in 2006.jpg|110px|thumb|left|A traditional Igbo hat made entirely from [[wool]].]]Women carried their babies on their backs with a strip of clothing binding the two with a knot at her chest. This baby carrying technique was and still is practiced by many people groups across Africa, including the Igbo. This method has been modernized in the form of the child carrier. Both men and women wore wrappers.[[File:Igba nkwu ceremony 07.jpg|thumb|Igba nkwu, Igbo traditional marriage]] [[File:Igbo Traditional marriage.jpg|thumb|Igbo Traditional Marriage attire]] === Calendar (Iguafo Igbo) === In the traditional Igbo calendar, a week has 4 days (''Eke'', ''Orie'', ''Afọ'', ''Nkwọ''), seven weeks make one month, a month has 28 days and there are 13 months in a year. In the last month, an extra day is added. The names of the days have their roots in the mythology of the Kingdom of Nri. It was believed that Eri, the sky-born founder of the Nri kingdom, had gone on a journey to discover the mystery of time. On his journey he had saluted and counted the four days by the names of the spirits that governed them, and so the names of the spirits (''eke'', ''orie'', ''afọ'' and ''Nkwo'') became the days of the week. {{col-begin}}{{col-2}} {| class="wikitable" !No. || Months (Ọnwa) || Gregorian equivalent |- |1 || '''Ọnwa Mbụ''' || (3rd week of February) |- |2 || '''Ọnwa Abụa''' || (March) |- |3 || '''Ọnwa Ife Eke''' || (April) |- |4 || '''Ọnwa Anọ''' || (May) |- |5 || '''Ọnwa Agwụ''' || (June) |- |6 || '''Ọnwa Ifejiọkụ''' || (July) |- |7 || '''Ọnwa Alọm Chi''' || (August to early September) |- |8 || '''Ọnwa Ilo Mmụọ''' || (Late September) |- |9 || '''Ọnwa Ana''' || (October) |- |10 || '''Ọnwa Okike''' || (Early November) |- |11 || '''Ọnwa Ajana''' || (Late November) |- |12 || '''Ọnwa Ede Ajana''' || (Late November to December) |- |13 || '''Ọnwa Ụzọ Alụsị''' || (January to early February)<ref>{{cite book |last1=Onwuejeogwu |first1=M. Angulu |title=An Igbo Civilization: Nri Kingdom & Hegemony |date=1981 |publisher=Ethnographica |isbn=978-978-123-105-6 }}{{page needed|date=January 2024}}</ref><ref>{{cite web |url=http://www.free-press-release.com/news/200802/1204305180.html |title=Eze Nri - Igu-Aro Festival - 1008th AD |publisher=Free-Press-Release Inc. |date=February 29, 2008 |access-date=2010-04-06 |archive-date=2009-07-24 |archive-url=https://web.archive.org/web/20090724025818/http://www.free-press-release.com/news/200802/1204305180.html |url-status=dead }}</ref> |} {{col-break}} An example of a month: '''''Ọnwa Mbụ''''' {| class="wikitable" |- !Eke || Orie || Afọ || Nkwọ |- ||||| 1 || 2 |- |3 || 4 || 5 || 6 |- |7 || 8 || 9 || 10 |- |11 || 12 || 13 || 14 |- |15 || 16 || 17 || 18 |- |19 || 20 || 21 || 22 |- |23 || 24 || 25 || 26 |- |27 || 28|||| |} {{col-end}} ==== Naming after market days ==== Newborn babies were sometimes named after the day of the week when born. This is no longer the fashion. Names such as '''Mgbeke''' (maiden [born] on the day of Eke), Mgborie (maiden [born] on the Orie day) are commonly seen among the Igbo people. For males, '''Mgbe''' is replaced by '''Nwa''' or <nowiki>'''</nowiki>Okoro<nowiki>'''</nowiki>(Igbo: Child [of]). Examples of this are Solomon Okoronkwo and Nwankwo Kanu, two popular footballers. h5zrisfhuij7t1iyz8satxvdvsgvzqd 5 330057 2818492 2818441 2026-07-18T21:20:38Z Codename Noreste 2969951 /* Process */ reply ([[mw:c:Special:MyLanguage/User:JWBTH/CD|CD]]) 2818492 wikitext text/x-wiki == Notice to colloquium == What is the sence of noticing community about that? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:10, 8 June 2026 (UTC) : A notice would be posted at the inactive SSM's user talk page, and a separate notification at the village pump listing the inactive support staff member(s). [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:46, 25 June 2026 (UTC) ::And the reasoning behind why whole community should know, there is inactive staff? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:08, 2 July 2026 (UTC) ::: This is standard practice as the stewards have done this similar procedure per [[m:Admin activity review]]. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:06, 2 July 2026 (UTC) == Inactive curator template == Just a note if this policy is agreet the template should be fixed. No it counts with 2 years. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:12, 8 June 2026 (UTC) : I'm not sure what you are trying to explain. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:47, 25 June 2026 (UTC) ::I am saying that if the inactive period is changed, the {{tl|Inactive curator}} template text ''"no edits or no logged actions for 2 years"'' should be changed to the appropriate one. This is just a notice, not to forget to do so. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:10, 2 July 2026 (UTC) ::: I see, thank you for explaining. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:05, 2 July 2026 (UTC) == Communication with the SSM and deadlines == A notification on the user's user page is a decent way to communicate with support staff. If they don't respond, it's clear that there's no point in waiting any longer and their rights have been revoked. On the contrary, if they respond, they suspect that they should start working on Wikiversity, but it may happen that they won't, i.e. SSM will respond, but they will continue to be inactive, so they will have another year of "peace". I would probably reduce the inactivity time to '''8 months''' (i.e. 6 months + 2 months, which may take to creat a custodian), but I would leave the response time at a '''month or more'''. I assume that sometimes the reason for inactivity is health problems or personal problems, and in such situations a person is usually not very reactive - i.e. they don't manage to respond quickly to all the requests that come to them. Another reason may be the busy work schedule of university teachers, who, for example, are on the job for 4 months during exams. This means, yes, you have been inactive for a while for some reason and then someone invites you to return to activity, but you are sick, or you are writing a scientific article, grant report, etc. and you don't have much time right now. Here, it would perhaps require standardized posts for all SSM roles, where a notice would be written that according to the policy, a SSM cannot be inactive for a given period. ''Then a question whether they will resume activity within 2 months.'' Yes - rights retained, no/no answer - rights removed within a month. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:29, 8 June 2026 (UTC) : I still think we should leave the timeframe as one year to maintain consistency with some other projects. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 01:40, 25 June 2026 (UTC) ::Well, why not. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 07:25, 25 June 2026 (UTC) == wording == This may sound pendantic... the prhase "are considered inactive if they have made no edits and logged actions within one year." Does this mean that any edit [[w:Logical conjunction|AND]] a log action are required in one year? Does it mean that either one is required in that time frame, but not necessarily both? [[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 04:39, 10 July 2026 (UTC) : This means an edit and a logged action, together. I am not sure what is the point of desysopping a custodian who makes edits but otherwise does not use their permission(s). [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 23:45, 10 July 2026 (UTC) ::I updated the draft language to remove any ambiguity. [[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 17:18, 16 July 2026 (UTC) == 'crat actions == For reference, please see the en.wikipedia section at [[meta:Admin activity review/Local inactivity policies]]. <blockquote>Bureaucrats: same as the Inactive administrators rule above, plus if a bureaucrat does not participate in bureaucrat activity for over three years, their rights may also be removed.</blockquote> As someone who uses this tool, I can tell you that these rights are infrequently used. If more than 12 months passes with no need to flip a custodian bit (a not particularity rare occurrence) it would result in every 'crat here having the bit simultaneously removed. Comments? [[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 17:25, 16 July 2026 (UTC) : Should this be a part of this policy, a notice would be recommended explaining that some bureaucrat haven't used their bit involving bureaucrat actions, and a week should be given whether they would like to retain it or not, or otherwise resign at [[m:SRP]]. [[User:Codename Noreste|<span style="color: blue">Codename Noreste</span>]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 19:56, 16 July 2026 (UTC) : Pinging bureaucrats [[User:Atcovi|Atcovi]], [[User:Dave Braunschweig|Dave Braunschweig]], [[User:Jtneill|Jtneill]] and [[User:Koavf|Koavf]] to this discussion. [[User:Codename Noreste|<span style="color: blue">Codename Noreste</span>]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 19:57, 16 July 2026 (UTC) ::I am fine with CN's proposal. I think a more sensible policy would be to collapse bureaucrat and sysop rights together, so if someone hasn't changed users rights (bureaucrat) <em>or</em> blocked a user, deleted a page, deleted a rev, imported a page, etc. (sysop rights), then that person's advanced user rights would fall under the inactivity policy. ―[[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> 23:26, 16 July 2026 (UTC) == Process == *For item #2 of process I would recommend requiring both talk page notification and also "email this user." If someone is inactive they might not notice the talk page edit. *For #3 we should make clear that a steward request can only be made after a community discussion has concluded. The draft policy seems to skip over the step of review by the community, how long the review should remain open, and other important due process details. [[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 22:24, 16 July 2026 (UTC) : For number 3, the discussion should last about the same as the inactivity response timeframe, but I don't think there should be voting, as this is about inactivity. [[User:Codename Noreste|<span style="color: blue">Codename Noreste</span>]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:20, 18 July 2026 (UTC) 7ms56es03z6t8q4k3duwcvhw8b2fke5 2818493 2818492 2026-07-18T21:22:20Z Codename Noreste 2969951 /* Process */ reply ([[mw:c:Special:MyLanguage/User:JWBTH/CD|CD]]) 2818493 wikitext text/x-wiki == Notice to colloquium == What is the sence of noticing community about that? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:10, 8 June 2026 (UTC) : A notice would be posted at the inactive SSM's user talk page, and a separate notification at the village pump listing the inactive support staff member(s). [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:46, 25 June 2026 (UTC) ::And the reasoning behind why whole community should know, there is inactive staff? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:08, 2 July 2026 (UTC) ::: This is standard practice as the stewards have done this similar procedure per [[m:Admin activity review]]. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:06, 2 July 2026 (UTC) == Inactive curator template == Just a note if this policy is agreet the template should be fixed. No it counts with 2 years. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:12, 8 June 2026 (UTC) : I'm not sure what you are trying to explain. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:47, 25 June 2026 (UTC) ::I am saying that if the inactive period is changed, the {{tl|Inactive curator}} template text ''"no edits or no logged actions for 2 years"'' should be changed to the appropriate one. This is just a notice, not to forget to do so. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:10, 2 July 2026 (UTC) ::: I see, thank you for explaining. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:05, 2 July 2026 (UTC) == Communication with the SSM and deadlines == A notification on the user's user page is a decent way to communicate with support staff. If they don't respond, it's clear that there's no point in waiting any longer and their rights have been revoked. On the contrary, if they respond, they suspect that they should start working on Wikiversity, but it may happen that they won't, i.e. SSM will respond, but they will continue to be inactive, so they will have another year of "peace". I would probably reduce the inactivity time to '''8 months''' (i.e. 6 months + 2 months, which may take to creat a custodian), but I would leave the response time at a '''month or more'''. I assume that sometimes the reason for inactivity is health problems or personal problems, and in such situations a person is usually not very reactive - i.e. they don't manage to respond quickly to all the requests that come to them. Another reason may be the busy work schedule of university teachers, who, for example, are on the job for 4 months during exams. This means, yes, you have been inactive for a while for some reason and then someone invites you to return to activity, but you are sick, or you are writing a scientific article, grant report, etc. and you don't have much time right now. Here, it would perhaps require standardized posts for all SSM roles, where a notice would be written that according to the policy, a SSM cannot be inactive for a given period. ''Then a question whether they will resume activity within 2 months.'' Yes - rights retained, no/no answer - rights removed within a month. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:29, 8 June 2026 (UTC) : I still think we should leave the timeframe as one year to maintain consistency with some other projects. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 01:40, 25 June 2026 (UTC) ::Well, why not. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 07:25, 25 June 2026 (UTC) == wording == This may sound pendantic... the prhase "are considered inactive if they have made no edits and logged actions within one year." Does this mean that any edit [[w:Logical conjunction|AND]] a log action are required in one year? Does it mean that either one is required in that time frame, but not necessarily both? [[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 04:39, 10 July 2026 (UTC) : This means an edit and a logged action, together. I am not sure what is the point of desysopping a custodian who makes edits but otherwise does not use their permission(s). [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 23:45, 10 July 2026 (UTC) ::I updated the draft language to remove any ambiguity. [[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 17:18, 16 July 2026 (UTC) == 'crat actions == For reference, please see the en.wikipedia section at [[meta:Admin activity review/Local inactivity policies]]. <blockquote>Bureaucrats: same as the Inactive administrators rule above, plus if a bureaucrat does not participate in bureaucrat activity for over three years, their rights may also be removed.</blockquote> As someone who uses this tool, I can tell you that these rights are infrequently used. If more than 12 months passes with no need to flip a custodian bit (a not particularity rare occurrence) it would result in every 'crat here having the bit simultaneously removed. Comments? [[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 17:25, 16 July 2026 (UTC) : Should this be a part of this policy, a notice would be recommended explaining that some bureaucrat haven't used their bit involving bureaucrat actions, and a week should be given whether they would like to retain it or not, or otherwise resign at [[m:SRP]]. [[User:Codename Noreste|<span style="color: blue">Codename Noreste</span>]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 19:56, 16 July 2026 (UTC) : Pinging bureaucrats [[User:Atcovi|Atcovi]], [[User:Dave Braunschweig|Dave Braunschweig]], [[User:Jtneill|Jtneill]] and [[User:Koavf|Koavf]] to this discussion. [[User:Codename Noreste|<span style="color: blue">Codename Noreste</span>]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 19:57, 16 July 2026 (UTC) ::I am fine with CN's proposal. I think a more sensible policy would be to collapse bureaucrat and sysop rights together, so if someone hasn't changed users rights (bureaucrat) <em>or</em> blocked a user, deleted a page, deleted a rev, imported a page, etc. (sysop rights), then that person's advanced user rights would fall under the inactivity policy. ―[[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> 23:26, 16 July 2026 (UTC) == Process == *For item #2 of process I would recommend requiring both talk page notification and also "email this user." If someone is inactive they might not notice the talk page edit. *For #3 we should make clear that a steward request can only be made after a community discussion has concluded. The draft policy seems to skip over the step of review by the community, how long the review should remain open, and other important due process details. [[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 22:24, 16 July 2026 (UTC) : For number 3, the discussion should last about the same as the inactivity response timeframe, but I don't think there should be voting, as this is about inactivity. [[User:Codename Noreste|<span style="color: blue">Codename Noreste</span>]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:20, 18 July 2026 (UTC) : In that case, go ahead. I am not against these changes. [[User:Codename Noreste|<span style="color: blue">Codename Noreste</span>]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:22, 18 July 2026 (UTC) jw896oiky5gdvs2rl9or00hkgrfjm9b 2818498 2818493 2026-07-19T00:07:17Z Atcovi 276019 /* 'crat actions */ Reply 2818498 wikitext text/x-wiki == Notice to colloquium == What is the sence of noticing community about that? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:10, 8 June 2026 (UTC) : A notice would be posted at the inactive SSM's user talk page, and a separate notification at the village pump listing the inactive support staff member(s). [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:46, 25 June 2026 (UTC) ::And the reasoning behind why whole community should know, there is inactive staff? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:08, 2 July 2026 (UTC) ::: This is standard practice as the stewards have done this similar procedure per [[m:Admin activity review]]. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:06, 2 July 2026 (UTC) == Inactive curator template == Just a note if this policy is agreet the template should be fixed. No it counts with 2 years. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:12, 8 June 2026 (UTC) : I'm not sure what you are trying to explain. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:47, 25 June 2026 (UTC) ::I am saying that if the inactive period is changed, the {{tl|Inactive curator}} template text ''"no edits or no logged actions for 2 years"'' should be changed to the appropriate one. This is just a notice, not to forget to do so. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:10, 2 July 2026 (UTC) ::: I see, thank you for explaining. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:05, 2 July 2026 (UTC) == Communication with the SSM and deadlines == A notification on the user's user page is a decent way to communicate with support staff. If they don't respond, it's clear that there's no point in waiting any longer and their rights have been revoked. On the contrary, if they respond, they suspect that they should start working on Wikiversity, but it may happen that they won't, i.e. SSM will respond, but they will continue to be inactive, so they will have another year of "peace". I would probably reduce the inactivity time to '''8 months''' (i.e. 6 months + 2 months, which may take to creat a custodian), but I would leave the response time at a '''month or more'''. I assume that sometimes the reason for inactivity is health problems or personal problems, and in such situations a person is usually not very reactive - i.e. they don't manage to respond quickly to all the requests that come to them. Another reason may be the busy work schedule of university teachers, who, for example, are on the job for 4 months during exams. This means, yes, you have been inactive for a while for some reason and then someone invites you to return to activity, but you are sick, or you are writing a scientific article, grant report, etc. and you don't have much time right now. Here, it would perhaps require standardized posts for all SSM roles, where a notice would be written that according to the policy, a SSM cannot be inactive for a given period. ''Then a question whether they will resume activity within 2 months.'' Yes - rights retained, no/no answer - rights removed within a month. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:29, 8 June 2026 (UTC) : I still think we should leave the timeframe as one year to maintain consistency with some other projects. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 01:40, 25 June 2026 (UTC) ::Well, why not. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 07:25, 25 June 2026 (UTC) == wording == This may sound pendantic... the prhase "are considered inactive if they have made no edits and logged actions within one year." Does this mean that any edit [[w:Logical conjunction|AND]] a log action are required in one year? Does it mean that either one is required in that time frame, but not necessarily both? [[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 04:39, 10 July 2026 (UTC) : This means an edit and a logged action, together. I am not sure what is the point of desysopping a custodian who makes edits but otherwise does not use their permission(s). [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 23:45, 10 July 2026 (UTC) ::I updated the draft language to remove any ambiguity. [[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 17:18, 16 July 2026 (UTC) == 'crat actions == For reference, please see the en.wikipedia section at [[meta:Admin activity review/Local inactivity policies]]. <blockquote>Bureaucrats: same as the Inactive administrators rule above, plus if a bureaucrat does not participate in bureaucrat activity for over three years, their rights may also be removed.</blockquote> As someone who uses this tool, I can tell you that these rights are infrequently used. If more than 12 months passes with no need to flip a custodian bit (a not particularity rare occurrence) it would result in every 'crat here having the bit simultaneously removed. Comments? [[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 17:25, 16 July 2026 (UTC) : Should this be a part of this policy, a notice would be recommended explaining that some bureaucrat haven't used their bit involving bureaucrat actions, and a week should be given whether they would like to retain it or not, or otherwise resign at [[m:SRP]]. [[User:Codename Noreste|<span style="color: blue">Codename Noreste</span>]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 19:56, 16 July 2026 (UTC) : Pinging bureaucrats [[User:Atcovi|Atcovi]], [[User:Dave Braunschweig|Dave Braunschweig]], [[User:Jtneill|Jtneill]] and [[User:Koavf|Koavf]] to this discussion. [[User:Codename Noreste|<span style="color: blue">Codename Noreste</span>]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 19:57, 16 July 2026 (UTC) ::I am fine with CN's proposal. I think a more sensible policy would be to collapse bureaucrat and sysop rights together, so if someone hasn't changed users rights (bureaucrat) <em>or</em> blocked a user, deleted a page, deleted a rev, imported a page, etc. (sysop rights), then that person's advanced user rights would fall under the inactivity policy. ―[[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> 23:26, 16 July 2026 (UTC) :::I agree with Justin. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 00:07, 19 July 2026 (UTC) == Process == *For item #2 of process I would recommend requiring both talk page notification and also "email this user." If someone is inactive they might not notice the talk page edit. *For #3 we should make clear that a steward request can only be made after a community discussion has concluded. The draft policy seems to skip over the step of review by the community, how long the review should remain open, and other important due process details. [[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 22:24, 16 July 2026 (UTC) : For number 3, the discussion should last about the same as the inactivity response timeframe, but I don't think there should be voting, as this is about inactivity. [[User:Codename Noreste|<span style="color: blue">Codename Noreste</span>]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:20, 18 July 2026 (UTC) : In that case, go ahead. I am not against these changes. [[User:Codename Noreste|<span style="color: blue">Codename Noreste</span>]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:22, 18 July 2026 (UTC) 8ari4a5z74iz9kv8qo3djwji9q2flrc 0 330134 2818515 2815174 2026-07-19T09:55:03Z CorraleH 2903442 2818515 wikitext text/x-wiki {{Article info | first1 = Mónica | last1 = Bonilla | orcid1 = 0000-0002-4594-0093 | affiliation1 = Wikimedia Colombia | first2 = Johana | last2 = Botero | orcid2 = 0009-0006-0484-6291 | affiliation2 = Wikimedia Colombia | first3 = Bernardo | last3 = Caycedo | orcid3 = 0009-0008-1861-5034 | affiliation3 = Wikimedia Colombia | first4 = Manuel | last4 = Franco-Avellaneda | orcid4 = 0000-0002-0895-8219 | affiliation4 = Wikimedia Colombia | first5 = Nathaly | last5 = Montoya | orcid5 = 0009-0004-3539-5599 | affiliation5 = Wikimedia Colombia | submitted = 2025-11-14 | published = 2026-06-10 | correspondence1 = | journal = WikiJournal of Humanities | w1 = | license = {{CC-BY-SA work}} | abstract = }} [[File:Banner wikiconf wiki.png|thumb|alt=Wikiconferencia Colombia 2025|Wikiconferencia Colombia 2025|link=https://meta.wikimedia.org/wiki/Wikiconferencia_Colombia_2025|320px]] ==Foreword== The Proceedings of the [[meta:Wikiconferencia_Colombia_2025|WikiConf Colombia 2025]], held on November 14, 2025, in Bogotá, Colombia, bring together abstracts dedicated to the study of Wikipedia and other wiki projects as significant phenomena within the contemporary scientific, educational, and information space. Organized by Wikimedia Colombia, with the support of the Wikimedia Foundation, the conference gathered participants from Colombia and abroad and marked an important step in fostering a scholarly community focused on interdisciplinary research of open knowledge and wiki environments. The contributions, published under the Creative Commons Attribution-ShareAlike 4.0 International license, reflect a wide range of research topics, including digital commons, free culture, historical memory, biodiversity, and Indigenous languages, while also emphasizing the role of Wikipedia in transforming prevailing digital paradigms and advancing the principles of free culture. WikiConf Colombia 2025 constitutes the annual meeting of the Wikimedia community in Colombia. This year’s edition, held in Bogotá, aims to foster critical discussions on transforming prevailing digital paradigms and advancing the principles of free culture. The conference convenes a diverse range of participants, including local community members, [[meta:Wikimedia Colombia|Wikimedia Colombia]] (WMCO) staff, national scholarship recipients, and guests, to exchange experiences, address emerging challenges, and collaboratively envision more open and participatory futures for the Wikimedia movement in Colombia. The conference seeks to reflect on digital commons as collaborative practices that challenge both the privatization of the digital sphere and uncritical forms of automation. It aims to open a space for imagining freer and more solidaristic futures in which open culture, historical memory, biodiversity, and Indigenous languages occupy a central place. Digital commons are woven through diversity: they emerge in collaboration with Indigenous, Afro-descendant, rural, and urban communities, through the exchange between ancestral knowledge and open, emerging technologies. WikiConf Colombia 2025 serves as a moment of recognition, collective learning, and the creation of new alliances. ==Abstracts== # [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/Opening: Wikimedia Colombia and Universidad del Rosario|Opening: Wikimedia Colombia and Universidad del Rosario]] # [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/Roundtable: the commons in dispute: sovereignty and free knowledge from Wikimedia|Roundtable: the commons in dispute: sovereignty and free knowledge from Wikimedia]] # [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/Joo'uya Waashajaaiwaa Wanüiki - teaching Wayuunaiki with digital media|Joo'uya Waashajaaiwaa Wanüiki - teaching Wayuunaiki with digital media]] # [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/How to build a Wikipedia article from zero and not die during the process? a situated experience|How to build a Wikipedia article from zero and not die during the process? a situated experience]] # [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/Free metaphor|Free metaphor]] # [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/Oral libraries in the midst of a knowledge system based on the written word|Oral libraries in the midst of a knowledge system based on the written word]] # [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/Roundtable: the path toward community environmental monitoring|Roundtable: the path toward community environmental monitoring]] # [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/Panel: AI and digital commons|Panel: AI and digital commons]] # [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/We are not communists, we are commoners|We are not communists, we are commoners]] # [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/Decentralized Web3 governance for the Wikimedia community?|Decentralized Web3 governance for the Wikimedia community?]] # [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/OpenStreetMap and Wikimedia: two worlds full of data that tell stories|OpenStreetMap and Wikimedia: two worlds full of data that tell stories]] # [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/Congressional and presidential elections are coming up: what can we do as the Wikimedia Colombia community?|Congressional and presidential elections are coming up: what can we do as the Wikimedia Colombia community?]] # [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/Wikimedia in the teaching of phylogenomics in Colombia|Wikimedia in the teaching of phylogenomics in Colombia]] # [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/Inclusion and sustainability at Wikimania Nairobi 2025: a narrative|Inclusion and sustainability at Wikimania Nairobi 2025: a narrative]] # [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/#Fracking: tracing climate change disinformation on social media|#Fracking: tracing climate change disinformation on social media]] # [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/Roundtable: linguistic diversity and digital fabric|Roundtable: linguistic diversity and digital fabric]] # [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/Electoral API: how to transform public information into citizen knowledge? open data for democracy|Electoral API: how to transform public information into citizen knowledge? open data for democracy]] # [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/Stories, experiences, and images of the mothers of victims of false positives: process to build the memorial for women belonging to the MAFAPO collective|Stories, experiences, and images of the mothers of victims of false positives: process to build the memorial for women belonging to the MAFAPO collective]] # [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/The river that we are|The river that we are]] # [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/Lightning talk: echoes of a mural|Lightning talk: echoes of a mural]] # [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/Restrictions and possibilities of Colombia's public (but not) open data|Restrictions and possibilities of Colombia's public (but not) open data]] # [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/Co-creation of a translation methodology using Wikibooks|Co-creation of a translation methodology using Wikibooks]] # [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/Superchange: the universe where climate heroes are born|Superchange: the universe where climate heroes are born]] # [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/Photographic co-creation workshop: self-representation and alterity in Abya Yala|Photographic co-creation workshop: self-representation and alterity in Abya Yala]] <!-- [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/Hackathon|Hackathon]] [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/Poster session|Poster session]] --> 7xo0696i97ldkpq9wer7ys966vnn775 0 330631 2818482 2818471 2026-07-18T15:03:46Z Grimes2 2895877 heptachord 2818482 wikitext text/x-wiki ==Basic tetrachords== 1½1, ½11, 11½ <score> \new Staff \with { \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { d1 e -\tweak HorizontalBracketText.text "semitone" \startGroup f\stopGroup g} </score> <score> \new Staff \with { \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { e1 -\tweak HorizontalBracketText.text "semitone" \startGroup f\stopGroup g a} </score> <score> \new Staff \with { \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { c1 d e -\tweak HorizontalBracketText.text "semitone" \startGroup f\stopGroup} </score> and the tritone<br> 111 <score> \new Staff \with { \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { f1 g a b} </score> Two consecutive tetrachords build a heptachord. ==Heptachord== <score> \new Staff \with { \consists Ambitus_engraver \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { a1^"kitmum" b -\tweak HorizontalBracketText.text "semitone" \startGroup c\stopGroup d e-\tweak HorizontalBracketText.text "semitone"\startGroup f\stopGroup g } </score> ==7 heptachord modes== The 7 modal patterns for the Babylonian heptachords are: {| class="wikitable sortable" style="text-align: center; |+ Babylonian tonal system ! Mode !! Pattern !! Center note || Semitones || Tritones |- | ''kitmum'' || 1½11½1 || D || 10 || 0 |- | ''pītum'' || ½11½11 || E || 10 || 0 |- | ''qablītum'' || 11½11½ || C || 10 || 0 |- | ''išartum'' || 1½111½ || G || 10 || 1 |- | ''embūbum'' || ½111½1 || A || 10 || 1 |- | ''nīd qablim'' || 111½11 || B || 11 || 2 |- | ''nīš tuḫrim'' || 11½111 || F || 11 || 2 |} <score> \new Staff \with { \consists Ambitus_engraver \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { a1^"kitmum" b -\tweak HorizontalBracketText.text "semitone" \startGroup c\stopGroup d e-\tweak HorizontalBracketText.text "semitone"\startGroup f\stopGroup g } </score> <score> \new Staff \with { \consists Ambitus_engraver \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { g1^"pītum" a b -\tweak HorizontalBracketText.text "semitone" \startGroup c\stopGroup d e-\tweak HorizontalBracketText.text "semitone"\startGroup f\stopGroup } </score> <score> \new Staff \with { \consists Ambitus_engraver \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { b1^"qablītum" -\tweak HorizontalBracketText.text "semitone" \startGroup c\stopGroup d e-\tweak HorizontalBracketText.text "semitone"\startGroup f\stopGroup g a} </score> <score> \new Staff \with { \consists Ambitus_engraver \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c { e1^"išartum" -\tweak HorizontalBracketText.text "semitone" \startGroup f\stopGroup g a b-\tweak HorizontalBracketText.text "semitone"\startGroup c\stopGroup d} </score> <score> \new Staff \with { \consists Ambitus_engraver \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { d1^"embūbum" e -\tweak HorizontalBracketText.text "semitone" \startGroup f\stopGroup g a b-\tweak HorizontalBracketText.text "semitone"\startGroup c\stopGroup} </score> <score> \new Staff \with { \consists Ambitus_engraver \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { f,1^"nīd qablim" g a b -\tweak HorizontalBracketText.text "semitone" \startGroup c\stopGroup d e} </score> <score> \new Staff \with { \consists Ambitus_engraver \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { c1^"nīš tuḫrim" d e -\tweak HorizontalBracketText.text "semitone" \startGroup f\stopGroup g a b} </score> ==2 consecutive heptachords== <score> \new Staff \with { \consists Ambitus_engraver \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { a1^"kitmum" b -\tweak HorizontalBracketText.text "semitone" \startGroup c\stopGroup d e-\tweak HorizontalBracketText.text "semitone"\startGroup f\stopGroup g a-\tweak HorizontalBracketText.text "semitone" \startGroup bes\stopGroup c d-\tweak HorizontalBracketText.text "semitone" \startGroup es\stopGroup f} </score> ==7 consecutive heptachords== 7 consecutive heptachords build the 7 Babylonian modes: [[File:Kitmum.svg|thumb|left|1000px]] [[File:Pitum.svg|thumb|left|1000px]] [[File:Qablitum.svg|thumb|left|1000px]] [[File:Isartum.svg|thumb|left|1000px]] [[File:Embubum.svg|thumb|left|1000px]] [[File:Nid quablim.svg|thumb|left|1000px]] [[File:Nis tuhrim.svg|thumb|left|1000px]] {{clear}} ===6 cyclic consecutive heptachords=== [[File:Circle of fourths.svg|thumb|left|Circle of fourths, ''kitmum'', 36 notes, 60 semitones = Anu 𒀭]] {{clear}} ==Tuning== A tuning procedure ‘loosening’ (TU.LU) in Music of Mesopotamia for a 7-stringed instrument based on a transposition to D/D♭: <score> \new Staff \with { \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { d1^"išartum" e -\tweak HorizontalBracketText.text "semitone" \startGroup f\stopGroup g a b-\tweak HorizontalBracketText.text "semitone"\startGroup c\stopGroup } </score> <score> \new Staff \with { \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { d1^"kitmum" e -\tweak HorizontalBracketText.text "semitone" \startGroup f\stopGroup g a-\tweak HorizontalBracketText.text "semitone"\startGroup bes\stopGroup c } </score> <score> \new Staff \with { \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { d1^"embūbum" -\tweak HorizontalBracketText.text "semitone" \startGroup es\stopGroup f g a-\tweak HorizontalBracketText.text "semitone"\startGroup bes\stopGroup c } </score> <score> \new Staff \with { \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { d1^"pītum" -\tweak HorizontalBracketText.text "semitone" \startGroup es\stopGroup f g -\tweak HorizontalBracketText.text "semitone"\startGroup as\stopGroup bes c } </score> <score> \new Staff \with { \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { des1^"nīd qablim" es f g-\tweak HorizontalBracketText.text "semitone"\startGroup as\stopGroup bes c } </score> <score> \new Staff \with { \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { des1^"nīš tuḫrim" es f -\tweak HorizontalBracketText.text "semitone"\startGroup ges\stopGroup as bes c } </score> <score> \new Staff \with { \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { des1^"qablītum" es f -\tweak HorizontalBracketText.text "semitone"\startGroup ges\stopGroup as bes-\tweak HorizontalBracketText.text "semitone"\startGroup ces\stopGroup } </score> ‘tightening’ (GÍD.I) <score> \new Staff \with { \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { d1^"išartum" e -\tweak HorizontalBracketText.text "semitone" \startGroup f\stopGroup g a b-\tweak HorizontalBracketText.text "semitone"\startGroup c\stopGroup } </score> <score> \new Staff \with { \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { d1^"qablītum" e fis -\tweak HorizontalBracketText.text "semitone"\startGroup g\stopGroup a b-\tweak HorizontalBracketText.text "semitone"\startGroup c\stopGroup } </score> <score> \new Staff \with { \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { d1^"niš tuĥrim" e fis -\tweak HorizontalBracketText.text "semitone"\startGroup g\stopGroup a b cis } </score> <score> \new Staff \with { \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { d1^"nīd qablim" e fis gis-\tweak HorizontalBracketText.text "semitone"\startGroup a\stopGroup b cis } </score> <score> \new Staff \with { \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { dis1^"pītum" -\tweak HorizontalBracketText.text "semitone" \startGroup e\stopGroup fis gis -\tweak HorizontalBracketText.text "semitone"\startGroup a\stopGroup b cis } </score> <score> \new Staff \with { \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { dis1^"embūbum" -\tweak HorizontalBracketText.text "semitone" \startGroup e\stopGroup fis gis ais-\tweak HorizontalBracketText.text "semitone"\startGroup b\stopGroup cis } </score> <score> \new Staff \with { \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { dis1^"kitmum" eis -\tweak HorizontalBracketText.text "semitone" \startGroup fis\stopGroup gis ais-\tweak HorizontalBracketText.text "semitone"\startGroup b\stopGroup cis } </score> ==Further reading== * {{cite conference | first = Leon | last = Crickmore | title = A New Light on the Babylonian Tonal System | conference = Proceedings of the International Conference of Near Eastern Archaeomusicology (ICONEA 2008), The British Museum, London, December 4–6, 2008, editors: Richard Dumbrill and Irving Finkel | publisher = Iconea Publications | url = https://musicircle.net/wp-content/uploads/2018/08/Crickmore-Iconea20081.pdf | place = London | year = 2008 | pages = 11-22 }} * {{cite web | title=Babylonian Tonal System | website=Allan Pettersson Fanpage | url=https://pettersson-fanpage.de/Babylon/Babylon.html | access-date=19 January 2026}} [[Category:Music theory]] cy6g7av7meaimktfu9icejy21ao4mui 2818483 2818482 2026-07-18T16:00:46Z Grimes2 2895877 /* 7 heptachord modes */ 2818483 wikitext text/x-wiki ==Basic tetrachords== 1½1, ½11, 11½ <score> \new Staff \with { \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { d1 e -\tweak HorizontalBracketText.text "semitone" \startGroup f\stopGroup g} </score> <score> \new Staff \with { \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { e1 -\tweak HorizontalBracketText.text "semitone" \startGroup f\stopGroup g a} </score> <score> \new Staff \with { \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { c1 d e -\tweak HorizontalBracketText.text "semitone" \startGroup f\stopGroup} </score> and the tritone<br> 111 <score> \new Staff \with { \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { f1 g a b} </score> Two consecutive tetrachords build a heptachord. ==Heptachord== <score> \new Staff \with { \consists Ambitus_engraver \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { a1^"kitmum" b -\tweak HorizontalBracketText.text "semitone" \startGroup c\stopGroup d e-\tweak HorizontalBracketText.text "semitone"\startGroup f\stopGroup g } </score> ==7 heptachord modes== The 7 modal patterns for the Babylonian heptachords are: {| class="wikitable sortable" style="text-align: center; |+ Babylonian tonal system ! Mode !! Pattern !! Center note || Semitones || Tritones |- | ''kitmum'' || 1½11½1 || D || 10 || 0 |- | ''pītum'' || ½11½11 || E || 10 || 0 |- | ''qablītum'' || 11½11½ || C || 10 || 0 |- | ''išartum'' || 1½111½ || G || 10 || 1 |- | ''embūbum'' || ½111½1 || A || 10 || 1 |- | ''nīd qablim'' || 111½11 || B || 11 || 2 |- | ''nīš tuḫrim'' || 11½111 || F || 11 || 2 |} <score> \new Staff \with { \consists Ambitus_engraver \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { a1^"kitmum" b -\tweak HorizontalBracketText.text "semitone" \startGroup c\stopGroup d e-\tweak HorizontalBracketText.text "semitone"\startGroup f\stopGroup g } </score> <score> \new Staff \with { \consists Ambitus_engraver \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { g1^"qablītumpītum" a b -\tweak HorizontalBracketText.text "semitone" \startGroup c\stopGroup d e-\tweak HorizontalBracketText.text "semitone"\startGroup f\stopGroup } </score> <score> \new Staff \with { \consists Ambitus_engraver \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { b1^"pītum" -\tweak HorizontalBracketText.text "semitone" \startGroup c\stopGroup d e-\tweak HorizontalBracketText.text "semitone"\startGroup f\stopGroup g a} </score> <score> \new Staff \with { \consists Ambitus_engraver \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c { e1^"išartum" -\tweak HorizontalBracketText.text "semitone" \startGroup f\stopGroup g a b-\tweak HorizontalBracketText.text "semitone"\startGroup c\stopGroup d} </score> <score> \new Staff \with { \consists Ambitus_engraver \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { d1^"embūbum" e -\tweak HorizontalBracketText.text "semitone" \startGroup f\stopGroup g a b-\tweak HorizontalBracketText.text "semitone"\startGroup c\stopGroup} </score> <score> \new Staff \with { \consists Ambitus_engraver \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { f,1^"nīd qablim" g a b -\tweak HorizontalBracketText.text "semitone" \startGroup c\stopGroup d e} </score> <score> \new Staff \with { \consists Ambitus_engraver \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { c1^"nīš tuḫrim" d e -\tweak HorizontalBracketText.text "semitone" \startGroup f\stopGroup g a b} </score> ==2 consecutive heptachords== <score> \new Staff \with { \consists Ambitus_engraver \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { a1^"kitmum" b -\tweak HorizontalBracketText.text "semitone" \startGroup c\stopGroup d e-\tweak HorizontalBracketText.text "semitone"\startGroup f\stopGroup g a-\tweak HorizontalBracketText.text "semitone" \startGroup bes\stopGroup c d-\tweak HorizontalBracketText.text "semitone" \startGroup es\stopGroup f} </score> ==7 consecutive heptachords== 7 consecutive heptachords build the 7 Babylonian modes: [[File:Kitmum.svg|thumb|left|1000px]] [[File:Pitum.svg|thumb|left|1000px]] [[File:Qablitum.svg|thumb|left|1000px]] [[File:Isartum.svg|thumb|left|1000px]] [[File:Embubum.svg|thumb|left|1000px]] [[File:Nid quablim.svg|thumb|left|1000px]] [[File:Nis tuhrim.svg|thumb|left|1000px]] {{clear}} ===6 cyclic consecutive heptachords=== [[File:Circle of fourths.svg|thumb|left|Circle of fourths, ''kitmum'', 36 notes, 60 semitones = Anu 𒀭]] {{clear}} ==Tuning== A tuning procedure ‘loosening’ (TU.LU) in Music of Mesopotamia for a 7-stringed instrument based on a transposition to D/D♭: <score> \new Staff \with { \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { d1^"išartum" e -\tweak HorizontalBracketText.text "semitone" \startGroup f\stopGroup g a b-\tweak HorizontalBracketText.text "semitone"\startGroup c\stopGroup } </score> <score> \new Staff \with { \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { d1^"kitmum" e -\tweak HorizontalBracketText.text "semitone" \startGroup f\stopGroup g a-\tweak HorizontalBracketText.text "semitone"\startGroup bes\stopGroup c } </score> <score> \new Staff \with { \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { d1^"embūbum" -\tweak HorizontalBracketText.text "semitone" \startGroup es\stopGroup f g a-\tweak HorizontalBracketText.text "semitone"\startGroup bes\stopGroup c } </score> <score> \new Staff \with { \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { d1^"pītum" -\tweak HorizontalBracketText.text "semitone" \startGroup es\stopGroup f g -\tweak HorizontalBracketText.text "semitone"\startGroup as\stopGroup bes c } </score> <score> \new Staff \with { \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { des1^"nīd qablim" es f g-\tweak HorizontalBracketText.text "semitone"\startGroup as\stopGroup bes c } </score> <score> \new Staff \with { \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { des1^"nīš tuḫrim" es f -\tweak HorizontalBracketText.text "semitone"\startGroup ges\stopGroup as bes c } </score> <score> \new Staff \with { \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { des1^"qablītum" es f -\tweak HorizontalBracketText.text "semitone"\startGroup ges\stopGroup as bes-\tweak HorizontalBracketText.text "semitone"\startGroup ces\stopGroup } </score> ‘tightening’ (GÍD.I) <score> \new Staff \with { \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { d1^"išartum" e -\tweak HorizontalBracketText.text "semitone" \startGroup f\stopGroup g a b-\tweak HorizontalBracketText.text "semitone"\startGroup c\stopGroup } </score> <score> \new Staff \with { \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { d1^"qablītum" e fis -\tweak HorizontalBracketText.text "semitone"\startGroup g\stopGroup a b-\tweak HorizontalBracketText.text "semitone"\startGroup c\stopGroup } </score> <score> \new Staff \with { \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { d1^"niš tuĥrim" e fis -\tweak HorizontalBracketText.text "semitone"\startGroup g\stopGroup a b cis } </score> <score> \new Staff \with { \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { d1^"nīd qablim" e fis gis-\tweak HorizontalBracketText.text "semitone"\startGroup a\stopGroup b cis } </score> <score> \new Staff \with { \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { dis1^"pītum" -\tweak HorizontalBracketText.text "semitone" \startGroup e\stopGroup fis gis -\tweak HorizontalBracketText.text "semitone"\startGroup a\stopGroup b cis } </score> <score> \new Staff \with { \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { dis1^"embūbum" -\tweak HorizontalBracketText.text "semitone" \startGroup e\stopGroup fis gis ais-\tweak HorizontalBracketText.text "semitone"\startGroup b\stopGroup cis } </score> <score> \new Staff \with { \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { dis1^"kitmum" eis -\tweak HorizontalBracketText.text "semitone" \startGroup fis\stopGroup gis ais-\tweak HorizontalBracketText.text "semitone"\startGroup b\stopGroup cis } </score> ==Further reading== * {{cite conference | first = Leon | last = Crickmore | title = A New Light on the Babylonian Tonal System | conference = Proceedings of the International Conference of Near Eastern Archaeomusicology (ICONEA 2008), The British Museum, London, December 4–6, 2008, editors: Richard Dumbrill and Irving Finkel | publisher = Iconea Publications | url = https://musicircle.net/wp-content/uploads/2018/08/Crickmore-Iconea20081.pdf | place = London | year = 2008 | pages = 11-22 }} * {{cite web | title=Babylonian Tonal System | website=Allan Pettersson Fanpage | url=https://pettersson-fanpage.de/Babylon/Babylon.html | access-date=19 January 2026}} [[Category:Music theory]] 0i2aj31dmy4s674i6g00561xng5jvd4 2818484 2818483 2026-07-18T16:01:20Z Grimes2 2895877 /* 7 heptachord modes */ 2818484 wikitext text/x-wiki ==Basic tetrachords== 1½1, ½11, 11½ <score> \new Staff \with { \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { d1 e -\tweak HorizontalBracketText.text "semitone" \startGroup f\stopGroup g} </score> <score> \new Staff \with { \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { e1 -\tweak HorizontalBracketText.text "semitone" \startGroup f\stopGroup g a} </score> <score> \new Staff \with { \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { c1 d e -\tweak HorizontalBracketText.text "semitone" \startGroup f\stopGroup} </score> and the tritone<br> 111 <score> \new Staff \with { \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { f1 g a b} </score> Two consecutive tetrachords build a heptachord. ==Heptachord== <score> \new Staff \with { \consists Ambitus_engraver \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { a1^"kitmum" b -\tweak HorizontalBracketText.text "semitone" \startGroup c\stopGroup d e-\tweak HorizontalBracketText.text "semitone"\startGroup f\stopGroup g } </score> ==7 heptachord modes== The 7 modal patterns for the Babylonian heptachords are: {| class="wikitable sortable" style="text-align: center; |+ Babylonian tonal system ! Mode !! Pattern !! Center note || Semitones || Tritones |- | ''kitmum'' || 1½11½1 || D || 10 || 0 |- | ''pītum'' || ½11½11 || E || 10 || 0 |- | ''qablītum'' || 11½11½ || C || 10 || 0 |- | ''išartum'' || 1½111½ || G || 10 || 1 |- | ''embūbum'' || ½111½1 || A || 10 || 1 |- | ''nīd qablim'' || 111½11 || B || 11 || 2 |- | ''nīš tuḫrim'' || 11½111 || F || 11 || 2 |} <score> \new Staff \with { \consists Ambitus_engraver \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { a1^"kitmum" b -\tweak HorizontalBracketText.text "semitone" \startGroup c\stopGroup d e-\tweak HorizontalBracketText.text "semitone"\startGroup f\stopGroup g } </score> <score> \new Staff \with { \consists Ambitus_engraver \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { g1^"qablītum" a b -\tweak HorizontalBracketText.text "semitone" \startGroup c\stopGroup d e-\tweak HorizontalBracketText.text "semitone"\startGroup f\stopGroup } </score> <score> \new Staff \with { \consists Ambitus_engraver \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { b1^"pītum" -\tweak HorizontalBracketText.text "semitone" \startGroup c\stopGroup d e-\tweak HorizontalBracketText.text "semitone"\startGroup f\stopGroup g a} </score> <score> \new Staff \with { \consists Ambitus_engraver \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c { e1^"išartum" -\tweak HorizontalBracketText.text "semitone" \startGroup f\stopGroup g a b-\tweak HorizontalBracketText.text "semitone"\startGroup c\stopGroup d} </score> <score> \new Staff \with { \consists Ambitus_engraver \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { d1^"embūbum" e -\tweak HorizontalBracketText.text "semitone" \startGroup f\stopGroup g a b-\tweak HorizontalBracketText.text "semitone"\startGroup c\stopGroup} </score> <score> \new Staff \with { \consists Ambitus_engraver \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { f,1^"nīd qablim" g a b -\tweak HorizontalBracketText.text "semitone" \startGroup c\stopGroup d e} </score> <score> \new Staff \with { \consists Ambitus_engraver \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { c1^"nīš tuḫrim" d e -\tweak HorizontalBracketText.text "semitone" \startGroup f\stopGroup g a b} </score> ==2 consecutive heptachords== <score> \new Staff \with { \consists Ambitus_engraver \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { a1^"kitmum" b -\tweak HorizontalBracketText.text "semitone" \startGroup c\stopGroup d e-\tweak HorizontalBracketText.text "semitone"\startGroup f\stopGroup g a-\tweak HorizontalBracketText.text "semitone" \startGroup bes\stopGroup c d-\tweak HorizontalBracketText.text "semitone" \startGroup es\stopGroup f} </score> ==7 consecutive heptachords== 7 consecutive heptachords build the 7 Babylonian modes: [[File:Kitmum.svg|thumb|left|1000px]] [[File:Pitum.svg|thumb|left|1000px]] [[File:Qablitum.svg|thumb|left|1000px]] [[File:Isartum.svg|thumb|left|1000px]] [[File:Embubum.svg|thumb|left|1000px]] [[File:Nid quablim.svg|thumb|left|1000px]] [[File:Nis tuhrim.svg|thumb|left|1000px]] {{clear}} ===6 cyclic consecutive heptachords=== [[File:Circle of fourths.svg|thumb|left|Circle of fourths, ''kitmum'', 36 notes, 60 semitones = Anu 𒀭]] {{clear}} ==Tuning== A tuning procedure ‘loosening’ (TU.LU) in Music of Mesopotamia for a 7-stringed instrument based on a transposition to D/D♭: <score> \new Staff \with { \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { d1^"išartum" e -\tweak HorizontalBracketText.text "semitone" \startGroup f\stopGroup g a b-\tweak HorizontalBracketText.text "semitone"\startGroup c\stopGroup } </score> <score> \new Staff \with { \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { d1^"kitmum" e -\tweak HorizontalBracketText.text "semitone" \startGroup f\stopGroup g a-\tweak HorizontalBracketText.text "semitone"\startGroup bes\stopGroup c } </score> <score> \new Staff \with { \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { d1^"embūbum" -\tweak HorizontalBracketText.text "semitone" \startGroup es\stopGroup f g a-\tweak HorizontalBracketText.text "semitone"\startGroup bes\stopGroup c } </score> <score> \new Staff \with { \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { d1^"pītum" -\tweak HorizontalBracketText.text "semitone" \startGroup es\stopGroup f g -\tweak HorizontalBracketText.text "semitone"\startGroup as\stopGroup bes c } </score> <score> \new Staff \with { \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { des1^"nīd qablim" es f g-\tweak HorizontalBracketText.text "semitone"\startGroup as\stopGroup bes c } </score> <score> \new Staff \with { \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { des1^"nīš tuḫrim" es f -\tweak HorizontalBracketText.text "semitone"\startGroup ges\stopGroup as bes c } </score> <score> \new Staff \with { \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { des1^"qablītum" es f -\tweak HorizontalBracketText.text "semitone"\startGroup ges\stopGroup as bes-\tweak HorizontalBracketText.text "semitone"\startGroup ces\stopGroup } </score> ‘tightening’ (GÍD.I) <score> \new Staff \with { \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { d1^"išartum" e -\tweak HorizontalBracketText.text "semitone" \startGroup f\stopGroup g a b-\tweak HorizontalBracketText.text "semitone"\startGroup c\stopGroup } </score> <score> \new Staff \with { \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { d1^"qablītum" e fis -\tweak HorizontalBracketText.text "semitone"\startGroup g\stopGroup a b-\tweak HorizontalBracketText.text "semitone"\startGroup c\stopGroup } </score> <score> \new Staff \with { \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { d1^"niš tuĥrim" e fis -\tweak HorizontalBracketText.text "semitone"\startGroup g\stopGroup a b cis } </score> <score> \new Staff \with { \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { d1^"nīd qablim" e fis gis-\tweak HorizontalBracketText.text "semitone"\startGroup a\stopGroup b cis } </score> <score> \new Staff \with { \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { dis1^"pītum" -\tweak HorizontalBracketText.text "semitone" \startGroup e\stopGroup fis gis -\tweak HorizontalBracketText.text "semitone"\startGroup a\stopGroup b cis } </score> <score> \new Staff \with { \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { dis1^"embūbum" -\tweak HorizontalBracketText.text "semitone" \startGroup e\stopGroup fis gis ais-\tweak HorizontalBracketText.text "semitone"\startGroup b\stopGroup cis } </score> <score> \new Staff \with { \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { dis1^"kitmum" eis -\tweak HorizontalBracketText.text "semitone" \startGroup fis\stopGroup gis ais-\tweak HorizontalBracketText.text "semitone"\startGroup b\stopGroup cis } </score> ==Further reading== * {{cite conference | first = Leon | last = Crickmore | title = A New Light on the Babylonian Tonal System | conference = Proceedings of the International Conference of Near Eastern Archaeomusicology (ICONEA 2008), The British Museum, London, December 4–6, 2008, editors: Richard Dumbrill and Irving Finkel | publisher = Iconea Publications | url = https://musicircle.net/wp-content/uploads/2018/08/Crickmore-Iconea20081.pdf | place = London | year = 2008 | pages = 11-22 }} * {{cite web | title=Babylonian Tonal System | website=Allan Pettersson Fanpage | url=https://pettersson-fanpage.de/Babylon/Babylon.html | access-date=19 January 2026}} [[Category:Music theory]] bz47wq04q370yasjlzfzrug1d3hm2ia 2818485 2818484 2026-07-18T16:03:11Z Grimes2 2895877 /* 7 heptachord modes */ 2818485 wikitext text/x-wiki ==Basic tetrachords== 1½1, ½11, 11½ <score> \new Staff \with { \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { d1 e -\tweak HorizontalBracketText.text "semitone" \startGroup f\stopGroup g} </score> <score> \new Staff \with { \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { e1 -\tweak HorizontalBracketText.text "semitone" \startGroup f\stopGroup g a} </score> <score> \new Staff \with { \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { c1 d e -\tweak HorizontalBracketText.text "semitone" \startGroup f\stopGroup} </score> and the tritone<br> 111 <score> \new Staff \with { \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { f1 g a b} </score> Two consecutive tetrachords build a heptachord. ==Heptachord== <score> \new Staff \with { \consists Ambitus_engraver \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { a1^"kitmum" b -\tweak HorizontalBracketText.text "semitone" \startGroup c\stopGroup d e-\tweak HorizontalBracketText.text "semitone"\startGroup f\stopGroup g } </score> ==7 heptachord modes== The 7 modal patterns for the Babylonian heptachords are: {| class="wikitable sortable" style="text-align: center; |+ Babylonian tonal system ! Mode !! Pattern !! Center note || Semitones || Tritones |- | ''kitmum'' || 1½11½1 || D || 10 || 0 |- | ''pītum'' || ½11½11 || E || 10 || 0 |- | ''qablītum'' || 11½11½ || C || 10 || 0 |- | ''išartum'' || 1½111½ || G || 10 || 1 |- | ''embūbum'' || ½111½1 || A || 10 || 1 |- | ''nīd qablim'' || 111½11 || B || 11 || 2 |- | ''nīš tuḫrim'' || 11½111 || F || 11 || 2 |} <score> \new Staff \with { \consists Ambitus_engraver \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { a1^"kitmum" b -\tweak HorizontalBracketText.text "semitone" \startGroup c\stopGroup d e-\tweak HorizontalBracketText.text "semitone"\startGroup f\stopGroup g } </score> <score> \new Staff \with { \consists Ambitus_engraver \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { g1^"qablītum" a b -\tweak HorizontalBracketText.text "semitone" \startGroup c\stopGroup d e-\tweak HorizontalBracketText.text "semitone"\startGroup f\stopGroup } </score> <score> \new Staff \with { \consists Ambitus_engraver \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { b1^"pītum" -\tweak HorizontalBracketText.text "semitone" \startGroup c\stopGroup d e-\tweak HorizontalBracketText.text "semitone"\startGroup f\stopGroup g a} </score> <score> \new Staff \with { \consists Ambitus_engraver \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c { e1^"embūbum" -\tweak HorizontalBracketText.text "semitone" \startGroup f\stopGroup g a b-\tweak HorizontalBracketText.text "semitone"\startGroup c\stopGroup d} </score> <score> \new Staff \with { \consists Ambitus_engraver \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { d1^"išartum" e -\tweak HorizontalBracketText.text "semitone" \startGroup f\stopGroup g a b-\tweak HorizontalBracketText.text "semitone"\startGroup c\stopGroup} </score> <score> \new Staff \with { \consists Ambitus_engraver \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { f,1^"nīd qablim" g a b -\tweak HorizontalBracketText.text "semitone" \startGroup c\stopGroup d e} </score> <score> \new Staff \with { \consists Ambitus_engraver \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { c1^"nīš tuḫrim" d e -\tweak HorizontalBracketText.text "semitone" \startGroup f\stopGroup g a b} </score> ==2 consecutive heptachords== <score> \new Staff \with { \consists Ambitus_engraver \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { a1^"kitmum" b -\tweak HorizontalBracketText.text "semitone" \startGroup c\stopGroup d e-\tweak HorizontalBracketText.text "semitone"\startGroup f\stopGroup g a-\tweak HorizontalBracketText.text "semitone" \startGroup bes\stopGroup c d-\tweak HorizontalBracketText.text "semitone" \startGroup es\stopGroup f} </score> ==7 consecutive heptachords== 7 consecutive heptachords build the 7 Babylonian modes: [[File:Kitmum.svg|thumb|left|1000px]] [[File:Pitum.svg|thumb|left|1000px]] [[File:Qablitum.svg|thumb|left|1000px]] [[File:Isartum.svg|thumb|left|1000px]] [[File:Embubum.svg|thumb|left|1000px]] [[File:Nid quablim.svg|thumb|left|1000px]] [[File:Nis tuhrim.svg|thumb|left|1000px]] {{clear}} ===6 cyclic consecutive heptachords=== [[File:Circle of fourths.svg|thumb|left|Circle of fourths, ''kitmum'', 36 notes, 60 semitones = Anu 𒀭]] {{clear}} ==Tuning== A tuning procedure ‘loosening’ (TU.LU) in Music of Mesopotamia for a 7-stringed instrument based on a transposition to D/D♭: <score> \new Staff \with { \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { d1^"išartum" e -\tweak HorizontalBracketText.text "semitone" \startGroup f\stopGroup g a b-\tweak HorizontalBracketText.text "semitone"\startGroup c\stopGroup } </score> <score> \new Staff \with { \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { d1^"kitmum" e -\tweak HorizontalBracketText.text "semitone" \startGroup f\stopGroup g a-\tweak HorizontalBracketText.text "semitone"\startGroup bes\stopGroup c } </score> <score> \new Staff \with { \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { d1^"embūbum" -\tweak HorizontalBracketText.text "semitone" \startGroup es\stopGroup f g a-\tweak HorizontalBracketText.text "semitone"\startGroup bes\stopGroup c } </score> <score> \new Staff \with { \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { d1^"pītum" -\tweak HorizontalBracketText.text "semitone" \startGroup es\stopGroup f g -\tweak HorizontalBracketText.text "semitone"\startGroup as\stopGroup bes c } </score> <score> \new Staff \with { \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { des1^"nīd qablim" es f g-\tweak HorizontalBracketText.text "semitone"\startGroup as\stopGroup bes c } </score> <score> \new Staff \with { \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { des1^"nīš tuḫrim" es f -\tweak HorizontalBracketText.text "semitone"\startGroup ges\stopGroup as bes c } </score> <score> \new Staff \with { \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { des1^"qablītum" es f -\tweak HorizontalBracketText.text "semitone"\startGroup ges\stopGroup as bes-\tweak HorizontalBracketText.text "semitone"\startGroup ces\stopGroup } </score> ‘tightening’ (GÍD.I) <score> \new Staff \with { \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { d1^"išartum" e -\tweak HorizontalBracketText.text "semitone" \startGroup f\stopGroup g a b-\tweak HorizontalBracketText.text "semitone"\startGroup c\stopGroup } </score> <score> \new Staff \with { \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { d1^"qablītum" e fis -\tweak HorizontalBracketText.text "semitone"\startGroup g\stopGroup a b-\tweak HorizontalBracketText.text "semitone"\startGroup c\stopGroup } </score> <score> \new Staff \with { \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { d1^"niš tuĥrim" e fis -\tweak HorizontalBracketText.text "semitone"\startGroup g\stopGroup a b cis } </score> <score> \new Staff \with { \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { d1^"nīd qablim" e fis gis-\tweak HorizontalBracketText.text "semitone"\startGroup a\stopGroup b cis } </score> <score> \new Staff \with { \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { dis1^"pītum" -\tweak HorizontalBracketText.text "semitone" \startGroup e\stopGroup fis gis -\tweak HorizontalBracketText.text "semitone"\startGroup a\stopGroup b cis } </score> <score> \new Staff \with { \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { dis1^"embūbum" -\tweak HorizontalBracketText.text "semitone" \startGroup e\stopGroup fis gis ais-\tweak HorizontalBracketText.text "semitone"\startGroup b\stopGroup cis } </score> <score> \new Staff \with { \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { dis1^"kitmum" eis -\tweak HorizontalBracketText.text "semitone" \startGroup fis\stopGroup gis ais-\tweak HorizontalBracketText.text "semitone"\startGroup b\stopGroup cis } </score> ==Further reading== * {{cite conference | first = Leon | last = Crickmore | title = A New Light on the Babylonian Tonal System | conference = Proceedings of the International Conference of Near Eastern Archaeomusicology (ICONEA 2008), The British Museum, London, December 4–6, 2008, editors: Richard Dumbrill and Irving Finkel | publisher = Iconea Publications | url = https://musicircle.net/wp-content/uploads/2018/08/Crickmore-Iconea20081.pdf | place = London | year = 2008 | pages = 11-22 }} * {{cite web | title=Babylonian Tonal System | website=Allan Pettersson Fanpage | url=https://pettersson-fanpage.de/Babylon/Babylon.html | access-date=19 January 2026}} [[Category:Music theory]] p1k30h9l5zbka3ohy6d2764gmd8p8dd 2818486 2818485 2026-07-18T16:04:41Z Grimes2 2895877 /* 7 heptachord modes */ 2818486 wikitext text/x-wiki ==Basic tetrachords== 1½1, ½11, 11½ <score> \new Staff \with { \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { d1 e -\tweak HorizontalBracketText.text "semitone" \startGroup f\stopGroup g} </score> <score> \new Staff \with { \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { e1 -\tweak HorizontalBracketText.text "semitone" \startGroup f\stopGroup g a} </score> <score> \new Staff \with { \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { c1 d e -\tweak HorizontalBracketText.text "semitone" \startGroup f\stopGroup} </score> and the tritone<br> 111 <score> \new Staff \with { \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { f1 g a b} </score> Two consecutive tetrachords build a heptachord. ==Heptachord== <score> \new Staff \with { \consists Ambitus_engraver \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { a1^"kitmum" b -\tweak HorizontalBracketText.text "semitone" \startGroup c\stopGroup d e-\tweak HorizontalBracketText.text "semitone"\startGroup f\stopGroup g } </score> ==7 heptachord modes== The 7 modal patterns for the Babylonian heptachords are: {| class="wikitable sortable" style="text-align: center; |+ Babylonian tonal system ! Mode !! Pattern !! Center note || Semitones || Tritones |- | ''kitmum'' || 1½11½1 || D || 10 || 0 |- | ''pītum'' || ½11½11 || E || 10 || 0 |- | ''qablītum'' || 11½11½ || C || 10 || 0 |- | ''išartum'' || 1½111½ || G || 10 || 1 |- | ''embūbum'' || ½111½1 || A || 10 || 1 |- | ''nīd qablim'' || 111½11 || B || 11 || 2 |- | ''nīš tuḫrim'' || 11½111 || F || 11 || 2 |} <score> \new Staff \with { \consists Ambitus_engraver \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { a1^"kitmum" b -\tweak HorizontalBracketText.text "semitone" \startGroup c\stopGroup d e-\tweak HorizontalBracketText.text "semitone"\startGroup f\stopGroup g } </score> <score> \new Staff \with { \consists Ambitus_engraver \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { b1^"pītum" -\tweak HorizontalBracketText.text "semitone" \startGroup c\stopGroup d e-\tweak HorizontalBracketText.text "semitone"\startGroup f\stopGroup g a} </score> <score> \new Staff \with { \consists Ambitus_engraver \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { g1^"qablītum" a b -\tweak HorizontalBracketText.text "semitone" \startGroup c\stopGroup d e-\tweak HorizontalBracketText.text "semitone"\startGroup f\stopGroup } </score> <score> \new Staff \with { \consists Ambitus_engraver \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { d1^"išartum" e -\tweak HorizontalBracketText.text "semitone" \startGroup f\stopGroup g a b-\tweak HorizontalBracketText.text "semitone"\startGroup c\stopGroup} </score> <score> \new Staff \with { \consists Ambitus_engraver \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c { e1^"embūbum" -\tweak HorizontalBracketText.text "semitone" \startGroup f\stopGroup g a b-\tweak HorizontalBracketText.text "semitone"\startGroup c\stopGroup d} </score> <score> \new Staff \with { \consists Ambitus_engraver \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { f,1^"nīd qablim" g a b -\tweak HorizontalBracketText.text "semitone" \startGroup c\stopGroup d e} </score> <score> \new Staff \with { \consists Ambitus_engraver \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { c1^"nīš tuḫrim" d e -\tweak HorizontalBracketText.text "semitone" \startGroup f\stopGroup g a b} </score> ==2 consecutive heptachords== <score> \new Staff \with { \consists Ambitus_engraver \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { a1^"kitmum" b -\tweak HorizontalBracketText.text "semitone" \startGroup c\stopGroup d e-\tweak HorizontalBracketText.text "semitone"\startGroup f\stopGroup g a-\tweak HorizontalBracketText.text "semitone" \startGroup bes\stopGroup c d-\tweak HorizontalBracketText.text "semitone" \startGroup es\stopGroup f} </score> ==7 consecutive heptachords== 7 consecutive heptachords build the 7 Babylonian modes: [[File:Kitmum.svg|thumb|left|1000px]] [[File:Pitum.svg|thumb|left|1000px]] [[File:Qablitum.svg|thumb|left|1000px]] [[File:Isartum.svg|thumb|left|1000px]] [[File:Embubum.svg|thumb|left|1000px]] [[File:Nid quablim.svg|thumb|left|1000px]] [[File:Nis tuhrim.svg|thumb|left|1000px]] {{clear}} ===6 cyclic consecutive heptachords=== [[File:Circle of fourths.svg|thumb|left|Circle of fourths, ''kitmum'', 36 notes, 60 semitones = Anu 𒀭]] {{clear}} ==Tuning== A tuning procedure ‘loosening’ (TU.LU) in Music of Mesopotamia for a 7-stringed instrument based on a transposition to D/D♭: <score> \new Staff \with { \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { d1^"išartum" e -\tweak HorizontalBracketText.text "semitone" \startGroup f\stopGroup g a b-\tweak HorizontalBracketText.text "semitone"\startGroup c\stopGroup } </score> <score> \new Staff \with { \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { d1^"kitmum" e -\tweak HorizontalBracketText.text "semitone" \startGroup f\stopGroup g a-\tweak HorizontalBracketText.text "semitone"\startGroup bes\stopGroup c } </score> <score> \new Staff \with { \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { d1^"embūbum" -\tweak HorizontalBracketText.text "semitone" \startGroup es\stopGroup f g a-\tweak HorizontalBracketText.text "semitone"\startGroup bes\stopGroup c } </score> <score> \new Staff \with { \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { d1^"pītum" -\tweak HorizontalBracketText.text "semitone" \startGroup es\stopGroup f g -\tweak HorizontalBracketText.text "semitone"\startGroup as\stopGroup bes c } </score> <score> \new Staff \with { \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { des1^"nīd qablim" es f g-\tweak HorizontalBracketText.text "semitone"\startGroup as\stopGroup bes c } </score> <score> \new Staff \with { \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { des1^"nīš tuḫrim" es f -\tweak HorizontalBracketText.text "semitone"\startGroup ges\stopGroup as bes c } </score> <score> \new Staff \with { \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { des1^"qablītum" es f -\tweak HorizontalBracketText.text "semitone"\startGroup ges\stopGroup as bes-\tweak HorizontalBracketText.text "semitone"\startGroup ces\stopGroup } </score> ‘tightening’ (GÍD.I) <score> \new Staff \with { \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { d1^"išartum" e -\tweak HorizontalBracketText.text "semitone" \startGroup f\stopGroup g a b-\tweak HorizontalBracketText.text "semitone"\startGroup c\stopGroup } </score> <score> \new Staff \with { \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { d1^"qablītum" e fis -\tweak HorizontalBracketText.text "semitone"\startGroup g\stopGroup a b-\tweak HorizontalBracketText.text "semitone"\startGroup c\stopGroup } </score> <score> \new Staff \with { \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { d1^"niš tuĥrim" e fis -\tweak HorizontalBracketText.text "semitone"\startGroup g\stopGroup a b cis } </score> <score> \new Staff \with { \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { d1^"nīd qablim" e fis gis-\tweak HorizontalBracketText.text "semitone"\startGroup a\stopGroup b cis } </score> <score> \new Staff \with { \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { dis1^"pītum" -\tweak HorizontalBracketText.text "semitone" \startGroup e\stopGroup fis gis -\tweak HorizontalBracketText.text "semitone"\startGroup a\stopGroup b cis } </score> <score> \new Staff \with { \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { dis1^"embūbum" -\tweak HorizontalBracketText.text "semitone" \startGroup e\stopGroup fis gis ais-\tweak HorizontalBracketText.text "semitone"\startGroup b\stopGroup cis } </score> <score> \new Staff \with { \consists Horizontal_bracket_engraver \remove "Time_signature_engraver" \remove "Bar_engraver" } \relative c' { dis1^"kitmum" eis -\tweak HorizontalBracketText.text "semitone" \startGroup fis\stopGroup gis ais-\tweak HorizontalBracketText.text "semitone"\startGroup b\stopGroup cis } </score> ==Further reading== * {{cite conference | first = Leon | last = Crickmore | title = A New Light on the Babylonian Tonal System | conference = Proceedings of the International Conference of Near Eastern Archaeomusicology (ICONEA 2008), The British Museum, London, December 4–6, 2008, editors: Richard Dumbrill and Irving Finkel | publisher = Iconea Publications | url = https://musicircle.net/wp-content/uploads/2018/08/Crickmore-Iconea20081.pdf | place = London | year = 2008 | pages = 11-22 }} * {{cite web | title=Babylonian Tonal System | website=Allan Pettersson Fanpage | url=https://pettersson-fanpage.de/Babylon/Babylon.html | access-date=19 January 2026}} [[Category:Music theory]] i47yy0zglhzi88y612ceso495vg9d85 6 330632 2818473 2026-07-18T13:51:36Z Young1lim 21186 {{Information |Description=Carry Lookahead Adders 2A traditional (20260718 - 20260717) |Source={{own|Young1lim}} |Date=2026-07-18 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2818473 wikitext text/x-wiki == Summary == {{Information |Description=Carry Lookahead Adders 2A traditional (20260718 - 20260717) |Source={{own|Young1lim}} |Date=2026-07-18 |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}} n8w07hwxe8qx4d71791isg737mm7yzn 6 330633 2818474 2026-07-18T13:52:11Z Young1lim 21186 {{Information |Description=Carry Lookahead Adders 2B simplified (20260718 - 20260717) |Source={{own|Young1lim}} |Date=2026-07-18 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2818474 wikitext text/x-wiki == Summary == {{Information |Description=Carry Lookahead Adders 2B simplified (20260718 - 20260717) |Source={{own|Young1lim}} |Date=2026-07-18 |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}} sss77myajjziftja5vcp35kum7t33ij 6 330634 2818476 2026-07-18T14:02:42Z Young1lim 21186 {{Information |Description=C04.SA0: Address and Dereference Operators (20260718 - 20260717) |Source={{own|Young1lim}} |Date=2026-07-18 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2818476 wikitext text/x-wiki == Summary == {{Information |Description=C04.SA0: Address and Dereference Operators (20260718 - 20260717) |Source={{own|Young1lim}} |Date=2026-07-18 |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}} i2tybb0fmhf3m3uiipx5opl8df2xnhi 6 330635 2818478 2026-07-18T14:06:02Z Young1lim 21186 {{Information |Description=Laurent.5: Permutation 6C (20260718 - 20260717) |Source={{own|Young1lim}} |Date=2026-07-18 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2818478 wikitext text/x-wiki == Summary == {{Information |Description=Laurent.5: Permutation 6C (20260718 - 20260717) |Source={{own|Young1lim}} |Date=2026-07-18 |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}} l0531m6d1te6avl1d1gjxnnxnfee30j 3 330637 2818513 2026-07-19T09:17:51Z MathXplore 2888076 delete1 ([[m:User:ZbVl/VD|Vandoom]]) 2818513 wikitext text/x-wiki == 2026-07-19 == <div class="mw-content-ltr" dir="ltr" style="text-align: left" lang="en">[[File:Information.svg|25px|alt=Information icon]] Hello. Apologies for writing this in English, but I wanted to let you know that one or more of [[Special:Contributions/&#126;2026-40651-77|your recent contributions]] have been undone because you removed content without adequately explaining why. In the future, it would be helpful to others if you described your changes to <span style="white-space:nowrap">Wikiversity</span> with an accurate [[:m:en:Help:Edit summary|edit summary]]. If this was a mistake, don't worry; the removed content has been restored. If you would like to experiment, please use the sandbox. Thanks. </div><!-- Glow-delete1 @ 1784452663879.7s --><nowiki></nowiki> [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 09:17, 19 July 2026 (UTC) 4iccfxfpkkfj51ola5j5aux6zduabon 2 330638 2818514 2026-07-19T09:34:39Z Ijustwantotalk34 3101985 I JUST UPLOADED IT 2818514 wikitext text/x-wiki <blockquote> === Writers Introduction: Greetings, I am Libby a passionate 11-year-old equestrian aspire from NSW now moved to South Australia who's love and education for horses has expanded over the course of 2 years. and I'm going to be honest with you, I've only had ever 1 horse riding lesson and rides at carnivals but don't please don't underestimate me 2 years of complete obsession and having a experienced mother and grandmother in the show horse world when younger really gets you somewhere in learning how to ride without riding a horse, a lot of people including Olympians say you can't learn how to ride with only reading and learning from social media but I deny that dearly my first lesson was in 2026 Jan with a welcoming women, Jackie. Below explains what I think to be a lot of what I know. I've stuck to only writing 2 parts of how to ride but maybe (Means when I feel like doing it) I'll add more such as the canter, bottom seat etc. Anyway the lesson was so fun Jackie said I was ready for a canter, in her 20+ years of teaching she said "I think you're ready for a canter" little did I know that me and Sparkles broke a stables record I was the first person ever In her entire lesson program to canter on their very first horse riding lesson there-and that was my first lesson ever so I am still very proud of myself I know that Almighty God gave me this present even though I never get to ride I'm being very excitingly patient whilst I wait for the next opportunity I get to ride so YAYYYYYY! a bit of a niche quote from me: "Patience brings records to your name" - Libby C. === </blockquote> == '''Upper seat while pony is in walking gait''': == You need your back comfortably straight or else you'll look like a sloth and it won't be very good for your other natural riding aids needed to make your pony excellent and happy, once you’ve achieved that it's now time to put your shoulders back and relaxed to be aligned with you're ears, elbows hip and heel-Arms by the side of your body with your elbows bent enough so that your arms are in line with your pony's ears your middle and index fingers holding the reins with your thumbs on top. Chin up eyes above the horses ears-not looking down or directly on the horses ears, that will cause a distraction. Move your hips with the horses motion for example; if the horse is moving forward your hips go forward with the horses motion and the same when the horse is using its hindlegs to make a backward motion. (Make sure you lock your thighs comfortably without squeezing so that your legs don’t move while you're doing this) == Upper seat while pony is in trotting gait: == Make sure when you're doing your rising trot that you don’t slam your bottom back down to the saddle and slouch so that it dosent upset your pony and distract him. whilst your bringing your body down, you don’t fully sit back down in the saddle when rising, EVER! you should only brush your bottom on the saddle slightly like a paint brush It's also important that you don’t just go up and down like a straight line that’s a big No-No; from when I was speaking about only brushing the saddle slightly like a paint brush I was saying instead of just going up , down of a straight line you have to rise like a half circle on the saddle let me explain- when you first go up pretend there is a little bridge there where you have to put your hip area over it and once you’ve crossed it make sure your hip area hasn’t gone on the pommel of the saddle or over. ivooqa0fz0fsmz2oh1a81h2l7l743ey 2 330639 2818516 2026-07-19T11:22:00Z Solstag 64708 Ni! 2818516 wikitext text/x-wiki This page '''will''' document the [https://wikimedia.eventyay.com/wm/wikimania2026/talk/PHHMB3/ eponymous workshop taking place at Wikimania 2026]. == Failure, fun and responsibility == == Get on with Wikiversity == == A constituency for Wikiversity == 4yojh53y4ce0zfc4oi2aprl5zmnyhx8